-wave superconductor: Application to the Cooper-pair charge-density wave in underdoped cuprates

Model of phase fluctuations in a latticed-wave superconductor: Application to the Cooper-paircharge-density wave in underdoped cuprates

Ashot Melikyan and Zlatko TešanovićDepartment of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USAsReceived 7 September 2004; revised manuscript received 21 April 2005; published 9 June 2005d

We introduce and study anXY-type model of thermal and quantum phase fluctuations in a two-dimensionalcorrelated latticed-wave superconductor based on the QED3 effective theory of high-temperature supercon-ductors. General features of and selected results obtained within this model were reported earlier in an abbre-viated formatsZ. Tešanović, e-print cond-mat/0405235d. The model is geared toward describing not only thelong distance but also theintermediatelength-scale physics of underdoped cuprates. In particular, we elucidatethe dynamical origin and investigate specific features of the charge-density wave of Cooper pairs, which weargue is the state behind the periodic charge-density modulation discovered in recent scanning-tunneling-microscopy experiments. We illustrate how Mott-Hubbard correlations near half-filling suppress superfluiddensity and favor an incompressible state which breaks translational symmetry of the underlying atomic lattice.We show how the formation of the Cooper pair charge-density wave in such a strongly quantum fluctuatingsuperconductor can naturally be understood as an Abrikosov-Hofstadter problem in a type-IIdual supercon-ductor, with the role of the dual magnetic field played by the electron density. The resulting Abrikosov latticeof dual vortices translates into a periodic modulation of the Bogoliubov–de GennessBdGd gap function and theelectronic density. We numerically study the energetics of various Abrikosov-Hofstadter dual vortex arrays andcompute their detailed signatures in the single-particle local tunneling density of states. A 434 checkerboard-type modulation pattern naturally arises as an energetically favored ground state at and near thex=1/8 dopingand produces the local density of states in good agreement with experimental observations. The leading-orderbehavior of nodal BdG fermions remains unaffected.

DOI: 10.1103/PhysRevB.71.214511 PACS numberssd: 74.20.Rp, 74.40.1k, 74.72.2h

I. INTRODUCTION

Several recent experiments1–3 support the proposal thatthe pseudogap state in underdoped cuprates should beviewed as a phase-disordered superconductor.4 The effectivetheory based on this viewpoint was derived in Ref. 5: Onestarts with the observation6 that in a phase fluctuating cupratesuperconductor, the Cooper pairing amplitude is large androbust, resulting in a short coherence lengthj,kF

−1 andsmall, tight cores forsingly quantizedsantidvortices. As aresult, the phase fluctuations are greatly enhanced, withhc/2e vortex and antivortex excitations, their cores contain-ing hardly any electrons, quantum tunneling from place toplace with the greatest of ease, scrambling off-diagonal orderin the process—incidentally, this is the obvious interpretationof the Nernst effect experiments.2 Simultaneously, thelargely inert pairing amplitude takes on a dual responsibilityof suppressing multiply quantizedsantidvortices, which re-quire larger cores and cost more kinetic energy, while con-tinuously maintaining the pseudogap effect in the single-electron excitation spectrum. The theory of Ref. 5 uses thislarge d-wave pairing pseudogapD to set the stage uponwhich the low-energy degrees of freedom, identified as elec-trons organized into Cooper pairs and BdG nodal fermions,and fluctuatinghc/2e vortex-antivortex pairs, mutually inter-act via two emergent noncompact Us1d gauge fields,vm andam. v anda couple to electronic charge and spin degrees offreedom, respectively, and mediate interactions which are re-sponsible for the three major phases of the theory:5,7 Ad-wave superconductor, an insulating spin-density wave

sSDW, which at half-filling turns into a Mott-Hubbard-Neelantiferromagnetd, and an intermediate “algebraic Fermi liq-uid,” a non-Fermi-liquid phase characterized by critical,power-law correlations of nodal fermions.

In the context of the above physical picture, the recentdiscovery in scanning tunneling microscopysSTMdexperiments8–10 of the “electron crystal,” manifested by aperiodic modulation of the local density of statessDOSd, andthe subsequent insightful theoretical analysis11 of this modu-lation in terms of thepair density wave, comes not entirelyunexpected. Such modulation originates from the chargeBerry phase term involvingv0,

5,12 the timelike component ofvm, and the long-distance physics behind it bears some re-semblance to that of the elementary bosons, like4He sRef.13d. As the quantum phase fluctuations become very strong,they occasion a suppression of the compressibility of theunderlying electron system, via the phase-particle numberuncertainty relationDwDN*1, whose effective theory mani-festation is precisely the above charge Berry phase. Once theoff-diagonal order disappears, the system inevitably turns in-compressible and the diagonal positional order sets in, lead-ing to a Mott insulating state. The resulting charge-densitywave of Cooper pairssCPCDWd12 causes a periodic modu-lation of the electron density and the size of the pseudogapDand induces a similarly modulated local tunneling DOS. Inthis context, the observed “electron crystal” state9 should beidentified as a CPCDW.

While the above CPCDW scenario is almost certainlyqualitatively correct, the ultimate test of the theory iswhether it can explain and predict some of thespecific de-

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tails of the modulation patterns as they are actually observedin cuprates. This brings us to the main theme of this paper.Typically, when constructing an effective low-energy theoryof a condensed-matter system, we are solely concerned withthe long-distance, low-energy behavior. In the present case,however, this will not suffice. The modulation in question isassociated with length and energy scales which areinterme-diate, between the short-distance scale physics of a singlelattice spacing and the ultimate long-distance behavior. Ouraim should thus be to construct a description which will bevalid not only over very long length scales but also on thescale of several lattice spacings, which are the periodicitiesobserved in experiments.8–10 Furthermore, we ideally shouldbe aiming for a “bosonized” version of the theory, withinwhich a mean-field-type approximation for the CPCDW statecould be gainfully formulated. A natural question that needsto be answered first is what should be the objects that playthe role of these bosons?

One choice is to designatereal-spacepairs of electronssor holesd as such “elementary” bosons and to endow themwith some “manageable,” i.e., pairwise and short-ranged, ef-fective interactions. Such a model indeed generically leads toa phase diagram with compressible superfluid and incom-pressible Wigner crystal states just as our long-distance ar-gument has suggested. This picture of real-space pairs ariseswithin the SOs5d-based theory,14,15 where the low-energysector of the theory assumes the form of hard-core plaquettebosons with nearest- and next-nearest-neighbor interactions.When one is dealing with extremely strongly boundrealspace s-wave ord-wave pairs, this is undoubtedly the naturalchoice. In our view, however, in cuprates one is faced notwith the real-space pairs but with themomentum spaceCoo-per pairs. Apparently, one encounters here an echo of thegreat historical debate on Blatt-Schafroth versus BCSpairs—while certain long-distance features are the same inboth limits, many crucial properties are quite different.16 Tobe sure, the Cooper pairs in cuprates are not far from thereal-space boundary; the coupling is strong, the BCS coher-ence length is short, and the fluctuations are greatly en-hanced. Still, there is a simple litmus test that places cupratessquarely on the BCS side: they ared-wave superconductorswith nodal fermions.

This being the case, constructing a theory with Cooperpairs as “elementary” bosons turns into a daunting enterprise.Cooper pairs in nodald-wave superconductors are highlynonlocal objects in the real space, and the effective theory interms of their center-of-mass coordinates will reflect thisnonlocality in an essential way, with complicated intrinsi-cally multibody, extended-range interactions. The basic ideabehind the QED3 theory5 is that in these circumstances therole of “elementary” bosons should be accorded tovorticesinstead of Cooper pairs. Vortices in cuprates, with their smallcores, are simple real-space objects and the effective theoryof quantum fluctuating vortex-antivortex pairs can be writtenin the form that is local and far simpler to analyze. In thisdual language, the formation of the CDW of Cooper pairstranslates—via the charge Berry phase discussed above—tothe familiar Abrikosov-Hofstadter problem in a dualsuperconductor.12,17 The solution of this problem intimatelyreflects the nonlocal character of Cooper pairs and their in-

teractions, and the specific CPCDW modulation patterns thatarise in such theory are generallydifferent from those of areal-space pair density wave. These two limits, the Cooperversus the real-space pairs, correspond to two different re-gimes of a dual superconductor, reminiscent of the stronglytype-II versus strongly type-I regimes in ordinary supercon-ductors. This difference is fundamental and, while both de-scriptions are legitimate, only one has a chance of beingrelevant for cuprates.

A well-informed reader will immediately protest that thekey tenet of the QED3 theory is that wecannotwrite down auseful “bosonized” version of the theory at all—the nodalBdG fermions in a fluctuatingd-wave superconductor mustbe kept as an integral part of the quantum dynamics in un-derdoped cuprates. This, while true, mostly reflects the cen-tral role of nodal fermions in thespin channel. In contrast,the formation of CPCDW is predominantly achargesectoraffair and there, provided the theory is reexpressed in termsof vortex-antivortex fluctuations—i.e., properly “dualized”—the effect of nodal fermions is less singular and definitelytreatable. This gives one hope that a suitably “bosonized”dual version of the charge sector might be devised which willprovide us with a faithful representation of underdoped cu-prates. This is the main task we undertake in this paper.

To this end, following a brief review of the QED3 theoryin Sec. II, we propose in Sec. III a simple but realisticXY-type model of a thermally phase-fluctuatingd-wave su-perconductor. Starting from this model, we derive its effec-tive Coulomb gas representation in terms of vortex-antivortex pairs. This representation is employed in Sec. IVto construct an effective action for quantum fluctuations ofvortex-antivortex pairs and derive its field theory representa-tion in terms of a dual type-II superconductor, incorporatingthe effect of nodal fermions. In Sec. V, we discuss somegeneral features of this dual type-II superconductor and re-late our results to the recent STM experiments.8–10 This sec-tion is written in the style that seeks to elucidate basic con-cepts at the expense of overbearing mathematics; we hopethe presentation can be followed by a general reader. Afterthis, we plunge in Sec. VI into a detailed study of the dualAbrikosov-Hofstadter-like problem which arises in themean-field approximation applied to a dual superconductorand which regulates various properties of the CPCDW state.This particular variant of the Abrikosov-Hofstadter problemarises from the said charge Berry phase effect: In the dualrepresentation, quantum bosons representing fluctuatingvortex-antivortex pairs experience an overalldual magneticfield, generated by Cooper pairs, whose flux per plaquette ofthe sduald CuO2 lattice is set by doping:f =p/q=s1−xd /2.Finally, a brief summary of our results and conclusions arepresented in Sec. VII.

II. BRIEF REVIEW OF THE QED 3 THEORY

The purpose of this section is mainly pedagogical: Beforewe move on to our main topic, we provide some backgroundon the QED3 effective theory of the pairing pseudogap inunderdoped cuprates. This will serve to motivate our interestin constructing anXY-type model of fluctuatingd-wave su-

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perconductors and to make a casual reader aware of the con-nection between the theoretical notions discussed in the restof this paper and the actual physics of real electrons in CuO2planes. The readers well versed in the art of construction ofeffective field theories or those already familiar with the ap-proach of Ref. 5 can safely jump directly to Sec. III.

The effective theory5 of a strongly fluctuatingdx2−y2-wavesuperconductor represents the interactions of fermions withhc/2e vortex-antivortex excitations in terms of two noncom-pact emergent gauge fields,vm and am

i . The LagrangianL=L f +L0 is

CFD0 + iv0 − ieA0 +sD + iv − ieAd2

2m− mGC − iDCTs2hC

+ c.c. +L0fv,ag, s1d

whereC=sc↑ ,c↓d, si’s are the Pauli matrices,D is the am-plitude of thedx2−y2 pairing pseudogap,Am is the externalelectromagnetic gauge field,Dm=]m+ i2am

i ssi /2d is theSUs2d covariant derivative, andh;Dx

2−Dy2. L0fv ,ag is gen-

erated by the Jacobian of the Franz-Tešanović sFTd singulargauge transformation,

expS−E0

b

dtE d2rL0fvm,ami gD

=E dVnoA,B

E Dwsr ,td 3 2−Nldfsw/pd − snA + nBdg

3dfsFi/pd − GA,Bi g, s2d

where 2pnA,B=]3]wA,B, w=]3v, Fli = 1

2elmnFmni ,

Fmni =]man

i −]nami +2ei jkam

j ank, GA,B

i =Vsndi jmj, mj

=(0,0,snA−nBd), Vsnd rotates the spin quantization axisfrom z to n, andedVn is the integral over such rotations.

Its menacing appearance notwithstanding, the physics be-hind s1d is actually quite clear. InL f, which is just the effec-tive d-wave pairing Lagrangian, the original electronscssxd,with the spin quantized along anarbitrary direction n,

have been turned into topological fermionsC=sc↑ ,c↓dthrough the application of the FT transformation:sc↑ , c↓d→ (expsiwAdc↑ ,expsiwBdc↓), where expfiwAsxd+ iwBsxdgequals expfiwsxdg, the center-of-mass fluctuating supercon-ducting phasessee Ref. 5 for detailsd. The main purpose ofthis transformation is to strip the awkward phase factorexpfiwsxdg from the center-of-mass gap function, leaving be-hind ad-wave pseudogap amplitudeD and two gauge fields,v andai, which mimic the effects of phase fluctuations. TheJacobian given byL0fv ,aig s2d insures that the fluctuationsin continuousfields v and ai faithfully represent configura-tions of discretesantidvortices. Note that topological fermi-ons do not carry a definite charge and are neutral on theaverage—they do, however, carry a definite spinS= 1

2, re-flecting the fact that the spin SUs2d symmetry remains intactin a spin-singlet superconductor. As a consequence, the spin-

density operatorSz= c↑c↑− c↓c↓=c↑c↑−c↓c↓ is an invariantof the FT transformation.

The two gauge fields,v and ai, are the main dynamicalagents of the theory. They describe the interactions of fermi-onic Bogoliubov–de GennessBdGd quasiparticles withvortex-antivortex pair excitations in the fluctuating supercon-ducting phasewsxd. A Us1d gauge fieldvm is the quantumfluctuating superflow and enters the nonconserved chargechannel. In the presence of the external electromagnetic fieldAm, one hasvm→vm−eAm everywhere inL f s1d sbut not inL0d, to maintain the local gauge symmetry of Maxwell elec-trodynamics. The JacobianL0 s2d is the exception since it isa purely mathematical object, generated by the change ofvariables from discretesantidvortex coordinates to continu-ous fieldsv sand aid. vm appearing inL0 contains only thevortex part of the superflow and is intrinsically gauge invari-ant. This point is of much importance since it is the state ofthe fluctuatingsantidvortices, manifested throughL0fv ,aig,which determines the state of our system as a whole, as wewill see momentarily.

By comparison tov, an SUs2d gauge fieldami is of a more

intricate origin. It encodes topological frustration experi-enced by BdG quasiparticles as they encirclehc/2e vortices;the frustration arises from the ±1 phase factors picked up byfermions moving through the spacetime filled with fluctuat-ing vortex-antivortex pairs. These phase factors, unlike thoseproduced by superflow and emulated byv, are insensitive tovorticity and depend only on the spacetime configuration ofvortex loops, not on their internal orientation: if we picture afixed set of closed loops ins2+1d-dimensional spacetime, wecan change the orientation of any of the loops any which waywithout affecting the topological phase factors, although theones associated with superflow will change dramatically.This crucial symmetry of the original problem dictates themanner in whichai couples to fermions in the above effec-tive theory.5 Its topological origin is betrayed by its couplingto a conservedquantum number—in spin-singlet supercon-ductors this is a quasiparticle spin—and the absence of directcoupling to the nonconserved charge channel. By strict rulesof the effective theory, such minimal coupling to BdG qua-siparticles is what endowsam

i with its ultimate punch at thelowest energies. While in this limitvm—which couples tofermions in the nonminimal, nongauge invariant way—turnsmassive and ultimately irrelevant,am

i ends up generating theinteracting infraredsIRd critical point of QED3 theory whichregulates the low-energy fermiology of the pseudogap state.

L s1d manifestly displays the Us1d charge and SUs2d spinglobal symmetries, and is useful for general considerations.To perform explicit calculations in the above theory, onemust face up to a bit of algebra that goes into computingL0fv ,aig. The main technical obstacle is provided by twoconstraints:sid the sources ofam

i should be an SUs2d gaugeequivalent of half-flux Dirac strings, andsii d these sourcesare permanently confined to the sources ofvm, which them-selves are also half-flux Dirac strings. The confined objectsformed by these two half-string species are nothing but thephysical fluctuatinghc/2e vortex excitations. These con-straints are the main source of technical hurdles encounteredin trying to reduce s1d to a more manageable form—nevertheless, it is imperative they be carefully enforced lestwe lose the essential physics of the original problem. The

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constraints can be solved by introducing two dual complexfields Fsxd= uFueif and Fssxd= uFsueifs to provide coherentfunctional integral representation of the half-strings corre-sponding tovm and am

i , respectively. In this dual language,the half-strings are simply the worldlines of dual bosonsFandFs propagating throughs2+1d-dimensional spacetime. Itis important to stress that this is just a mathematical tool usedto properly couple vortex-antivortex pair fluctuations to theelectronic degrees of freedom—readers less intimate with thetechnology of dualization will find it explained in detail inthe Appendix. The confinement of the two species of half-strings intohc/2e santidvortices is accomplished by demand-ing uFsxdu= uFssxdu. This finally results inL0 expressed as

expS−E0

b

dtE d2rL0fvm,ami gD

→E dVnE DfF,fs,Ad,kig 3 C−1fuFug

3expHE d3xsi2Ad ·w + i2ki · Fi − LddJ , s3d

whereLdfF ,Ad,kig is a dual Lagrangian given by

md2uFu2 + us]m − i2pAdmdFu2 +

g

2uFu4 + uFu2s]mfs − 2pVi3km

i d2

s4d

andCfuFug is a normalization factor determined by

E dVnDfai,fs,kigexphi2ki · Fi + uFu2s]mfs − 2pVi3km

i d2j.

s5d

Here Ad and ki are the charge and spin dual gauge fields,respectively, having been introduced to enforce the twodfunctions in s2d. Notice thatAd and ki couple tov and ai

as expsi2Ad·w+ i2ki ·Fid=expfi2Ad·s]3vd+ ikli elmns]man

i

−]nami +2ei jkam

j ankdg s3d.

At this point, one should take note of the following con-venient property ofL s1d ands3d, one that will prove handyrepeatedly as we go along:Ad andki provide—via the abovecoupling tov and ai—the only link of communication be-tween the fermionic matter LagrangianL f and the vortex-antivortex state JacobianL0. This way of formulating thetheory is more than a mere mathematical nicety: It is particu-larly well-suited for the strongly fluctuating superconductor,which nonetheless doesnot belong to the extreme tightlybound limit of real-space pairs, when they can be viewed as“elementary” bosons. Such an extreme limit of “preformed”pairs or pair “molecules” is frequently invoked, inappropri-ately in our view, to describe the cuprates. Were the cupratesin this extreme limit, there would beno gapless nodal qua-siparticle excitations, the situation manifestly at odds withthe available experimental information.19 In addition, suchdemise of nodal fermions would rob the effective low-energytheory of anyspin degrees of freedom, a most undesirabletheoretical feature in the context of all we know about thecuprates.

In a superconductor, the vortex-antivortex pairs are boundand the dual bosons representing vortex loop worldlines arein the “normal” si.e., nonsuperfluid stated: kFl=kFsl=0 sseethe Appendixd. Consequently, both dual gauge fieldsAd andki are massless; this is as it should be in the dual normalstate. This implies that upon integration overAd andki, bothvm and am

i will be massive: L0→M02v0

2+M'2 v2+sv→ad,

where M02~M'

2 ,1/jd, the dual correlation length. As ad-vertized above, the massive character ofv immediatelymakes the system a physical superconductor: If we introducea static external source vector potentialAexsqd, the responseto this perturbation is determined by the fermionic “mean-field” stiffness originating fromL f and the “intrinsic” stiff-ness set byL0, whichever issmaller. In a strongly fluctuatingsuperconductor, this in practice meansL0. The massM'

2 sx,Td of the Doppler gauge fieldv in L0 is effectively thehelicity modulus of such a superconductor and determines itssuperfluid densityrs, reduced far below what would followfrom L f alone. In a similar vein, the massM0

2sx,Td of v0 setsthe compressibility of the systemxc, again far smaller thanthe value for the noninteracting system of electrons at thesame density,

rs , limq→0

d2E0

dAexsqddAexs− qd=

P fj jsqdM'

2

P fj jsqd + M'

2 , M'2 ,

xc , limq→0

d2E0

dA0exsqddA0

exs− qd=

P frrsqdM0

2

P frrsqd + M0

2 , M02, s6d

where E0 is the ground-state energy andP fj jsq ;x,Td and

P frrsq ;x,Td are the current-current and density-density re-

sponses of the fermionic LagrangianL f, respectively.Evidently, the “intrinsic” stiffness ofv as it appears inL0

encodes at the level of the effective theory the strong Mott-Hubbard short-range correlations which are the root cause ofreduced compressibility and superfluid density and enhancedphase fluctuations in underdoped cuprates. This is entirelyconsistent with the spirit of the effective theory and shouldbe contrasted with the application of a Gutzwiller-type pro-jector to a mean-field BCS state, to which it is clearly supe-rior: In the latter approach, one is suppressing on-site doubleoccupancy by a local projector mimicking correlations thatarealready included at the level of the renormalized “mean-field” fermionic LagrangianL f in s1d, i.e., they are built-ininto P f

j j andP frr s6d. In this approach, the superconductivity

is always present atT=0 as long as the doping is finite, albeitwith a reduced superfluid density. In contrast, within the ef-fective theorys1d, the microscopic Mott-Hubbard correla-tions areadditionally echoed by the appearance of well-defined vortex-antivortex excitations, whose core size is keptsmall by a large pairing pseudogapD. Such vortex-antivortexexcitations further suppress superfluid density from its renor-malized “mean-field” value and ultimately destroy the off-diagonal long-range ordersODLROd altogether—even atT=0 and finite doping—all the while remaining sharply de-fined with no perceptible reduction inD.

As the vortex-antivortex pairs unbind, the superconduc-tivity is replaced by the dual superfluid order:kFlÞ0,


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kFslÞ0, and vm and ami turn massless:L0→c0js=v0d2

+cjs=3vd2+sv→ad, wherejsx,T→0d is the superconduct-ing correlation length,c0 andc are numerical constants, andwe have used radiation gauge= ·v=0. Physically, the mass-less character ofvm and am

i describes the admixing offreequantum vortex-antivortex excitations into the ground stateof the system which started as ad-wave superconductor.Now, it is clear from Eq.s6d that the response toAexsqd isentirelydetermined byL0. The massless character ofv in L0implies the vanishing of the helicity modulus and superfluiddensity, despite the fact that the contribution fromL f to bothremains finite and hardly changes through the transition.Similarly, the response toA0

exsqd vanishes as well, sincev0 isnow massless. Thus, the system simultaneously loses super-fluidity srs→0d and turns incompressiblesxc,c0juqu2→0dfor doping smaller than some criticalxc.

Note that the theory5 predicts a universal relation betweenthe superfluid densityrs and the fluctuation diamagnetic sus-ceptibility xdia in underdoped cuprates, asT→0. Right be-fore the superconductivity disappears, forx.xc, the discus-sion surrounding Eq.s6d implies rs,M'

2 ,1/jd. Similarly,in the region of strong superconducting fluctuations forx,xc, M'

2 in s6d is replaced bycjq2 as is clear from theprevious paragraph. The prefactorcj is nothing but the dia-magnetic susceptibilityxdia,cj. Assuming that the super-conducting correlation length and its dual are proportional toeach other,j,jd, one finally obtainsxdia,rs

−1. In the casewhere strong phase fluctuations deviate from “relativistic”behavior, the dynamical critical exponentzÞ1 needs to beintroduced and the above expressions generalized tors,M'

2 ,1/jdz, xdia,j2−z, and xdia,rs

sz−2d/z. Experimentalobservation of such a universal relation betweenrs andxdiawould provide powerful evidence for the dominance ofquantumphase fluctuations in the pseudogap state of under-doped cuprates.

In the “dual mean-field” approximationkFl=kFsl=F,one finds

L0 =1

4p2uFu2s] 3 vd2 + L0

afaig,

e−ed3xL0a= C−1E Dki expHE d3xsi2ki · Fi + RfkigdJ ,

s7d

whereed3xR; lnedVn expf−4p2ed3xuFu2skmi nid2g.

L0afaig appears somewhat unwieldy, chiefly through its

nonlocal character. The nonlocality is the penalty we pay forstaying faithful to the underlying physics: While the topo-logical origin ofai demands coupling to the conserved SUs2dspin three-currents, its sources are permanently confined toDirac half-strings of the superflow Doppler fieldv, which byits very definition is a noncompact Us1d gauge fieldsseebelowd. The manifestly SUs2d invariant forms1d subjected tothese constraints is therefore even more elegant than it isuseful.

The situation, however, can be remedied entirely by ajudicious choice of gauge:am

1 =am2 =0, am

3 =am. In this “spin-axial” gauge—which amounts to selecting a fixedsbut arbi-traryd spin quantization axis—the integration overki can beperformed andL0

a s7d reduces to a simplelocal Maxwellian,

L0fv,ag → 1

4p2uFu2s] 3 vd2 +

1

4p2uFu2s] 3 ad2, s8d

whereF is the dual order parameter of the pseudogap state,i.e., the condensate of loops formed by vortex-antivortex cre-ation and annihilation processes ins2+1d-dimensional Eu-clidean spacetime. This is the quantum version of theKosterlitz-Thouless unbound vortex-antivortex pairs and isdiscussed in detail in the Appendix. The effective LagrangianL=L f +L0fv ,ag of the quantum fluctuatingd-wave super-conductor finally takes the form

CFD0 + iv0 − ieA0 +sD + iv − ieAd2

2m− mGC − iDCTs2hC

+ c.c. +L0fv,ag, s9d

where Dm→]m+ iams3, other quantities remain as definedbelow Eq.s1d, andL0fv ,ag of the pseudogap state is givenby Eq. s8d. More generally, the full spin-axial gauge expres-sion for L0fv ,ag in s9d, valid both in the pseudogapskFlÞ0d and superconductingskFl=0d states, is

expS−E0

b

dtE d2rL0fvm,amgD→E DfF,fs,Ad,kg 3 C−1fuFug

3expHE d3xsi2Ad ·w + i2k · F − LddJ , s10d

where the dual LagrangianLdfF ,Ad,kg is

md2uFu2 + us]m − i2pAdmdFu2 +

g

2uFu4 + uFu2s]mfs − 2pkmd2,

s11d

and normalization factorCfuFug equals

E Dfa,fs,kgexphi2k · F + uFu2s]mfs − 2pkmd2j. s12d

Again, Ad and k are the charge and spin dual gaugefields, respectively, which, in the spin-axial gauge,

couple to v and a as expsi2Ad·w+ i2k ·Fd=expfi2Ad·s]3vd+ iklelmns]man−]namdg s10d.

The above is just the standard form of the QED3 theorydiscussed earlier.5 It resurfaces here as a particular gaugeedition of a more symmetric, but also far more cumbersomedescription—fortunately, in contrast to its high-symmetryparent, the Lagrangians9d itself is eminently treatable.20

A committed reader should note that the ultimate noncom-pact Us1d gauge theory form ofL s9d arises as a naturalconsequence of the constraints described above. Once thesources ofv anda are confined into physicalhc/2e vortices,


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the noncompact Us1d character is the shared fate for both.While we do have the choice of selecting the singular gaugein which to represent the Berry gauge fielda, there is nosimilar choice forv, which must be a noncompact Us1dgauge field. It is then only natural to use the FT gauge andrepresenta as a noncompact Us1d field as well, in order tostraightforwardly enforce the confinement of their respectivesources.21 The noncompactness in this context reflects noth-ing but an elementary property of a phase-fluctuating super-conductor: The conservation of a topological vortex charge] ·n=0=] ·nA,B.22

We end this section with the following remarks: In ex-plicit calculations withs9d or its lattice equivalent, it is oftenuseful to separate the low-energy nodal BdG quasiparticleexcitations from the rest of the electronic degrees of freedomby linearizingL f near the nodes. The nodal fermionscs,a,

wherea=1, 1, 2, and 2is a node index, can then be arrangedinto N four-component BdG-Dirac spinors following theconventions of Ref. 5, whereN=2 for a single CuO2 layer.These massless Dirac-like objects carry no overall chargeand are at zero chemical potential—reflecting the fact thatpairing in the particle-particle channel always pins thed-wave nodes to the true chemical potential—and can bethought of as the particle-hole excitations of the BCS“vacuum.” They, however, can be polarized and their polar-ization will renormalize the fluctuations of the Dopplergauge fieldv, a point which will be emphasized later. Fur-thermore, the nodal fermions carry spinS= 1

2 and interactstrongly with the Berry gauge fielda to which they are mini-mally coupled. In contrast, the rest of the electronic degreesof freedom, which we label “antinodal” fermions,cs,kabl,

where kabl=k12l, k21l, k12l, and k21l, are combined intospin-singlet Cooper pairs and do not contribute significantlyto the spin channel. On the other hand, these antinodal fer-mions have finite density and carry the overall electriccharge. Their coupling tov is the dynamical driving forcebehind the formation of the CPCDW. Meanwhile, the nodalfermions are not affected by the CPCDW at the leadingorder—their low-energy effective theory is still the symmet-ric QED3 even though the translational symmetry is brokenby the CPCDW.12

The presence of these massless Dirac-like excitations inL f is at the heart of the QED3 theory of cuprates.5 QED3theory is an effective low-energy description of a fluctuatingdx2−y2-wave superconductor, gradually losing phase coher-ence by progressive admixing of quantum vortex-antivortexfluctuations into its ground state. All the while, even as theODLRO is lost, the amplitude of the BdG gap function re-mains finite and largely undisturbed as the doping is reducedtoward half-filling, as depicted in Fig. 1. How is this pos-sible? A nodaldx2−y2-wave superconductor—in contrast to itsfully gapped cousin or a conventionals-wavesuperconductor—possessestwo fundamental symmetries inits ground state; the familiar one is just the presence of theODLRO, shared by all superconductors. In addition, there isa more subtle symmetry of its low-energy fermionic spec-trum which is exclusively tied to the presence of nodes. Thissymmetry isemergent, in the sense that it is not a symmetryof the full microscopic Hamiltonian but only of its low-

energy nodal sector, and is little more than the freedom in-trinsic to arranging two-component nodal BdG spinors intofour-component massless Dirac fermions5—by analogy tothe field theory, we call this symmetrychiral. The QED3theory first formulates and then answers in precise math-ematical terms the following question: can a nodaldx2−y2-wave superconductor lose ODLRO but nonetheless re-tain the BdG chiral symmetry of its low-energy fermionicspectrum?5 In conventional BCS theory, the answer is astraightforward “no:” as the gap goes to zero, all vestiges ofthe superconducting state are erased and one recovers aFermi-liquid normal state. However, in a strongly quantumfluctuating superconductor considered here, which losesODLRO via vortex-antivortex unbinding, the answer is a re-markable “yes.” The chirallysymmetric, IR critical phase ofQED3 is the explicit realization of this new, non-Fermi-liquidground state of quantum matter, with the phase order of adx2−y2-wave superconductor gone but the chiral symmetry ofnodal fermionic excitations left in its wake. While these ex-citations are incoherent, being strongly scattered by themassless Berry gauge field, the BdG chiral symmetry of thelow-energy sector remains intact.24 Importantly, as interac-tions get too strong—the case in point being the cuprates atvery low dopings—the BdG chiral symmetry will be even-tually spontaneously broken, leading to antiferromagnetismand possibly other fully gapped states.5,7 Nonetheless, theBdG chiral symmetry breaking is not fundamentally tied tothe loss of ODLRO: different HTS materials are likely tohave differentT=0 phase diagrams with larger or smallerportions of a stable critical “normal” state—described by thesymmetric QED3—sandwiched between a superconductor

FIG. 1. sColor onlined. The phase diagram of cuprates followingfrom the QED3 theory. Under theTNernst dome, the Nernst effectexperimentssRef. 2d indicate strong vortex-antivortex fluctuations;this is where the conditions for the applicability of the theory aremet sRef. 23d. As the doping is reduced, the superconductingground state is followed by an “algebraic” Fermi liquid, a non-Fermi-liquid state described by the symmetric phase of QED3 andcharacterized by critical, power-law correlations of nodal fermionssRef. 24d. At yet lower doping, the BdG chiral symmetry is broken,resulting in an incommensurate antiferromagnetsSDWd, whicheventually morphs into the Neel antiferromagnet at half-filling. De-pending on material parameterssanisotropy, strength of residual in-teractions, degree of interlayer coupling, etc.d of a particular HTSsystem, the transition between a superconductor and a SDW couldbe a direct one, without the intervening chiral symmetric groundstate. The way CPCDW, the main subject of this paper, fits into thisphase diagram is detailed later in the text.


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and an antiferromagnetsSDWd sFig. 1d. In all cases, pro-vided our starting assumption of the predominantly pairingnature of the pseudogap is correct, the symmetric QED3

emerges as the underlying effective theory of underdopedcuprates, echoing the role played by Landau Fermi-liquidtheory in conventional metals.

III. THERMAL (“CLASSICAL”) PHASE FLUCTUATIONS

Our first goal is to introduce anXY model type represen-tation of thermalsor “classical”d phase fluctuations. To thisend, we first observe that in cuprates, we are dealing with ad-wave superconductor on a tightly bound two-dimensionalCuO2 lattice sthe black lattice in Fig. 2d. The simplest start-ing point is the latticed-wave superconductorsLdSCd modeldiscussed in detail by Vafeket al.18 The model describesfermions hopping between nearest-neighbor siteski j l on asquare lattice with a renormalized matrix elementt* and con-tains a nearest-neighbor spin-singlet pairing term with aneffective coupling constantleff adjusted to stabilize thedx2−y2

state with the maximum pairing gapD,

HLdSC= − t* oki j l,s

cis† cjs + o

ki j lDi jfci↑

† cj↓† − ci↓

† cj↑† g + sc.c.d

+ s1/leffdoki j l

uDi j u2 + s¯d, s13d

wheres¯d denotes various residual interaction terms.cis† and

cis are creation and annihilation operators of someeffectiveelectron states, appropriate for energy scales below andaroundD, which already includerenormalizations generatedby integration of higher-energy configurations, particularlythose associated with strong Mott-Hubbard correlations. Thiseffective LdSC model is phenomenological but can be justi-fied within a more microscopic approach, an example beingthe one based on ast-Jd-style effective Hamiltonian. Thecomplex gap function is defined on the bonds of the CuO2sblackd lattice: Di j =Aij expsiui jd. The amplitudeAij is frozenbelow the pseudogap energy scaleD and equalsAij = ±Dalong horizontalsverticald bonds. Thus, thed-wave characterof the pairing has been incorporated directly intoAij from thestart.

What remains are the fluctuations of the bond phaseui j . Itis advantageous to represent thesebondphase fluctuations interms ofsitefluctuations by identifying expsiui jd→expsiwkd,wherewk is a site phase variable associated with the middleof the bondki j l. Consequently, we are now dealing with theset ofsitephaseswk located at the vertices of the blue latticein Fig. 2. Note that the blue lattice has twice as many sites asthe original CuO2 lattice or its dual. This is an importantpoint and will be discussed shortly.

We can now integrate over the fermions in the LdSCmodel and generate various couplings among phase factorsexpsiwkd residing on different “blue” sites. The result is theminimal siteXY-type model Hamiltonian representing a fluc-tuating classical latticed-wave superconductor,

HXYd = − Jo

nn

cosswi − w jd − J1ornnn

cosswi − w jd

− J2 obnnn

cosswi − w jd, s14d

where onn runs over nearest neighbors on the blue latticewhile ornnn and obnnn run over red and black next-nearest-neighbor bonds depicted in Fig. 2, respectively. The cou-plings beyond next-nearest neighbors are neglected25 in Eq.s14d. Such more distant couplings are not important for thebasic physics which is of interest to us here, and their effectcan be simulated by a judicious choice ofJ1 and J2—thereader should be warned that the situation changes whenT→0, as will be discussed in the next section. KeepingJ1andJ2, however, is essential. If onlyJ is kept in Eq.s14d, thetranslational symmetry of the blue lattice would be left intactand we would find ourselves in a situation more symmetricthan warranted by the microscopic physics of ad-wave su-perconductor. In this sense,J1 andJ2 or, more precisely, theirdifference J1−J2Þ0, are dangerously irrelevant couplingsand should be included in any approach which aims to de-scribe the physics at intermediate length scales.

FIG. 2. sColor onlined CuO2 lattice represented by thick blacklines svertices are copper atoms and oxygens are in the middle ofeach bondd. The nodes of the blue latticesthin solid linesd representsites of our “effective” XY model s14d, with the phase factorexpsiwid located at each blue vertex. The dual of the copper latticeis shown in red dashed lines. The nearest-neighbor coupling on theblue lattice is denoted byJ sblue linkd and the two next-nearestneighbor couplings areJ1 sred linkd andJ2 sblack linkd. Generically,J1ÞJ2 and the translational symmetry of the blue lattice is brokenby the “checkerboard” array of “red” and “black” plaquettes, as isevident from the figure. This doubles up the unit cell, which thencoincides with that of the originalsblackd CuO2 lattice, as it shouldbe.


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In general, near half-filling,J, J1, and J2 tend to be allpositive and mutually different. This ensures that the groundstate is indeed ad-wave superconductorsthe reader shouldrecall thatd-wave signs had already been absorbed intoAij ’sso the ground state is anXY ferromagnet instead of anantiferromagnet—the antiferromagnet would be an extendeds-wave stated. The explicit values ofJ, J1, and J2 can becomputed in a particular model. In this paper, we treat themas adjustable parameters.

A. Vortex-antivortex Coulomb gas representation ofHXYd

The fact thatJ1ÞJ2 breaks translational symmetry of theblue lattice. This is as it should be since the blue lattice hastwice as many sites as the original CuO2 lattice. This is animportant point to which we will return shortly. Now, how-ever, let us assume for the moment thatJ1=J2. This assump-tion is used strictly for pedagogical purposes. Then, the bluelattice has a full translational symmetry and it is straightfor-ward to derive the effective Coulomb gas representation forfluctuating vortex-antivortex pairs. Here we follow the deri-vation of Ref. 26. The cosine functions ins14d are expandedto second order and one obtains the effective continuumtheory,

Hcont=1

2JE d2r u = wsr du2 + s¯d, s15d

whereJ=J+J1+J2 andwsr d is the continuum version of thesite phasewi on the blue lattice.s¯d denotes higher-orderterms in the expansion of the cosine function. As usual, thesuperfluid velocity partvsr d= =wsr d is separated into a regu-lar sXY spin-waved part and a singularsvortexd part s=wdv,

vsr d = = xsr d + 2psz3 = d E d2r8nsr 8dGsr,r8d, s16d

where x is a regular free field andnsr d is the density oftopological vortex charge:nsr d=oidsr −r i

vd−o jdsr −r jad,

with r iv and r j

a being the positions of vortex and antivortexdefects, respectively. We are limiting ourselves to ±1 vortexcharges but higher charges are easily included.Gsr ,r8d is the2D electrostatic Green’s function which satisfies

=2Gsr,r8d = dsr − r 8d. s17d

Far from systems boundaries, the solution ofs17d is

Gsr,r8d =1

2plnsur − r 8u/ad + C, s18d

wherea is the UV cutoff, of the order of thesblued latticespacing andC is an integration constant, to be associatedwith the core energy.

This ultimately gives the vortex part of the Hamiltonianas

Hv = 2p2JE d2rd2r8d2r9nsr 8dnsr 9d = Gsr,r8d · = Gsr,r9d.

s19d

After integration by parts, this results in the desired Coulombgas representation ofHv,

− pJE d2rd2r8nsr dnsr 8dlnsur − r 8u/ad + EcE d2rn2sr d,

s20d

whereEc,C is the core energy. Note that the integrals in thefirst part ofs20d include only the regions outside vortex coressthe size of which is,a2d.

How is the above derivation affected if we now restoreJ1ÞJ2, as is the case in real cuprates? The long-distance partremains the same sinceJ1−J2 enters only at theOsk4d orderof the gradient expansion of cosine functions ins14d, i.e.,J1−J2Þ0 affects only the terms denoted bys¯d in Hconts15d. So, the strength of the Coulomb interaction between

santidvortices inHv s20d is still given by J=J+J1+J2. Thecore energy changes, however. This change can be tracedback tos17d ands18d; depending on whether the vortex coreis placed in a redsi.e., containing the red cross in Fig. 2d ora black plaquette of the blue lattice, the constant of integra-tion C will generally be different. This is just a reflection ofthe fact that forJ1ÞJ2 the original translational symmetry ofthe blue lattice is broken down to thecheckerboardpatternas is obvious from Fig. 2. So, Eq.s18d must be replaced by

Gsr,r8d =1

2plnsur − r 8u/ad + Crsbd. s21d

Gsr ,r8d is just the electrostatic potential at pointr pro-duced by a vortex charge atr 8. Since not all locations forr 8are equivalent, there are two constants of integrationCr ÞCb sCr −Cb,J1−J2d, corresponding to whetherr 8 is inthe red or black plaquettes of the blue latticesFig. 2d. Re-tracing the steps leading to the Coulomb gas representation,we finally obtain

Hvd = − pJE d2r E d2r8nsr dnsr 8dlnsur − r 8u/ad

+ Ecr E d2rn2sr d + Ec

bE d2rn2sr d, s22d

whereEcrsbd are the core energies ofsantidvortices located in

redsblackd plaquettes andEcr −Ec

b,J1−J2. Ecrsbd are treated as

adjustable parameters, chosen to best reproduce the energet-ics of the original Hamiltonians14d. The explicit lattice ver-sion of s20d follows from Ref. 27, where a duality transfor-mation and a Migdal-style renormalization procedure havebeen applied to theXY model,

Hvd = − pJ o

rÞr8

ssr dssr 8dlnsur − r 8u/ad + Ecr orPR

s2sr d

+ Ecb o

rPBs2sr d. s23d

In s23d, ssr d=0, ±1s±2, ±3, . . .d andr are summed over thered sRd and blacksBd sites of the lattice dual to the bluelattice in Fig. 2. Equations23d can be viewed as a convenientlattice regularization ofs22d. It is rederived in the Appendixwith the help of the Villain approximation.

We can also recastHv in a form which interpolates be-tween the continuums22d and lattices23d Coulomb gases.


214511-8

For convenience, we now limit our attention to only ±1 vor-tex chargessvortices and antivorticesd since these are therelevant excitations in the pseudogap regime. The underlyinglattice in s23d is “smoothed out” into a periodic potentialVsr d whose minima are located in the plaquettes of the bluelattice sat vertices of the dual blue latticed. The precise formof Vsr d can be computed in a continuum model of realisticcuprates involving local atomic orbitals, self-consistent com-putations of the pseudogapD, etc. and is not important forour purposes; only its overall symmetry matters and the factthat it is sufficiently “smooth.” This finally produces

Hvd = − pJE d2r E d2r8nsr dnsr 8dlnsur − r 8u/ad

+E d2rVsr drsr d, s24d

where nsr d is the vortex charge density as before andrsr d=oidsr −r i

vd+o jdsr −r jad is the vortex particle densityir-

respectiveof vortex charge. IfEcr =Ec

b, the periodicity ofVsr dcoincides with that of the blue lattice—the value at the mini-mum is basicallyEc

r =Ecb. On the other hand, ifEc

r ÞEcb, as is

the case in real cuprates,Vsr d has acheckerboardsymmetryon the blue lattice, withtwo different kinds of local extrema:the red one, at energy set byEc

r , and a black one, at energyEcb

sthis is all depicted in Fig. 2d.The most important consequence of these two different

local extrema ofVsr d is that there are two special locationsfor the position of vortices; on the original CuO2 lattice ofFig. 2 these correspond to vortices either residing in itsplaquettes or at its vertices. On general grounds, we expectone of these positions to be the absolute minimum while theother assumes the role of either a locally stable minimum ora saddle point with an unstable direction. WhetherEc

r ,Ecb or

the other way around and whether the higher energy is alocal minimum or a saddle point, however, can be answeredwith certainty only within a model more microscopic thanthe one used here. In particular, a specific analysis of theelectronic structure of a strongly correlated vortex core isnecessary, which goes far beyond theXY-type model used inthis paper. For our purposes,Ec

r and Ecb can be treated as

adjustable parameters and it suffices only to know that themicroscopic physics selects either the red or the blackplaquettes in Fig. 2 as the favored vortex core sites and rel-egates the other to either the metastable or a weakly unstablestatus.

IV. QUANTUM PHASE FLUCTUATIONS

The previous discussion pertains to the thermalsor clas-sicald 2D XY model. The key difference from the usual Cou-lomb gas representation for vortex-antivortex pairs turnedout to be the inequivalence of vortex core positions on theblue latticesFig. 2d, which served to recover the translationalsymmetry of the original CuO2 lattice. This effect was rep-resented by the blue lattice potentialVsr d in the vortexHamiltonians24d.

To obtain thequantumversion of the fluctuation problem

or an effectives2+1dD XY-type model, we need to includethe imaginary time dependence inwi →wistd. In general, thiswill result in vortex-antivortex pairs propagating throughimaginarysEuclideand time, as vortex cores move from oneplaquette to another through a sequence of quantum tunnel-ing processes. There are two aspects of such motion: Thefirst is the tunneling of the vortex core. This is a “micro-scopic” process, in the sense that a detailed continuummodel, with self-consistently computed core structure, isneeded to describe it quantitatively. For the purposes of thispaper, all we need to know is that one of the final compo-nents characterizing such a tunneling process is the “mass”M of the vortex core as it moves through imaginary time.28

This implies a term

1

2E

0

b

dtoi

MSdr ivsad

dtD2

s25d

in the quantum-mechanical vortex action, wherer ivsadstd is

the position of theith santidvortex in the CuO2 plane at timet. Note thatM is here just another parameter in the theorywhich, like J, J1, and J2, is more microscopic than ourmodel.28 However, the microscopic physics of the cupratesclearly points toM being far smaller than in conventionalsuperconductors. Cuprates are strongly coupled systems,with a short coherence lengthj,kF

−1 and the vortex core ofonly several lattice spacings in size. Consequently, there areonly a few electrons “inside a core” at any given time, mak-ing santidvortices “light” and highly quantum objects, withstrong zero-point fluctuations and vortex massM equal toonly a few electron masses. Furthermore, the core excitationsappear to be gapped29 by the combination of strong couplingand local Mott-Hubbard correlations.30–34This has an impor-tant implication for the second component of the core tun-neling process, the familiar Bardeen-Stephen form of dissi-pation, which is such an ubiquitous and dominating effect inconventional superconductors. In conventional systems, thecore is hundreds of nanometers in diameter and containsthousands of electrons.M is large in such a superconductorand, as a vortex attempts to tunnel to a different site, itsmotion is damped by these thousands of effectively normalelectrons, resulting in significant Bardeen-Stephen dissipa-tion and high viscosity. The motion of such a huge, stronglydamped object is effectively classical. Vortices in under-doped cuprates are precisely the opposite, with smallM, theBardeen-Stephen dissipation nearly absent, and a very lowviscosity. In this sense, the quantum motion of a vortex corein underdoped cuprates is closer to superfluid He than toother, conventional superconductors.35 We are therefore jus-tified in assuming that it is adequately described bys25d andignoring small vortex core viscosity for the rest of thispaper.36 The special feature of the cuprates, however, is thepresence of gapless nodal quasiparticles away from the coreswhich generate their own peculiar brand ofsweakddissipation—such an effect is discussed and included later inthis section.

The second aspect of the vortex propagation throughimaginary time involves the motion of the superflow velocityfield surrounding the vortexoutsideits core, i.e., in the re-


214511-9

gion of space where the magnitude ofD is approximatelyuniform. This is a long-range effect and, unlike the vortexcore energies, mass, or Bardeen-Stephen dissipation, exhibitscertain universal features, shared by all superconductors andsuperfluids. For example, in superfluid4He this effect wouldresult in a Magnus force acting on a vortex. In a supercon-ductor, the effect arises from the time derivative of the phasein the regions of uniformD. The origin of such time deriva-tive and its physical consequences are most easily appreci-ated by considering an ordinarysfictitiousd strongly fluctuat-ing s-wave superconductor with a phase factor expfifistdg ateachsite of the CuO2 lattice. For the reader’s benefit, wediscuss this case first as a pedagogical warm-up for what liesahead.

A. Pedagogical exercise: Quantum fluctuatings-wave superconductor

After performing the FT transformation and forgetting thedouble valuedness problemsignoring the Berry gauge fieldad since we are concentrating on the charge channel, thequantum action will contain a purely imaginary termsseeSec. IId,

i

2E

0

b

dtoi

fni↑std + ni↓stdg]tfi , s26d

wherenis; ciscis. It is useful to split the electron densityinto its average and the fluctuating parts:n=n↑+n↓→ n+dn.The average part is unimportant for the spin-wave phase dueto the periodicity offistd in the intervalf0,bg. In the vortexpart of the phase, however, this average density acts as amagnetic flux1

2n seen bysantidvortices.13 This first time de-rivative is just the charge Berry phase and it couples toshalfofd the total electronic charge density. After the fermions areintegrated out, the fluctuating part of the density will simplygenerate afi

2 term whose stiffness is set by the fermioniccompressibility.

For simplicity, we will first ignore the charge Berry phaseand setn=0 smod 2d. This results in as2+1dD XY model,

LXY =1

2K0o

i

fi2 − Jo

nn

cossfi − f jd, s27d

where K0 is effectively the fermionic charge susceptibilityscompressibilityd: K0,xc,kdni

2l. Again, we expand the co-sine to the leading ordersthe same results are obtained in theVillain approximation, see the Appendixd and separate theregular and singularsvortexd contributions to]mf→]mx+s]mfdv. As a result,LXY is transformed into

1

2Kms]mfd2 → iWm]mx + iWms]mfdv +

1

2Km

−1Wm2 , s28d

whereKm=sK0,J,Jd and W is a real Hubbard-Stratonovichthree-vector field. The integral over the free fieldx can becarried out, producing the constraint] ·W=0. The constraintis solved by demanding thatW=]3Ad, whereAd will soonassume the meaning of the dual gauge field. The vortex partis now manipulated into

LXY→ is] 3 Add · s]fdv +1

2Km

−1s] 3 Addm2

→ − 2piAd ·n +1

2Km

−1s] 3 Addm2 , s29d

where the integration by parts and]3 s]fdv=2pn have beenused andnm is the vorticity in thes2+1d-dimensional space-time svorticity, by its very nature, is conserved:] ·n=0d.Now, we use the standard transition from Feynman path in-tegrals to coherent functional integrals.37 A relativistic vortexboson complex fieldFsxd is introduced, whose worldlines ins2+1d-dimensional spacetime coincide with fluctuating vor-tex loops ssee the Appendix for a more detailed accountd.The first term ins29d is just the current of these relativisticvortex particles coupled to a vector potentialAd. Further-more, vortices have some intrinsic line actionedsS0, comingfrom core termssand/or lattice regularizationd which supplythe s2+1d kinetic term. In the end, one obtains a dual La-grangian,

Ld = us] + i2pAddFu2 + m2sr duFu2 +g

2uFu4 +

Km−1

2s] 3 Addm

2 ,

s30d

whereg.0 is a short-range repulsion describing the penaltyfor vortex core overlap. The mass term will in general bespatially modulated, reflecting the underlying lattice poten-tial, as in s24d. A detailed derivation of the above dual La-grangian can be found in the Appendix.

It is useful to underscore the following relation betweenthe steps that led tos30d and the formalism discussed in Sec.II, which is coached in the language closer to the originalelectrons: Only the familiar long-range Biot-Savartsantidvor-tex interactions, mediated by dual gauge fieldAd, arise fromthe fermionic action in Sec. II and Refs. 12 and 39. Thiscorresponds to large regions of space where the pseudogapDis large and approximately uniform. In contrast, smallsantidvortex core regions, whereD might exhibit significantnonuniformity, supply the core kinetic parts25d, vortex coreenergy, and the short-range core repulsion termsgd, whichare all stored in the “Jacobian” forv anda gauge fields. Thisis significant since it enables us to deal rather straightfor-wardly with the charge Berry phase, which now must berestored.

First, notice that the self-action ofAd, which was intro-duced in this section through a Hubbard-Stratonovich decou-pling in Eq. s28d, actually follows “microscopically” fromthe integration over the fermions and the gauge fieldv inSec. II and Ref. 12. ThereAd was introduced as a field en-forcing ad-function constraintdfs]3vd /p−ng but the phys-ics is precisely the same, the formal difference being just theorder in which we integrate the fermionic matter. Once werestore the Berry phase termsi /2ded3xns]tfdv back to theaction, we note that in the formalism of Sec. II and Ref. 12none of the fields in the dual Lagrangians30d couples to itdirectly. Instead, it isv0 that enters the Berry phase via thed-function constraint]tf→v0. The Berry phase term will


214511-10

then affects30d via the coupling ofv0 to the spatial part ofAd, i.e., the dual magnetic inductionBd= = 3Ad.

The above observations suggest that it is useful to sepa-rate out the saddle point part ofv asv→−iv+dv. The saddlepoint partv is determined by minimizing the total actionsthe“2” sign is chosen so thatv0 couples to the electron densityas a chemical potentiald. Of course, if there were no Berryphase term, we would havev=0. With the Berry phase termincluded,v=0 but v0 is now finite. Observe from Eqs.s1d,s2d of Ref. 12 that the saddle point equation forv0 simplyreduces to1

2n=Bd= = 3Ad. So, with the Berry phase in-cluded, the final form of the dual theory still remains givenby Eq. s30d but must be appended by the constraint1

2knsxdl=kBdsxdl.

If we now apply the dual mean-field approximation tos30d, one obtainsFmf given by

us= + i2pAddFu2 + m2sr duFu2 +g

2uFu4 +

K0−1

2Bd

2, s31d

with the constraint taking the formBdsr d= 12knsr dl. One im-

mediately recognizess31d as the Abrikosov-Hofstadter prob-lem for a dual type-II superconductor—we are in the type-IIregime38 since small compressibility implies large dual Gin-zburg parameterkd,1/ÎK0,1/Îxc—subjected to the over-all sduald magnetic field flux per plaquette of the dual lattice,f, given by f =p/q=s1−xd /2, wherex is doping. Note that athalf-filling, n=1⇒ f =1/2. Thesolution that minimizess31dat half-filling is a checkerboard array of vortices andantivortices in F, with the associated modulation inBdsr d= 1

2knsr dl, as discussed in Refs. 12 and 39. This is noth-ing but the dual equivalent of a CPCDW in ans-wave super-conductor at half-filling, with a twofold-degenerate array ofalternating enhanced and suppressed charge densities on sitesof the original lattice. An overwhelming analytic and numeri-cal support exists for this state being the actual ground stateof the negativeU Hubbard model,40 the prototypical theoret-ical toy model in this category. This gives us confidence thatthe mean-field approximation captures basic features of theproblem.

B. Quantum fluctuating d-wave superconductor

The above pedagogical exercise applies to an idealizedstrongly fluctuatings-wave superconductor and its associatedordinary s2+1dD XY model. What about ourd-wave latticesuperconductorsLdSCd and its effectiveXY-type models14d,with the bond phases mapped into site phases on the bluelattice? Again, we go from expsiui jd to expsiwkd as before.The cosine part ofs27d is replaced byHXY

d s14d and ishandled in the same way as for the classical case. This willultimately result in a modulated potentialVsr d of Eq. s24d. Inthe context of the dual theorys30d this will translate into aposition-dependent mass termm2sr duFsxdu2 in the dual La-grangians30d except now this mass term has the checker-board symmetry on the blue lattice, withmr

2 in the redplaquettes different frommb

2 in the black plaquettessmr

2−mb2,Ec

r −Ecbd. Including the constraint on the overall

dual inductionBd seems to complete the process.

Alas, the situation is a bit more involved. First, the timederivative part and the Berry phase are more complicated.The difficulty arises since thebond superconducting phaseui j of a LdSC couples in a more complicated way to thesitefermionic variables. However, we can still deal with this byenlisting the help of the FT gauge transformation whichscreens the long-range part ofui j by 1

2fi +12f j, wherefi are

the phases in the fermionic fields. After this transformationswe are again ignoringam, which we can put back in at theendd one obtainsi 1

2ni]tfi at each site of the CuO2 lattice.This translates intosi /8dni]tfi +si /8dnj]tf j for each bondki j l of the CuO2 lattice. This bond expression can be rewrit-ten as

i

8fni + njg

1

2s]tfi + ]tf jd +

i

8fni − njg

1

2s]tfi − ]tf jd

>i

8fni + njg]tui j + s¯d, s32d

wheres¯d contains higher-order derivatives and is typicallynot important in the discussion of critical phenomenasbutsee belowd. In the end, following our replacement of bondphases on a CuO2 lattice with site phases on the blue latticeui j →wk, we finally obtain the Berry phase term of our latticed-wave superconductor,

i

8fni + njg]twk → i

8fni + njg]twk +

i

8dDi j]twk. s33d

The expressions32d seems too good to be true and itis—the replacementui j → 1

2fi +12f j holds only far away from

vortices, when the phases change slowly between nearbybondsssitesd. In general, the bond phase of the LdSC will notcouple to the electron densities in a simple way suggested bys32d. Still, we have gained an important insight; its couplingto the overall electron density represented by the leadingterm in s33d is effectively exact. This follows directly fromthe electrodynamic gauge invariance which mandates thatboth regular and singular configurations of the phase musthave the same first time derivative in the action. For arbi-trarily smoothly varying phase, with no vortices present, thereplacementui j → 1

2fi +12f j is accurate to any desired degree

and the Berry phase given by the leading term ins33d fol-lows. Incidentally, this result is not spoiled by the higher-order terms, represented bys¯d in s32d, since they do notcontribute to theoverall dual magnetic field, by virtue ofni − nj =0. We, however, cannot claim with the same degreeof certainty thatdDi j is in a simple relationship to variationsin the electronic densities on sitesi and j ass32d would haveit.41 Instead, we should viewdDi j in s33d as a Hubbard-

Stratonovich field used to decouple theui j2 term in the quan-

tum action. This makesdDi j inherently abond field whosespatial modulation translates into variation in the pseudogapD and will be naturally related to the dual inductionBdbelow.42 This is different from our pedagogicals-wave exer-cise where the modulation inBd was directly related to thevariation in the electronic density.

The second source of complication is more intricate andrelates to the fact that, asT→0, we must be concerned about


214511-11

nodal fermions. In thes-wave case and the finite temperatured-wave XY-type models14d, we know that coupling con-stantsJ, J1, J2, and so on, are finite or at least can be made soin simple, reasonably realistic models. In general, we need tokeep only several of the nearby neighbor couplings to cap-ture the basic physics. There is a tacit assumption that suchexpansion is in fact well-behaved. In the quantumd-wavecase, however, the expansion in near-neighbor couplings isnot possible—atT=0 its presumed analytic structure is oblit-erated by gapless nodal fermions. Naturally, after the elec-trons are integrated out we will still be left with some effec-tive action for the phase degrees of freedomui jstd, but thisaction will be both nonlocal and nonanalytic in terms ofdifferences of phases on various bonds. The only option ap-pears to be to keep the gapless fermionic excitations in thetheory. This in itself of course is perfectly fine but it does notadvance our stated goal of “bosonizing” the CPCDW prob-lem in the least. Note that the issue here is not the Berrygauge fielda and its coupling to nodal fermions—even whenwe ignore coupling of vortices to spin by settinga→0 andconcentrate solely on the charge sector as we have done sofar, the gapless BdG quasiparticles still produce nonanalyticcontributions to the phase-only action. In short, what one hashere is more than a problem; it is a calamity. The implicationis nothing less than that there isno usableXY-type model forthe charge sector of a quantumd-wave superconductor, theoptimistic title of the present paper notwithstanding.

Fortunately, there is a way out of this conundrum. Whileindeed we cannot write down a simpleXY-like model for thefluctuating phases, it turns out that the dual version of thetheory can be modified in a relatively straightforward way toincorporate the nonanalyticity caused by nodal fermions. Tosee how this is accomplished, we go back to Sec. II and useour division of electronic degrees of freedom into low-energy nodalscs,ad and “high-energy” antinodal fermions,tightly bound into spin-singlet Cooper pairsscs,kabld. Imag-ine for a moment that we ignore nodal fermions altogether,by, for example, setting the number of Dirac nodal flavorsN→0 sin a single CuO2 planeN=2d. Exact integration overantinodal electronscs,kabl leads to anXY-type model pre-cisely of the type discussed at the beginning of this subsec-tion. Since antinodal electrons are fully gapped, the effectiveHamiltonian has the conjectured forms14d with finite cou-plingsJ, J1, andJ2, and with higher-order terms which decaysufficiently rapidly. The explicit values ofJ, J1, and J2 atT=0 might be difficult to determine but not more so than foran s-wave superconductor. Furthermore, since antinodal fer-mions carry all the charge density, the Berry phase term re-mains as determined earlier and so does the timelike stiffnessof wi. Derivation of the dual theory proceeds as envisioned,with all the short-range effects stored in vortex core termsand ultimately inm2sr d, and with only remaining long-rangeinteractions mediated by the dual gauge fieldAd. We nowrestore nodal fermionssNÞ0d; as already emphasized, theonly coupling of nodal fermions to the dual theory is via thegauge fieldsv anda. Since we are ignoring the spin channel,we can neglecta. As far asv is concerned, including nodalfermions leads to a nonlocal, nonanalytic correction to itsstiffness, expressed as

vmKmvm → vmsKmdmn + NQmndvn,

where Km is determined by antinodal electrons.Qmnsqdis the contribution from nodal fermions, linear inqand in general quite complicated. We give here a fewsimpler limits: Q00s0,vnd=cuvnu, Q0i =Qi0=0, Qxxsq ,0d=Qyysq ,0d=−cq., Qxysq ,0d=Qyxsq ,0d=−cq,sgnsqx,qyd,where q.=maxsuqxu , uqyud, q,=minsuqxu , uqyud, and c,p2/16Î2.44

In the language of dual theory, this translates into a modi-fied self-action for the dual gauge field,

1

2s] 3 AddmKm

−1s] 3 Addm

→ 1

2s] 3 AddmfKmdmn + NQmng−1s] 3 Addn, s34d

where fKmdmn+NQmng−1 is the tensor inverse ofKmdmn

+NQmn. This action is clearly more complicated than itsMaxwellian s-wave counterpart but nevertheless decidedlymanageable—most importantly, the part of the dual actioninvolving the dual boson fieldF remainsunaffected. Notethat nodal fermions inducesubleadingbut still long-rangedinteractions between vortices, in addition to the familiarBiot-Savart interactions. These interactions have square sym-metry on the blue lattice, reflecting their nodal origin. Fur-thermore, whenAd is integrated over, they will produce apeculiar dissipative term,uvnuv0

2 which describes the damp-ing of collective quantum vorticity fluctuations by gaplessnodal quasiparticles. The importance of this effect is second-ary relative to the mass term forv0 which is always gener-ated by antinodal fermions, basically because the density ofstatesNsEd for nodal fermions vanishes asE→0. However,given the smallness of traditional Bardeen-Stephen core vis-cosity, this “nodal” mechanism is an important source ofdissipation of vortex motion in underdoped cuprates. Sinceour focus in the present paper is dual mean-field theory, thisdissipative term will play no direct role.

Armed with the above analysis, we can finally write downthe Lagrangian of the quantumXY-type model describing aLdSC,

LXYd = io

i

f iwi +K0

2 oi

wi2 + HXY

d + Lnodal+ Lcore

= ioi

f iwi +K0

2 oi

wi2 − Jo

nn

cosswi − w jd

− J1ornnn

cosswi − w jd − J2 obnnn

cosswi − w jd

+ Lnodalfcosswi − w jdg + Lcore, s35d

where the sums run over sitesi of the blue lattice and thenotation is the same as below Eq.s14d, f i =

18snk+ njd with k, j

being the end sites of bondi, the timelike phase stiffnessK0results from the Hubbard-Stratonovich integration overdDi jas detailed above, andLcorecontains core terms coming froma small region around thesantidvortex whereD itself is sig-nificantly suppressed, an example beings25d. The explicit


214511-12

form of Lnodalfcosswi −w jdg in the XY model language is un-known but it is a nonanalytic, nonlocal functional ofcosswi −w jd; its effect will be incorporated once we arrive atthe dual description, following the above arguments. Notethat the short-range parts of cosine functions and time de-rivative in s35d will be subsumed intoLcore once we go tocontinuum or Villain representations of the problem, as de-scribed elsewhere in the paper. Furthermore, observe that athalf-filling the dual flux per plaquette of the blue lattice hasan intrinsic checkerboard pattern:f =1/2 forblack plaquettesand f =0 for red plaquettes. This is a direct consequence ofwstd being a bond phase, residing on the blue lattice sites inFig. 2, its Berry phase given by Eq.s33d. Thus, in the quan-tum problem, the Berry phasecombineswith the next-nearest-neighbor bondssfor J1ÞJ2d in breaking translationalsymmetry of the blue lattice down to the checkerboard pat-tern of Fig. 2. The average flux through the unit cell on theblue lattice isf =1/2,which is just as it should be since thereare two plaquettes of the blue lattice per single plaquette ofthe original black lattice and the dual flux per plaquette ofthe CuO2 lattice is f =1/2 athalf-filling.

It is now straightforward to derive a dual representation ofs35d by retracing the path that led froms27d to s30d. In thisfashion, we obtain the dual Lagrangian of thes2+1dD XYmodel appropriate for a strongly fluctuatingd-wave super-conductor near half-filling,45

Ld = us] + i2pAddFu2 + m2sr duFu2 +g

2uFu4

+1

2s] 3 AddmfKmdmn + NQmng−1s] 3 Addn + s¯d,

s36d

where]3Ad satisfies the constraint arising from the chargeBerry phase. The dual Lagrangian for ad-wave supercon-ductor s36d differs from its s-wave counterparts30d in thefollowing important respects:sid the periodicity of the masspotential is different and reflects the checkerboard periodic-ity on thesduald blue lattice set by our vortex core potentialVsr d, sii d the self-action for the dual gauge field contains anonlocal, nonanalytic contribution of nodal fermions,siii d themodulation of the dual fluxBd relates to the variation in thepseudogapdDi j sas opposed to the electronic densitydnid viathe constraintdBdsr d,s1/16dkdDi jsr dl, andsivd at dopingx,the overall dual flux through the unit cell of the blue latticeimposed by the Berry phase constraint isf =p/q=s1−xd /2,comprised of f =p/q=s1−xd /2 at a black plaquette andf =0 at a red one. We therefore expect the results for stableCPCDW states to be different from those of a hypotheticalstrongly fluctuatings-wave superconductor. Also, note thatK0 is now set byK0,kudDi j u2l, which is still related to fer-mion compressibility41 and still relatively small since the ba-sic aspect of our approach is that amplitude fluctuations inDi j are suppressed. This implies our dual superconductor re-mains in the type-II regimeskd,1/ÎK0@1d. Finally, s¯ddenotes higher-order terms which have been neglected. Forthose readers who find the above road tos36d perhaps a bit

too slick, we give a detailed step-by-step derivation in theAppendix.

An important point should be noted here: The neglectedhigher-other termss¯d involve higher-order kinetic terms,additional intermediate-range interactions, short-rangecurrent-current interactions, and numerous other contribu-tions. All are irrelevant in the sense of critical behavior. Inour problem, however, we are not interested only in the criti-cal behavior. In particular, we would like to determine thecharge modulation of the CPCDW on length scales which arenot extremely long compared to the CuO2 lattice constant.These additional terms could be important for this purpose.Particularly significant in this respect are the higher-ordercorrections to the Berry phases32d which, while not affect-ing the value of overallf, do influence the form of the effec-tive dual Abrikosov-Hofstadter problem that ensues.

The strategy of keeping a large number of otherwise irrel-evant terms is not a practical one. We will therefore intro-duce a simplification here which is actually quite natural andallows us to retain the essential realistic features of the origi-nal model and maintain the particle-hole symmetry as well.Consider a situation wherem2sr d in s36d is very stronglymodulated. This is a “tight-binding” limit, universally con-sidered appropriate for cuprates, and we can simply viewquantumsantidvortices as being able to hop only between thenearby plaquettes of the blue lattice, as is clearly implicit ins35d. Furthermore, we assume that the two extrema inm2sr dare separated by an energy scale larger than such hopping.This is a perfectly reasonable assumption since it illustratesan already important characteristic of our effective model,the fact that red and black plaquettes of the blue lattice areintrinsically not equivalentsby the form of the Berry phaseandJ1ÞJ2d. Under these circumstances, we can rewrite thedual Lagrangian in a tight-binding form,

Ld = or

us]t + i2pAd0dFru2 + ob

us]t + i2pAd0dFbu2

− okrr8l

trr expSi2pEr

r8ds ·AdDFr

*Fr8

− okrbl

trb expSi2pEr

b

ds ·AdDFr*Fb − sc.c.d

+ orSmr

2uFru2 +g

2uFru4D + o

bSmb

2uFbu2 +g

2uFbu4D

+1

2s] 3 AddmfKmdmn + NQmng−1s] 3 Addn + s¯d,

s37d

where vortex boson fieldsFrsbd reside on redsblackdplaquettes of the blue lattice,trr sbd is the vortex hopping be-tween the nearest red-red sblackd neighbors,

expsi2perr8sbdds·Add are the appropriate Peierls factors,

umr2−mb

2u@ trb score energies on red and black plaquettes aresignificantly differentd, and =3Ad in the last term is thelattice flux of the dual magnetic field, equal tof =p/q=s1−xd /2 per unit cell of the original CuO2 lattice. We have also


214511-13

assumed that it is the red plaquettes that are favored by vor-tex cores, making it unnecessary to includetbb explicitly. TheassumptionEr !Eb appears naturally warranted by the over-all symmetry of the problem but, should the details of mi-croscopic physics intervene and reverse the situation in favorof the black plaquettes, all one needs to do is exchange labelsr ↔b snote thatmr

2,0, mb2.0 in a dual superconductord.

The resulting tight-binding dual Hamiltonian,

Hd = − okrr8l

trr expSi2pEr

r8ds ·AdDFr

*Fr8

− okrbl

trb expSi2pEr

b

ds ·AdDFr*Fb − sc.c.d

+ orSmr

2uFru2 +g

2uFru4D + o

bSmb

2uFbu2 +g

2uFbu4D

+ s¯d, s38d

will be analyzed in two limits: fortrr @ trb2 / umr

2−mb2u, in which

case we can simply settrb→0, the effective Hofstadter prob-lem assumes the form equivalent to the standards-wave casedefined on the sites of the red latticesFig. 2d. Importantly,however, the relation between the modulation inBd anddDi jremainsdifferent from the s-wave case and peculiar to adwave, as explained above. We will call this limit an H1model. Similarly, in the opposite casetrr ! trb

2 / umr2−mb

2u, wecan settrr →0 and obtain an effective Hofstadter problemwith hoppings between red and black sites only. Note thatthis situation isnot equivalent to the standards-wave case: Inorder to hop from one red plaquette to another, a vortex mustgo through a black site, picking up a Peierls phase factordifferent from the one for a conventional direct red-to-redhop. In considering these Peierls factors, we must exercisecaution since the “assisted” hops between red plaquettes passdirectly through the dual fluxesf =p/q=s1−xd /2 located onblack sites. By infinitesimally displacing the said flux, one isback to the situation where all closed paths of hops are com-posed of fluxesf =p/q=s1−xd /2 through black andf =0through red plaquettes of the blue lattice. The resultingHamiltonian has an exact particle-hole symmetry aroundhalf-filling sx→−xd, as it should. We call this the H2 model.Finally, still in the limit trr ! trb

2 / umr2−mb

2u with trr →0, wecan “spread” the dual flux so it is uniformly distributedthroughout each blue lattice plaquette and equal tof =p/q=s1−xd /4. In this situation, dubbed an H3 model, thex→−x symmetry is obeyed only approximately, for smallxsnear half-fillingd, but this is all we are interested in. We haveestablished by explicit computations that the H3 model sat-isfies the particle-hole symmetry at some Hofstadter frac-tions f while it does not appear to do so at others; it is forthis reason that we tend to stay away from the H3 model inthis paper. The reader should note that the issue of how todeal with dual fluxes when hopping through black plaquettesarises only in tight-binding models H2 and H3sbut notH1d—the original dual Lagrangians36d is free of such issuesand has manifestx→−x symmetry. The down side, of

course, is that such a continuum theory is far more difficultto solve, both analytically and numerically.

The Hamiltonians38d, in its three editions H1, H2, andstoa limited extentd H3, defines probably the simplest dual ver-sion of the Hofstadter problem which appropriately builds inthed-wave character of the fluctuating lattice superconductorand the essential phenomenology of the pseudogap state inunderdoped cuprates. The extra termss¯d can still be usedfor fine-tuningsfor example,g→gr ,gb, additionaluFru2uFbu2repulsions, etc.d but the important particle-hole symmetryaround half-filling is already present without them. All thedetailed numerical calculations reported in the latter pages ofthis paper and described in Ref. 12 use the mean-field ver-sion of s38d and the three models based on it, H1strr @ trb

2 / umr2−mb

2ud and H2 and H3strr ! trb2 / umr

2−mb2ud, as the

point of departure.

V. DUAL SUPERCONDUCTOR AND COOPER PAIR CDWIN UNDERDOPED CUPRATES

The previous sections concentrated on the derivation ofthe effective quantumXY-type model for phase fluctuationsin underdoped cuprates and its dual counterparts36d ands37d. In this section, we pause to take stock of where we arewith respect to the real world and to consider some generalfeatures of the physical picture that emerges from Eqs.s36dand s37d. First, the dual superconductors36d describes thephysics of strongly fluctuating superconductors in terms of acomplex bosonic fieldF, which creates and annihilatesquantum vortex-antivortex pairs viewed as “charged” relativ-istic particlessvorticesd and antiparticlessantivorticesd—thisis depicted in Fig. 3. The conserved “charge” is just thevorticity associated with these topological defects, +1 forvortices and −1 for antivortices, and the gauge field coupledto it is nothing butAd. Physically,Ad describes the familiarlogarithmic interactions betweensantidvortices. We stressthat the dual description is just a convenient mathematicaltool: Its main advantage is that it allows a theorist to accessa strongly quantum fluctuating regime of a superconductor,where the superfluid densityrs at T=0 can be very small oreven vanish, while the pairing pseudogapD remains large.According to the theory of Ref. 5, this is the regime thatgoverns the properties of the pseudogap state in underdopedcuprates. This regime is entirely inaccessible by more con-ventional theoretical approaches which use the mean-fieldBCS state as the starting point around which to computefluctuation corrections.

The dual superconductor description predicts two basicphases of cuprates:kFl=0 is just the familiar superconduct-ing state. Quantum and thermal vortex-antivortex pair fluc-tuations are presentsand thuskuFu2lÞ0d but these pairs arealways bound, resulting in a finite, but considerably sup-pressed superfluid density. As vortex-antivortex pairs unbindat some dopingx=xc, the superfluid density goes to zero andthe superconducting state is replaced by its dual counterpart,kFlÞ0. The meaning of dual ODLRO is actually quitesimple: FinitekFl means that vortex loopsfloops of vortex-antivortex pairs being created and annihilated ins2+1d-dimensional spacetimeg can now be extended over the


214511-14

whole sample, i.e., the worldline of a dual relativistic bosoncan make its way from any point to infinityssee Fig. 3d. Thepresence of such unbound vortex-antivortex excitations di-rectly implies vanishing helicity modulus and thus the ab-sence of the Meissner effect andrs=0 ssee Sec. II for de-tailsd. The phase diagram of a dual superconductor as itapplies to cuprates is shown in Fig. 4.

In the dual mean-field approximation, just as in a conven-tional one, we ignore fluctuations inF and minimize theaction specified bys36d with respect to a complex functionFsr d. In a physical superconductorsx.xcd, Fsr d=0. Forx,xc, the minimum action corresponds to finiteFsr d. How-ever, since the charge Berry phase translates into a dual mag-netic field Bd, this finite Fsr d must containNd vortices,where Nd is the number of the dual flux quanta piercingthe system. Note that this number is nothing but half of thetotal number of electronsNd=N/2, the factor of12 being dueto the fact that we are considering Cooper pairs. In atight-binding representations37d this implies a dual flux

f =p/q=s1−xd /2 per each CuO2 unit cell. The presence ofsuch a vortex array inFsr d will be accompanied by spatialvariation in Bdsr d, which translates into the modulation ofthe pseudogapdDi j and thus into a CDW of Cooper pairssCPCDWd. Consequently, in the vocabulary of the dualmean-field approximation, the question of formation and thestructure of the CPCDW is equivalent to finding the Abriko-sov vortex array on a tight-binding lattice, i.e., theAbrikosov-Hofstadter problem defined bys37d ands38d. Theformation of CPCDW results in a modulation of the localtunneling DOS and one is led to identify the “electron crys-tal” state observed in STM experiments8–10 as the CPCDW.

Our task is to determine the specific pattern in which thevortices inFsr d arrange themselves to minimize the expec-tation value of the dual Hamiltonians38d. Once this isknown, we can determine the modulation in the dual induc-tion Bd and the structure of the CPCDW follows from theBd↔dDi j correspondence. In this section, we are interestedin general qualitative results and we therefore focus on theH1 model where such results are more transparent. The H2and H3 models turn out to be more opaque and have to bestudied numerically as soon as one is away from half-filling.

The solution of the Abrikosov-Hofstadter problem is ob-tained as follows: one first setsg=0 in s38d and finds theground state of the resulting quadratic Hofstadter Hamil-tonianHdsg=0d given by

FIG. 3. Quantum fluctuations of vortex-antivortex pairs.xyplane is the CuO2 layer andt axis shows direction of imaginarytime. Vorticessarrows pointing upwardsd and antivorticessarrowspointing downwardsd are always created and annihilated in pairs.Note that the structures arising from linked creation-annihilationevents form oriented loops carrying ±1 vorticity. These loops arejust the virtual particle-antiparticle creation and annihilation pro-cesses in the quantum vacuum of the relativistic bosonic fieldFsxd,as described in the text and the Appendix.F is our dual orderparameter: In a physical superconductor, such vorticity loops arefinite and kFl=0. Note that the intersections of such finite loopswith the xy plane at any given timet define a set of bound vortex-antivortex dipoles in the CuO2 layer. The superconducting order islost when vortex-antivortex pairs unbind and the average size of theabove vorticity loops diverges—some of the loops become as largeas the system size. This is the pseudogap state, withkFlÞ0. Thereader should note the following amusing aside: The above figurecan easily be adapted to depict the low-energyfermionicexcitationsof theorys1d. The creation and annihilation processes now describespin upsarrows pointing upwardsd and spin downsarrows pointingdownwardsd quasiparticle excitations from the BCS-type spin-singlet vacuum. The loops carry a well-defined spin and can bethought of as relativistic BdG Dirac particles/antiparticles—theirmassless character in a nodald-wave superconductor implies thepresence of loops of arbitrarily large size. These fermionic loopsmove in the background of fluctuating vortex loops discussedabove, their mutual interactions encoded in gauge fieldsv and a.This is a pictorial representation of the theorys1d.

FIG. 4. The schematic phase diagram of underdoped cuprates inthe theory of Refs. 5 and 12.T* denotes the pseudogap temperaturesT* ,Dd. Dual superconducting ordersfinite kFld implies the ab-sence of the true, physical superconductivity. Conversely, the ab-sence of dual orderskFl=0d corresponds to the physical supercon-ductor. Note that, since the CPCDW ground stateskFlÞ0d is acharge 2e insulator, it still exhibits strong superconducting fluctua-tions manifested by enhanced orbital diamagnetismssee Sec. IId.The shaded area represents the region of coexistence between astrongly fluctuating superconductor and a CPCDW state, which willgenerally occur in the dual theory. This region of “supersolid” be-havior is characterized bykFl=0 but finitekuFu2l, which is modu-lated within the CuO2 layer according to our theory of a dual su-perconductor. The precise size of a coexistence region, however, isdifficult to estimate from the mean-field theory and a more elabo-rate approach, including fluctuations inF andAd, needs to be em-ployed. Finally, the finite-temperature phase boundary of theCPCDW sdashed-dotted lined should not be taken quantitativelyapart from the fact that it is located belowT* .


214511-15

− okrr8l

trre2pier

r8ds·AdFr*Fr8 − o

krbltrbe2pier

bds·AdFr*Fb − sc.c.d

+ or

mr2uFru2 + o

b

mb2uFbu2. s39d

At flux f =p/q, the ground state isq-fold degenerate and wedenote it byFsqdsr d. One then turns on finiteg, forms a linearcombination of these degenerate statesoqaqFsqdsr d, and de-termines variationally the set of coefficientshaqj which mini-mizes the full Hamiltonians38d. With haqj fixed in this fash-ion, the only remaining degeneracy in the ground stateconsists of lattice translations and rotations. Once the groundstateFs0d is found, the magnetic inductionBd follows fromthe Maxwell equation,

dLd

dAd=

1

K0

D 3 dBdsr d − j F = 0, s40d

whereLd is given by Eq.s37d, dBd=D3A, and j F is thecurrent in the ground state of dual Hamiltonians39d with theuniform dual flux f. All quantities ins40d are defined on theblue lattice of Fig. 2:D is a lattice derivative,j F andAd arelink variables, andBd=D3Ad is a site variable. The detaileddefinitions of all these objects are given in the next section.The nonlocal, nonanalytic self-energy for the dual gaugefield in s37d was replaced by an effective Maxwellian

sKm→ Kmd—this approximation is valid for weak modulationdBd. We should stress that this way of determining the

ground stateFs0d and dual inductionBdsr d is valid only if K0

is sufficiently small so that the dual Ginzburg parameter

kd,1/ÎK0 is sufficiently large—in underdoped cupratesthis is a justified assumption since strong Mott-Hubbard cor-relations strongly suppress all charge-density fluctuations. Athigher modulation, i.e., for intermediate values ofkd, thenodal contribution becomes more significant and its intrinsicsquare symmetryfsee discussion around Eq.s34dg will act toorient dBd relative to the blue lattice. If this is the case, theinterplay between this “nodal” effect and the one arisingfrom the Abrikosov-Hofstadter problem itself can lead to in-teresting new patterns for the CPCDW state; a detailed studyof such an interplay is left for the future. Finally, fromdBdsr d;Bdsr d− f we obtaindDi j , which can be fed back intothe electronic structure via the expressions given in Sec. II.

Notice that the above method of solving the problem cor-responds to the strongly type-II regime of a dual supercon-ductor skd@1d. In this limit, the CPCDW pattern isprima-rily given by the dual supercurrentj F in the Maxwellequation s40d, itself determined by the solution to theAbrikosov-Hofstadter problems39d. The modulation indBdonly reflects this pattern of vortices inj F s40d, and the dualmagnetic energy is only a small fraction of the Abrikosov-Hofstadter condensation energy. Imagine now that we askthe following question: What are the interactions among vor-tices inF that have resulted inFs0d being the ground state ofthe Abrikosov-Hofstadter problem? This question is analo-gous to the one inquiring about the interactions that have ledto the triangular lattice of vortices in the continuum versionof the problem, i.e., the interactions inherent in the famed

Abrikosov participation ratiobA. In both cases, these are farfrom pairwise and short-ranged—they are in fact intrinsicallymultibody interactions, involving two-, three-, and multiple-body terms, all of comparable sizes and all long-ranged.43

They can be thought of as the interactions among the center-of-mass coordinates of Cooper pairs. This should be con-trasted with the picture of real-space pairs, interacting withsome simple, pairwise, and short-ranged interactions. Thepair density-wave patterns in this case are determined not bythe charge Berry phase and the Abrikosov-Hofstadter prob-lem but by the Wigner-style crystallization. This is preciselythe opposite limit of the Maxwell equations40d, in which itis the dual magnetic energy that determines the pattern ratherthan simply reflecting the one set by the Abrikosov-Hofstadter condensation energy, encoded inFs0d, and com-municated byj F. This is clearly seen in the Wigner crystallimit: consider an array of real-space pairs fixed in their po-sitions. Such particles carry a unit dual flux and are com-pletely invisible to vortices. As a result,j F=0 ands40d turnsinto the minimization of the dual magnetic energy. In thereal-space pair limit, this dual magnetic energy is nothingbut the original assumed interactions between the pairbosons:1

2oi jVsr i −r jd= 12 ed2rd2r8Bdsr dVsr −r 8dBdsr 8d, since

Bdsr d=oidsr −r id, wherehr ij are the pairs’ positions—this isjust the minimization of the potential energy in the Wignerproblem. Consequently, the two descriptions, that of theCooper pairs versus the real-space pairs, correspond to thetwo oppositelimits of s40d ands39d, the first to the stronglytype-II, the second to the strongly type-I limit of a dual su-perconductor. The density-wave patterns associated withthese two limits are generally different and can be distin-guished in experiments.

We now resume our discussion of the Abrikosov-Hofstadter problem. The simplest case, for all models, isf =1/2 or x=0. For the H1 model, thesantidvortices preferred plaquette locations andFr will be large compared toFb,which can be safely ignored. The resulting dual vortex arrayat half-filling is depicted in Fig. 5. The structure is a check-erboard with vortices inFr located on alternating blackplaquettes. There is a single dual vortex per two unit cells ofthe CuO2 lattice, as expected forf =1/2. Such close packingof dual vortices results in “empty” black plaquettes actuallybeing occupied by dual antivorticessmanifestation of the factthat our f =1/2 Hofstadter problem does not break the dualversion of time-reversal invarianced. There is, however,nomodulation in bond variablesdDi j located on vertices of theblue lattice due to its peculiar relation to dual magnetic in-ductionBd—all blue sites are in a perfectly symmetric rela-tion to the dual vortex-antivortex checkerboard pattern onblack plaquettes, as is obvious from Fig. 5. This impliesdDi j =0 and the pseudogapD remains uniform despiteFsr dÞ0. From this one would tend to conclude that therewill be no CPCDW and no modulation in the local DOS.Still, there is a clear lattice translational symmetry breakingin the dual sector as evident from Figs. 5 and 6. Conse-quently, if we go beyond the leading derivatives, for exampleby including corrections to the Berry phase discussed arounds26d, we expect that a weak checkerboard modulation willdevelop in quantities like the electron densitydni. The above


214511-16

is a special feature of the fluctuating LdSC which, however,is altered when we include the spin channel in our consider-ation. The Berry gauge fieldam must then be restored—itscoupling to nodal fermions induces antiferromagnetic order

at half-filling and thus breaks the above symmetry in theleading order by a commensurate spin-density wavesSDWd.7,5

As the system is doped,f decreases away from 1/2 ac-cording tof =p/q=s1−xd /2. The ground-state energy ofs38dis particularly low for dopings such thatq is a small integer,sintegerd2, or a multiple of 2, reflecting the square symmetryof the CuO2 planes. Such dopings are thus identified as “ma-jor fractions” in the sense of the Abrikosov-Hofstadter prob-lem. In the window of doping which is of interest in cu-prates, these fractions are 7/16, 4/9, 3/7, 6/13, 11/24,15/32, 13/32, 29/64, 27/64, . . .,corresponding to dopingsx=0.125s1/8d, 0.111s1/9d, 0.143s1/7d, 0.077s1/13d,0.083s1/12d, 0.0625s1/16d, 0.1875s3/16d,0.09375s3/32d, 0.15625s5/32d, etc. Other potentiallyprominent fractions, like 1/4, 1/3, 2/5, or 3/8, are associ-ated with dopings outside the underdoped regime of strongvortex-antivortex fluctuations. In general, we expect that par-ticularly low-energy states correspond to fractions such thatthe pattern of dual vortices inF can be easily accommodatedby the underlying CuO2 lattice. Furthermore, we expect thatthe quartic repulsion ins38d will favor the most uniformarray of dual vortices that can be constructed from theq-dimensional degenerate Hofstadter manifold. In the win-dow of dopings one deals with in cuprates, these conditionssingle out dopingx=0.125sf =7/16d as a particularly promi-nent fraction. Atx=0.125 sq=16d, the dual vortex patternand the accompanying modulation inBd can take advantageof a 434 elementary block which, when oriented along thexsyd direction, fits neatly into plaquettes of the dual lattice, asdepicted in Fig. 5. This 434 elementary block embeddedinto the original CuO2 lattice and containing seven dual vor-tices sf =7/16d is clearly the most prominent geometricalstructure among all the ones we have found in our study,both in its intrinsic simplicity and its favorable commensu-ration with the underlying atomic lattice. It is bound to beamong the highly energetically preferred states in under-doped cuprates, as it is indeed found in the next section.

Before we turn to the details of this energetics, we inves-tigate the signature in the electronic structure of the above434 elementary block. Such signature could be detected inthe STM experiments of the type performed in Refs. 8–10.To this end, we compute the dual inductionBd associatedwith the pattern of seven dual vortices in Fig. 5 and from itthe spatial modulation of thed-wave pseudogapdDi j on eachbond within the 434 supercell. WithD thus fixed, we evalu-ated the local density of BdG fermion states of our LdSCmodel. The results are shown in Figs. 7–9. Note that we doour computations in the “supersolid” region of the phasediagram in Fig. 4. This enables us to use the BdG formalismwith only moderate smearing in the coherence peaks comingfrom the gauge fieldsv and a and it also allows for ratherdirect comparison with experiments. The downside is that wehave to assume that the modulation profile inBd remains thesame as determined from our dual mean-field arguments.Considering the high symmetry of the elementary block inFig. 5, this appears to be a rather minor assumption.

VI. DUAL ABRIKOSOV-HOFSTADTER PROBLEM

In this section, we present the results of numerical analy-sis of the H1 model. Within this model, the values of the

FIG. 5. sColor onlined Left panel: the circles depict the check-erboard array of vortices inF forming the ground state of Hamil-tonian H1 at half-fillingsf =1/2 perplaquette of the atomic latticed.The empty black plaquettes are actually occupied by dual antivor-tices, brought into existence by a simple geometric constraint on thephase ofF in such a checkerboard array. The full green circlesdenote “dual vortex holes”sRef. 5d, i.e., the dual vortices that aremissing relative to the half-filling checkerboard pattern once thesystem is doped tox=1/8 sf =7/16d. The centers of these greencircles form a square which defines the 434 elementary block ofseven dual vorticessblack circlesd per 16 sites of the CuO2 latticediscussed in text. Right panel: the most general distribution ofmodulationsdDi j consistent with the vortex pattern just described.The corners of the elementary blocksgrayd, correspond to “dualvortex holes.” In general, there are sixsRef. 49d phenomenologicalparametersdDi j shown by different line styles. The number of in-equivalent sites of the black lattice is also sixssee Fig. 7d.

FIG. 6. sColor onlined The same as Fig. 5 except now the role ofred and black plaquettes is reversed. This corresponds to eitherEc

b,Ecr or thesantidvortex core location at the black plaquette being

a local minimum of the vortex lattice potentialVsr d. In the lattercase, the pictured array would be a metastable configuration of dualvortices atx=0 andx=1/8, ultimately unstable to the true groundstate at those dopings depicted in Fig. 5. The most general patternof dDi j is shown on the right. In this case, the “dual vortex holes”ssolid green circlesd correspond to the centers of squaressshown bydotted linesd that surround the corners of the elementary plaquettesgrayd. Note that in general there are sixsRef. 49d distinct param-eters controlling modulations of the pairing pseudogapDi j . Thenumber of nonequivalentsblackd sites, however, is threessee Fig.8d.


214511-17

matter fieldFsr d on the black sites are suppressed by verylargemb

2, andFsr d effectively lives on the lattice dual to theoriginal copper lattice, that is, on the red sites in Fig. 2.Therefore, the dual fluxes reside inside the red plaquettes,shown in Fig. 11. In this section, we will drop the subscriptdin order to make notation more compact and useAsr d for thevector potential corresponding to a uniform dual magneticflux equal tof =p/q. Modulation of the field, which will bedetermined numerically, is described bydAsr d. Withinmean-field approximation, our problem then is reduced tominimization of the following Ginzburg-Landau lattice func-tional:

Hd = − tor ,d

eiAr ,r+d+idAr ,r+dF*sr dFsr + dd

+ orSm2uFsr du2 +

g

2uFsr du4D +

kd2

2 or

dBd2sr d,

s41d

wherem2=mr2, d=h±x, ± yj, and the link variablesA are de-

fined as

Ar ,r+d =Er

r+d

dr ·A . s42d

Note that from this point on, we absorb the factors of 2p intothe definition ofA for numerical convenience and to con-form with the Abrikosov dimensionless notation.38 Themodulated part of the fluxdBdsr d is given by the circulationof the correspondingdA around each plaquette as

dAr ,r+x + dAr+x,r+x+y + dAr+x+y,r+y + dAr+y,r . s43d

The minimization ofHd with respect to the link variablesdAis equivalent to the solution of the following set of equations:

0 = 2tuFsr duuFsr + xdusinsAr ,r+x + dAr ,r+x + ar+x − ard

+ kd2fdBdsr d − dBdsr − ydg,

0 = 2tuFsr duuFsr + ydusinsAr ,r+y + dAr ,r+y + ar+y − ard

+ kd2fdBdsr − xd − dBdsr dg, s44d

wherear is the phase of the dual matter fieldFsr d. Since the

FIG. 7. sColor onlined The local density of statessLDOSd of the latticed-wave superconductor with the modulated bond gap functionDi j

corresponding to the 434 supercell structure of Fig. 5 at dopingx=1/8. Thecalculations are done within the LdSC model of phasefluctuations. Note that the fluctuations of the gauge fieldsv anda lead to small broadening of the peaks but produce no significant changes,as long as one is in the superconducting state. The parameters aret* =1.0, D=0.1, and the variation inD from the weakest to the strongestbond is ±25%. The portion enclosed within a rectangle is enlarged in the central panel. The numbers correspond to the locations of Cu atomswithin a 434 unit supercell on the CuO2 lattice depicted on the right panel. Note that the Cu atom labeled 1 coincides with the location ofthe “dual vortex hole” in Fig. 5sfull green circled whereD is the weakest, while the one labeled 6 corresponds to the Cu atom in the centerof Fig. 5 which is surrounded by the four strongestD’s. The radii of the red circles indicate the magnitude of LDOS atE=−0.27t* sdashedline in the central paneld. This pattern of LDOS is very robust in our calculations and is precisely the tight-binding analogue of thecheckerboard structure observed by Hanaguriet al. sRef. 9d snote that on a tight-binding lattice, the Fourier transforms at wave vectors2p /4a0 and 332p /4a0 are not independent since 332p /4a0 and −2p /4a0 are equivalentd. The symmetry of modulations inDi j corre-sponding to Fig. 5 implies that there are only six nonequivalent sites within the 434 unit cell. Note that the modulation pattern at energiesaboveD is actuallyreversedcompared to the pattern at energies belowD. Finally, the nodes remain effectively intact, in accordance withRef. 12. Finite LDOS at zero energy is entirely due to artificial broadening used to emulate finite experimental resolution. In the absence ofsuch broadening, the LDOS remains zero within numerical accuracyssee Fig. 9d.


214511-18

last terms in Eqs.s44d can be identified asx and y compo-nents of the lattice curl, these are the lattice analogs of thesduald Maxwell’s equations in two dimensions, providing ex-plicit lattice realization of Eq.s40d.

Before we present the results of the numerical computa-tion, we will discuss briefly the structure of the solutions thatshould be expected on general symmetry grounds. In thelimit of infinite kd, the gauge fielddA does not fluctuate, andonly Fsr d should be varied in minimizingHd. To understandwhy the solution for a general filling is inhomogeneous, con-sider first a case ofg=0. Then the functionalHd simplydescribes a particle on a tight-binding lattice moving in a

uniform magnetic fieldf. The corresponding Hamiltonian is

HHofFsr d = − t od=±x,±y

eiAr ,r+dFsr + dd + m2Fsr d.

The minimization of the functionalHd is closely related

to finding the ground states of HamiltonianHHof. Note thatalthough the dual magnetic field felt by the particles is per-fectly uniform, the gauge fieldAsr d is not, regardless of thegauge used. Indeed, if this were the case, the circulation ofthe vector potentialA around the primitive plaquette wouldbe zero by periodicity, which is not possible as the circula-tion is equal to the flux of the magnetic field 2pp/q throughthe plaquette. Thus, the HamiltonianHHof does not commutewith the usual lattice translation operators. Instead, as notedby Peierls long ago, magnetic translation operatorsTR, gen-erating lattice translations complemented by simultaneousgauge transformations, must be constructed in order to com-

mute with HHof.

Unlike the ordinary translations, operatorsTR do not com-mute. Rather, operatorsTR form a ray representation of thetranslation group. The theory for irreducible ray representa-tions of the translation group was constructed by Brown.46

Alternatively, one can use the magnetic translation group in-troduced by Zak,47 and use the ordinary representations ofthat group to classify the eigenstates of the Hamiltonian.

On the lattice, the classification of the states is simple, andfor completeness we demonstrate how the magnetic eigen-states can be constructed. Consider, for example, the LandaugaugeAx=0, Ay=2pxp/q, in which the unit cell spansqelementary plaquettes in thex direction. Then the Peierlsfactors Ar ,r+d, shown in Fig. 10, are periodic modulo 2pwith an enlargedq31 unit cell. The Hamiltonian now can bewritten as

HHofFsx,yd = m2Fsx,yd − tfFsx + 1,yd + Fsx − 1,yd

+ Fsx,y + 1de2pisp/qdx + Fsx,y − 1de−2pisp/qdxg.

The Hamiltonian, obviously, remains invariant under trans-formationsr → r + y and r → r +qx. Consequently, it can bediagonalized with the usual Bloch conditions

Fsr + qxd = eiqkxFsr d s45d

Fsr + yd = eikyFsr d, s46d

whereskx,kyd is the crystal momentum defined in a Brillouinzone s2p /qd32p. Using these conditions, we rewrite theequation for the eigenstates as

FIG. 8. sColor onlined Left panel: The local density of statessLDOSd of the latticed-wave superconductor with the modulated bond gapfunction Di j corresponding to the 434 supercell structure of Fig. 6 at dopingx=1/8. Theparameters used are the same as in Fig. 7. Rightpanel: the local density of states at energyE=−0.27t sdashed line in the central paneld. The radii of the circles are proportional to the LDOSat a given Cu atom. Note that inside the unit cell 434 there are threesrather than six, as in Fig. 7d nonequivalent sites. Even in the absenceof any underlying theory, this pattern of modulation appears to correspond to CPCDW simply by visual inspection.


214511-19

m2gsxd − tfgsx + 1d + gsx − 1d + 2gsxdcossky + 2pxp/qdg

= Egsxd, s47d

wheregsxd=Fsx,0d. Now we have a one-dimensional equa-tion for gsxd, wherex=0,1,2, . . . ,q−1, that has to be solvedwith the Bloch condition,

gsx + qd = eiqkxgsxd.

Thus the problem of diagonalization is reduced to the diago-nalization of aq3q matrix for eachk. Consequently, theeigenfunctions can be labeled by momentumk and the bandindex n=1,2, . . . ,q. Note, however, that we did not exhaustall the information contained in the magnetic translation op-erators, apart from translations byq lattice spacings in thexdirection. Consider an eigenstate described by crystal mo-mentumk within the Brillouin zone and characterized with

wave functiongsxd. Then functiong1sxd=gsx+1d is also an

eigenstate of the HamiltonianHHof with the same energy butwith momentumskx,ky+2pp/qd. By repeating this opera-tion, one finds thatq states with crystal momenta describedby the samekx but differentky,

ky,ky + 2pp/q,ky + 4pp/q, . . . ,ky + sq − 1d2pp/q,

all have the same energy. Sincep andq are mutually prime,this set coincides with

ky,ky + 2p/q,ky + 4p/q, . . . ,ky + sq − 1d2p/q.

Occasionally, in the theory of magnetic translation groups,this is expressed by using a reduced Brillouin zone of sizes2pd2/q2; then every band is said to beq-fold degenerate. As

FIG. 9. sColor onlined The local density of statessLDOSd of the latticed-wave superconductor with the modulated bond gap functionDi j

corresponding to the 434 supercell structure of Fig. 5 atx=1/8, computed under the assumption of perfect particle-hole symmetry forlow-energy fermionic excitations. The parameters used are the same as in Fig. 7, with the exception of the broadening, which has beensuppressed here in order to demonstrate that the nodes remain essentially unaffected by CPCDWsRef. 12d.

FIG. 10. sColor onlined Left panel: the Peierls link variablesAi j in Landau gauge forq=4. x denotes 2pp/q. BothAr ,r+x andAr ,r+y areperiodic in they direction. AlthoughAr ,r+y increases monotonically withx, expsiAr ,r+xd is periodic with the unit cell shown by the dashedrectangle. Right panel: the dispersionEskx,kyd of the lowest Hofstadter band forp/q=1/4 inunits of t* . There areq=4 ground states atkx=0andky=2p j /q, where j =0,1, . . . ,q−1. Forq=4, the energy of the ground states isE0=−2Î2t.


214511-20

a typical example, a dispersion forp/q=1/4 isshown in theright panel of Fig. 10.

In the gauge we just described, there areq degenerateminima described by wave functionsF jsx,yd that are locatedat kx=0 andky=2pmp/q, wherej =0,1,2, . . . ,sq−1d. There-fore, sufficiently close to the transition, the minimum of thefunctional Hd should be sought as a linear combination ofthe q degenerate statesF jsr d and our problem is equivalentto minimizing the Abrikosov participation ratio,

minor

uFsr du4

soruFsr du2d2, whereFsr d = o

j=1

q

CjF jsr d. s48d

Sincekx=0 for all F jsr d, any linear combination of functionsF jsr d is periodic in thex direction: Fsr +qxd=Fsr d. In ad-dition, for each of theq ground states,ky is a multiple of2p /q, and consequentlyFsr +qyd=Fsr d. From these twoproperties we find that any linear combinationo jCjF jsr dmust be periodic in theq3q unit cell.

The minimization problem can be formulated equivalentlyin terms of coefficientsCj,

Hd = E0oj

uCju2 + o G j1,j2,j3,j4s4d Cj1

* Cj2* Cj3

Cj4, s49d

whereE0 is the ground-state energy ofHHof following froms47d and

G j1,j2,j3,j4s4d = o

rF j1

* sr dF j2* sr dF j3

sr dF j4sr d. s50d

Note thats49d itself has Ginzburg-Landau form withq orderparametersCj. In our case, the form of the matrixGs4d isdictated by our “microscopic” HamiltonianHd and corre-sponds to the Abrikosov participation ratios48d. One couldin principle generalize the theory by considering a com-pletely general form ofGs4d compatible with the overall sym-metry requirements. Such a procedure is equivalent to intro-

ducing long-ranged quartic interactions with general kernelKsr ,r 8 ,r 9 ,r-d,

o Ksr ,r 8,r 9,r-dF*sr dF*sr 8dFsr 9dFsr-d.

Equations49d is the most direct and convenient representa-tion for numerical minimization ofHd in the limit of infinitekd. However, in order to allow for intermediate values ofkdand to be able to analyze the impact of various short-rangedterms describing interaction of dual fluxesdBdsr d, we alsohave opted to follow a slightly different route: The numericalresults that are presented below are produced bydirect nu-merical minimization of functionals41d with respect to boththe dual matter fieldF and the link variablesdAa by impos-ing periodic boundary conditions onNx3Ny blocks withvarying Nx andNy. At least in the vicinity of the transition,the largest unit cell one has to consider isq3q. It should bestressed that we found identical results using both ap-proaches whenever direct comparison is possiblessuffi-ciently largekd and no additional short-ranged interactionsbetween the fluxesd.

We perform numerical minimization of functionals41dwith respect to both the matter fieldFsr d and the link vari-ablesdAa by imposing periodic boundary conditions onn3m blocks with variablem andn. The number of indepen-dent variables grows as 3mn and the largest unit cells wewere able to consider are 838. We used the conjugate gra-dient minimization technique with as many as 104–105 ran-domly chosen different starting points.

We remind the reader that since the dual matter field vari-ablesFsr d in the H1 model live on the red sites, the dualfluxesdBdsr d reside inside the red plaquettes, shown in Fig.11. Note that each link of the original copper lattice is sharedby two red plaquettes and therefore the enhancement of thed-wave gap functionD should be interpreted as the averagebetween the fluxes at the neighboring dual plaquettes. Thus,at half-filling f = 1

2 there is no modulation inD and no charge-density modulation emerging from our model H1 even

FIG. 11. sColor onlined left panel: Any link of the black latticesthick solid linesd, which corresponds toDi j , is shared by two dualplaquettes shown in gray. Within the H1 model, therefore, the enhancement or suppression ofDi j is determined by the average of the fluxestrough the neighboring plaquettes of the dual latticesred dashed linesd. Central panel: distribution ofdBd at half filling. In units of t, theparameters used arekd

2=30.0,m2=−1.0, andg=2.0. The energy per site is −3.695. Right panel: distribution ofdBd at p/q=1/4. In units oft, the parameters used arekd

2=30.0,m2=−1.0, andg=2.0. The energy per site is −3.564.


214511-21

though there is a checkerboard pattern in the dual flux. Thecheckerboard pattern remains the same for the entire range ofparameters we were able to check. However, as explained inthe previous section, since there is a definite symmetrybreaking in the dual sector by the checkerboard array ofdBdsr d, higher-order derivatives and other terms not includedin our dual Lagrangian are expected to generate a weakcheckerboard modulation to accompanydBdsr d.

At field characterized byf =p/q=7/16, which within theH1 model corresponds to dopingx=1/8, thestructure of theconfiguration is considerably more complex. When restrictedto a 434 lattice, the resulting pattern is the square lattice of“crosses” separated by four unit cells, shown in Fig. 12. Thisconfiguration, however, is not the true global minimum. If alarger unit cell for modulations of the dual fluxes is allowed,the energy can be additionally lowered by<0.5% by distort-ing the ideal square pattern. The lowest energy we foundcorresponds to the quasitriangular lattice of crosses shown inthe right panel of Fig. 12. The lowest-energy state that wefind has the symmetry of this quasitriangular pattern for allkd

2 from 1.0 to 105.The smallness of the energy differences, involving only a

few percent of the overall Abrikosov-Hofstadter energyscale, indicates that the state that emerges victorious can bechanged by the additional short-range interactions and de-rivative terms which we have routinely neglected. Not sur-prisingly, therefore, the precise energetics of various low-energy Abrikosov-Hofstadter states is decided by details. Wefind that inclusion of termsuFsr du6 and dual density-densityinteractions uFsr du2uFsr +ddu2 with moderate coefficientsdoes not change the patterns we described. On the otherhand, the inclusion of the terms describing short-ranged in-teractions between the dual fluxes produces significant ef-

fects. An example of the typical pattern obtained by replac-ing the self-interaction term

or

1

2kd

2sdBdd2 s51d

by

or

1

2kd

2sdBdd2 + G0onn

dBdsidBds jd + G1onnn

dBdsidBds jd

s52d

is shown in Fig. 13. The parametersG0 and G1 in s52d arechosen to make the distribution of the dual flux somewhatsmoother than what is demanded bys51d only. This sufficesto bring the energy of the square pattern in Fig. 13 from justabove to just below that of the quasitriangular pattern of Fig.12. It is tempting to speculate that this slight additional“smoothening” of the dual flux represents the combined ef-fects of nodal fermions and Coulomb interactions present inreal systems. Note that the symmetry and the qualitative fea-tures of this pattern coincide with our 434 “elementary”block conjectured to be the likely CPCDW ground state inthe previous sectionssee Figs. 5 and 6d. Once dBdsid aretranslated intodDi j , the resulting pattern closely resemblesthe checkerboard distribution of the local density of states ofan “electron crystal” observed in the STM experiments, asillustrated in Fig. 7.

Let us now consider this observed STM checkerboard pat-tern in more detail, in light of our theory. Obviously, ouranalysis being restricted to the tight-binding lattice, we can-not describe the local density of statessLDOSd at positionsbetween the sites of Cu lattice—the peaks in our theoretical

FIG. 12. Left panel: The pattern of the dual fluxesdBd for p/q=7/16 withperiodic boundary conditions in 434 unit cell. The unit cell434 is repeated four times in horizontal and vertical directions for presentation purposes. The energy per site is −3.193 in units oft and theparameters of the model are the same as in the previous figure. The positivesnegatived values ofdBd are shown in blackswhited. Right panel:the same but with an 838 unit cell. The square lattice is distorted towards the triangular lattice. The energy per site is −3.208t and is lowerby 0.5% compared to the square arrangement.


214511-22

LDOS are always located on top of Cu atoms. In contrast,the STM measurements by Hanaguriet al.9 have much betterspatial resolution and can image the actual continuousatomic orbitals. Our tight-binding lattice results forLDOSsr ,Ed could be viewed as a “coarse-grained” represen-tation of the LDOSgsr ,Ed observed experimentally,

LDOSsR,Ed ~E gsr ,Eddr ,

where the integral extends over a square of sizea03a0 cen-tered around Cu lattice siteR. Equivalently, the true con-tinuum LDOS signal could be obtained by broadening ourlattice LDOS around each Cu lattice site.

A prominent feature of the checkerboard pattern in STMmeasurements that has received much attention is the pres-ence of a pronounced Fourier signal not only at wavevectors2p /a0s 1

4 ,0d and s2p /a0ds0, 14

d, which correspond to the4a034a0 periodicity just described, but also at 2p /a0s 3

4 ,0dand s2p /a0ds0, 3

4d. While it is tempting to associate the 3/4

peaks with an entirely independent type of order, we notethat for a periodic lattice of identical 4a034a0 tiles, the Fou-rier transform is adiscrete serieswith wavevectorsQnx,ny

= 14s2p /a0dsnx,nyd, wherenx andny are integers, irrespective

of how complex is the internal structure of each tile. For ageneral structure of the tile, all of the harmonicssnx,nyd arepresent, and there is noa priori reason for the Fourier coef-ficients withnx=3, ny=0 to be particularly small. The pres-ence of a large signal atQ3,0 is only natural, and no moreunexpected than the weakness of Fourier harmonics atQ2,0andQ0,2.

Next, in order to describe and compare the Fourier trans-forms of spatially broadened LDOS patterns in Figs. 7–9 and

tunneling LDOS observed in experiments,9 we introduce asimple model which approximates each bright spot inside theprimitive 4a034a0 tile as

hsr d = expFJScos2px

4a0+ cos

2py

4a0− 2DG .

While the specific functional form of the peak is not impor-tant, our choice is convenient sincegsr d has a period of4a034a0 and a Gaussian shape, centered at positionsr =s4Nxa0,4Nya0d, whereNx andNy are integers. The widthof the Gaussians is,a0/ÎJ.

The real-space tunneling LDOS pattern of Hanaguriet al.can now be represented by a function

gsr d = Ahsr d + Bod

hsr + dd + Cod8

hsr + d8d, s53d

where the first term represents the brightest peak of each tile,the term proportional toB represents the set of foursecond brightest peaks located atd= ± s1+edax, ±s1+eday,and the last term represents the weak peaks atd8= ±a0s1+edx±a0s1+edy ssee Fig. 14d. Parametere equalszero if the real-space LDOS peaks are centered at the Culattice sites, whilee=1/3 if themaxima of all peaks, exceptfor the central one, are displaced from their commensuratepositions on top of Cu atoms—this is how the experimentaldata were interpreted in Ref. 9. We will show that for a widerange of parametersJ,A,B,C, and arbitrary 0,e,1/3, theQ3,0 peaks are much stronger than peaks atQ2,0, although toexplain other features of Fig. 2 in Ref. 9, the “commensu-rate” choice e=0 appears to be more natural. For themomentum-space directionn=0 shown as black circles inFig. 2 of Ref. 9, the Fourier coefficients are

FIG. 13. Left panel: The pattern of the dual fluxesdBd for the Hofstadter-Abrikosov problem atf =7/16 sx=1/8d obtained for the838 unit cell. The parameters of the model arekd

2=30.0,G0=6.0, G1=−12.0,m2=−1.0, andg=2.0. The 434 checkerboard symmetry ofthe pattern is precisely what is shown in Fig. 7. Right panel: The distribution of modulations in the pairing pseudogapdDi j within theelementary tile of the “tartan” pattern shown in the left panel. This pattern has the symmetry of Fig. 5 with “dual vortex holes” occupyingthe corners of the elementary tile.


214511-23

gm,0 =1

16a2E0

4a0E0

4a0

dxdyGsx,yde−is2p/4a0dmx.

The integral is elementary and the result forgm,0 is

FA + 2B + s2B + 4Cdcos2ps1 + edm

4GI0sJdImsJde−2J,

s54d

where Im is the regular modified Bessel function.ImsJd,shown for several values ofJ in Fig. 14, is a monotonicallydecreasing function ofm. The nonmonotonic part ofgm,0,denoted bygm,0, is contained in the first factor ofs54d. Whene=0 scommensurate position of the peaksd, this factor is aperiodic function ofm with period 4, while fore=1/3 thesaid period is equal to 3. Table I summarizes the factorsgm,0for the two cases.

We start our analysis with the incommensurate casee=1/3. Thethird column of Table I implies that in this caseg1,0 and g2,0 are equal and smaller than the componentg3,0,as shown in the third column of the table. Experimentally,however, the Fourier component atQ1,0 sthe 1/4 peakd isroughly of the same magnitude asQ3,0, and it is the compo-nent atQ2,0 that is the weakest. To account for particularlysmallg2,0!g1,0, one has to select the value ofJ such that themonotonic functionImsJd decreases rapidly in the region be-tweenm=1 andm=2. Figure 14 indicates that this is the case

for 3,J. This choice, however, also dramatically suppressesthe Fourier transforms atm=3 andm=4, both of which arerather large in experiments.

While the qualitative features of the Fourier transformedexperimental LDOS could possibly be reconciled withe=1/3 by fine-tuning parametersA, B, andC, the situationmight in fact be better described by assuming thecommen-surate location of the peaks, in registry with Cu atomsse=0d; see the second column the Table I. In this case, har-monicm=2 isautomaticallysuppressed compared to the 1/4and 3/4 Fourier components. The suppression of the Fouriercomponent at 2/4 can be qualitatively understood as follows:This Fourier component is determined by the overlap of theLDOS signal and cosf2s2p /4a0dxg. Obviously, the maximaof the LDOS signal correspond to alternating maxima andminima of cosf2s2p /4a0dxg and destructive interference ofthe two functions occurs. For harmonics 1/4 and 3/4sand ofcourse for 4/4d, the overlaps are significant and their Fouriercoefficients are larger. Peaks corresponding to larger valuesof m are strongly reduced due to the monotonic dependenceImsJd of the Fourier transform onm. This suppression canserve as an estimate of parameterJ: visually, the last discern-ible peak in Fig. 2 of Ref. 9 appears atQ5,0, which placesJin the range of 5–10ssee the right panel of Fig. 14d. In theright panel of Fig. 14, the spatial Fourier transforms of theLDOS s53d with parametersJ=7, A=1, B=0.5, andC=0and commensurate placement of the peakse=0 are shown.This example illustrates that the major features of theFourier-transformed LDOS obtained by Hanaguriet al.9 arerobust properties of an elementary tile of size 4a034a0 andnine peaks occupyingcommensuratelocations at sites of theCu lattice as depicted in our Figs. 7–9: the large magnitudeof the 3/4 peak is simply a higher harmonic describing thecharacteristic intratile structure.

VII. CONCLUSIONS

Our main goal in this paper is to devise a more realisticdescription of a strongly quantum and thermally fluctuating

TABLE I. Nonmonotonic dependencegm,0 of the Fourier trans-formed LDOS gsr d defined ins53d.

m e=0 e=1/3

m;0 mod 4 A+4B+4C A+4B+4C

m;1 mod 4 A+2B A+B−2C

m;2 mod 4 A−4C A+B−2C

m;3 mod 4 A+2B A+4B+4C

FIG. 14. sColor onlined left panel: Functiongsr d used to emulate the LDOS signal with parametersJ=7.0, a=1.0, b=0.5, andc=0.2.Center panel: Regular modified Bessel functionImsJd shown as a function of its indexm for several values of fixedJ. Right panel: Fouriercoefficientsgsqx,0d for LDOS pattern shown in the left panel. The reader should compare this with the similar plot in Fig. 2 of Ref. 9.


214511-24

d-wave superconductor, based on the theory of Ref. 5. Sucha description applies not only to long-distance and low-energy properties, which are the primary domain of Ref. 5,but also to intermediate length scales, of order of severallattice spacings, and to energies up to the pseudogap scaleD.This enables us to use the theory to address the experimentalobservations of Refs. 8–10. The charge modulation observedin those experiments is attributed to the formation of theCooper pair CDW, the dynamical origin of which is in strongquantum fluctuations of vortex-antivortex pairs. These quan-tum superconducting phase fluctuations reflect enhancedMott-Hubbard correlations in underdoped cuprates as dopingapproaches zero. Quantum fluctuatinghc/2e santidvortices“see” a physical electron as a source of a half-quantumdualmagnetic flux and the theory of the CPCDW can be formu-lated as the Abrikosov-Hofstadter problem in a type-II dualsuperconductor.12 An XY-type model of such a dual super-conductor appropriate for a latticed-wave superconductor isconstructed, both for thermal and quantum phase fluctua-tions. The specific translational symmetry-breaking patternsthat arise from the dual Abrikosov-Hofstadter problem arediscussed for various dopingsx, which determines the dualflux per unit cell of the CuO2 lattice via f =p/q=s1−xd /2. Inturn, the spatial modulation of the dual magnetic inductionBd corresponding to these Abrikosov-Hofstadter patterns isrelated to the modulation in the gap function of the latticed-wave superconductor and is used to compute LDOS ob-served in STM experiments. A good agreement is found forx=1/8 sf =7/16d, which is the dominant fraction of theAbrikosov-Hofstadter problem in the window of dopingswhere our theory applies.

ACKNOWLEDGMENTS

The authors thank J.C. Davis, M. Franz, J.E. Hoffman, S.Sachdev, A. Sudbø, O. Vafek, A. Yazdani, and S.C. Zhang foruseful discussions and correspondence and S. Sachdev forgenerously sharing with us unpublished results of Ref. 17.A.M. also thanks P. Hirschfeld, B. Andersen, T. Nunner, andL.-Y. Zhu for numerous discussions. The final touches wereapplied to this paper while one of ussZ.T.d enjoyed the hos-pitality of the Aspen Center for Physics. This work was sup-ported in part by the NSF Grant No. DMR00-94981.

APPENDIX: TWO ALTERNATIVE DERIVATIONS OF THEDUAL ABRIKOSOV-HOFSTADTER HAMILTONIAN

In this appendix, we present two different and self-contained derivations of the dual Abrikosov-HofstadterHamiltonians37d and s38d, either one of which can serve asan alternative to the derivation given in the main text. Thefirst approach is somewhat more detailed and in a sense more“microscopic” since it uses a quantum vortex-antivortexHamiltonian as a springboard to derive the effective dualfield theory s36d. In turn, such a vortex-antivortex Hamil-tonian in principle can be derived from thesstill unknowndfully microscopic theory of cuprates. Incidentally, this deri-vation is thes2+1d-dimensional analogue of the 3D casepresented in the Appendix of Ref. 6. The second derivation

follows the familiar Villain approximation to theXY modeland applies it to our specific situation. The Villain approxi-mation is less “realistic” but provides a transparent and sys-tematic way of deriving dual representations ofXY-like mod-els.

In both derivations the starting point is the effectives2+1dD XY model of a quantum fluctuatingd-wave supercon-ductor:

LXYd = io

i

f iwi +K0

2 oi

wi2 − Jo

nn

cosswi − w jd

− J1ornnn

cosswi − w jd − J2 obnnn

cosswi − w jd

+ Lnodalfcosswi − w jdg + Lcore, sA1d

where wistd is the fluctuating phase on a site of the bluelattice in Fig. 2, the firstsimaginaryd term is the charge Berryphase corresponding to the overall fluxf through a plaquetteof the blue lattice,J is the nearest-neighborXY coupling,J1 s2d are the next-nearest-neighborXY couplings along redsblackd diagonals,Lnodal is the contribution of nodal fermi-ons, andLcore denotes core contributions arising from smallregions around vortices where the pairing pseudogap is sig-nificantly suppressed—this, for example, generally includesthe mass terms25d, the energy cost of core-core overlap, theBardeen-Stephen core dissipation, etc. The reader shouldbear in mind that the last effect is small in cuprates, as ex-plained in the main text, and will be neglected in the Appen-dix. Furthermore, we are neglecting vortex interactions withthe spin of low-energy nodal fermions, represented by theBerry gauge fielda; this is justified away from the criticalpoint. The results below are easily adapted to the extendeds-wave pairing symmetry. Similarly, both derivations arestraightforwardly applied to a yet simpler case, a fluctuatings-wave superconductor,

LXY = i foi

fi +K0

2 oi

fi2 − Jo

nn

cossfi − f jd + Lcore,

sA2d

which was used in the main text as a pedagogical example.Just as the starting points of two derivations coincide,

their final product, the effective dual theory at long and in-termediate length scales, will also turn out to be the same.

1. Hd from a “microscopic” vortex-antivortexHamiltonian

The quantum partition function of a phase fluctuating su-perconductor is

ZXYd =E Dwi expS−E

0

b

dtLXYd fwistdgD , sA3d

where the functional integraleDwistd runs over phase vari-ableswistd such that expfiwistdg is periodic in the intervaltP f0,bg.

The difficulty in computing Eq.sA3d is twofold: the factthatwistd is acompactphase variable, defined on an interval


214511-25

f0,2pd, rather than an ordinary real field taking values inf−` , +`g, and the cosine functions inLXY

d that couple phaseson different sites in a nonlinear fashion. To deal with theproblem, one approximates the cosines with quadraticforms. One popular approximation on a lattice is due to Vil-lain and will be discussed in the next subsection. Incontinuum, the approximation amounts to replacingcosswi −w jd→1−sa2/2ds=wd2+¯, where a is the latticespacing and wsxd is now a function in continuouss2+1d-dimensional spacetime. Its compact character is en-forced by writing]mwsxd→]mxsxd+f]mwsxdgv, wherex is anordinary real field andf]mwsxdgv is the part of the phaseassociated with vortices, defined via=3 f=wsr ,tdgv

=2poad(r −r avstd)−2poad(r −r a

astd), with hr avsadstdj being

santidvortex positions. Simultaneously with this decomposi-tion of ]mwsxd, eDwi is replaced by the functional integra-tions overxsxd and santidvortex positionshr a

vsadstdj. We areassuming here that thesantidvortices of topological charge±1 dominate the fluctuation behavior in the regime where theamplitude of the pseudogapD is large and stiff, allowing usto safely neglect topological defects corresponding to vortic-ity ±2, ±3, . . . due totheir higher core energies. This assump-tion simplifies the algebra considerably. Furthermore, this as-sumption is natural within the theory of Ref. 5: Theproliferation of defects of high topological charge is equiva-lent to strong amplitude fluctuations and the eventual col-lapse of the pseudogap—as long as we are in the pseudogapregime, the ±1santidvortices are the only relevant excita-tions. With these changes in place, the partition functionsA3d finally takes the form

ZXYd → o

Nv=0

`

oNa=0

`1

Nv!Na!E Dx p

a,g=1

Nv,NaEhr a

v s0dj=hr av sbdj

3Dr avstdE

hr gas0dj=hr g

asbdjDr g

astd

3expS−E d3xLXYd fx,hr a

vsadstdjgD , sA4d

whereed3x=e0bdted2r and the sethr a

vsadstdj containingNvsadsantidvortices att=0 coincides with the one att=b, to en-sure proper periodicity of expfiwsr ,tdg in imaginary time.Lastly,

LXYd fx,hr a

vsadstdjg = i f sr dwv +K0

2sx + wvd2 +

J

2s=x + = wvd2

+ Lnodal+ Lcored fhr a

vsadstdjg, sA5d

wherefsr d=oi f idsr −Rid, hRij are the sites of the blue lattice,

J=J+J1+J2, K0 has been rescaled bya2, and

wvsr ,td = oa

fr − r avstdg 3 z

ur − r avstdu2

· r av − o

g

fr − r gastdg 3 z

ur − r gastdu2

· r ga

=E d2r8sr − r 8d 3 z

usr − r 8du2·Jsr ,td,

=wvsr ,td = − oa

fr − r avstdg 3 z

ur − r avstdu2

+ og

fr − r gastdg 3 z

ur − r gastdu2

=

−E d2r8sr − r 8d 3 z

usr − r 8du2nsr ,td. sA6d

In Eq. sA6d, we found it useful to introduce vorticitydensity and current:nsr ,td=oad(r −r a

vstd)−ogd(r −r gastd),

Jsr ,td=oar avd(r −r a

vstd)−ogr gad(r −r g

astd), in terms ofwhich, when combined with vortex particle density andcurrent rsr ,td=oad(r −r a

vstd)+ogd(r −r gastd), j sr ,td

=oar avd(r −r a

vstd)+ogr gad(r −r g

astd), we can write

Lcored =

1

2oa

MSdr av

dtD2

dsr − r avd +

1

2og

MSdr ga

dtD2

dsr − r gad

+ Hcored , sA7d

where

Hcored = Vsr drsr ,td +

1

2rsr ,td E d2r8Vs2dsr ,r 8drsr 8,td

+1

2jksr ,td E d2r8Vkl

s2dsr ,r 8d j lsr 8,td

+1

2nsr ,td E d2r8Vs2dsr ,r 8dnsr 8,td

+1

2Jksr ,td E d2r8Vkl

s2dsr ,r 8dJlsr 8,td + s¯d. sA8d

sAntidvortex mass terms appearing in Eq.sA7d have beenintroduced already in Sec. IVfsee the discussion surroundingEq. s25dg, while Hcore

d represents a systematic expansion invortex core density, including single-core, two-core terms,and so on. All the short-range terms that arise from expand-ing the cosine functionssA1d in continuum limit have beenabsorbed intoHcore

d , as was our habit throughout the text—inparticular,Vsr d is just the vortex potential on the blue lattice,containing crucial information ond-wave pairing, which isextensively discussed in Sec. III.

We can now integrate outx, the regularsXY “spin-wave”dpart of the phase. The quadratic phase stiffness terms in Eq.sA5d are decoupled as

K0

2sx + wvd2 +

J

2s=x + = wvd2

→ iW0sx + wvd + iW · s=x + = wvd +1

2K0W0

2 +1

2JW2

sA9d

via the Hubbard-Stratonovich vector fieldW=sW0,Wd snote

that we have set the “dual speed of light”ÎJ/K0 to unityd.Integration by parts givesiW·]x→−is] ·Wdx and is fol-lowed by functional integration overxsxd, resulting in thelocal d-function constraintds] ·Wd. The constraint is solvedby introducing a noncompact gauge fieldAd such thatW=]3Ad, ensuring] ·W=] ·s]3Add=0. What remains of Eq.


214511-26

sA9d is further transformed by another partial integration,

is] 3 Add · s]wdv +1

2K0s] 3 Add0

2 +1

2Js] 3 Add'

2

→ − iAd · „] 3 s]wdv… +1

2K0s] 3 Add0

2 +1

2Js] 3 Add'

2 ,

sA10d

where s]3Add0,' denotes temporal and spatial componentsof ]3Ad, respectively. Now observe that Eq.sA6d implies]3 s]wdv=s2pn,2pJd. This allows us to finally write thepartition function of the quantum vortex-antivortex system as

Zvd = o

Nv=0

`

oNa=0

`1

Nv!Na!p

a,g=1

Nv,NaEhr a

v s0dj=hr av sbdj

Dr avstd

3 Ehr g

as0dj=hr gasbdj

Dr gastdexpS−E d3xLv

dD , sA11d

whereLvd equals

1

2oa

MSdr av

dtD2

dsr − r avd +

1

2og

MSdr ga

dtD2

dsr − r gad + Lnodal

+ Hcored − 2piAd0n − i2psAd

s0d + Add ·J +1

2K0s] 3 Add0

2

+1

2Js] 3 Add'

2 , sA12d

and=3Ads0d=Bd

s0d= fsr dz.EquationssA11d andsA12d are an important result of this

Appendix. We recognizeZvd as equivalent to a partition func-

tion of two species of nonrelativistic quantum bosons ex-pressed in the Feynman path integral representation over par-ticle worldline trajectories. These vortex and antivortexbosons have identical massM and carry dual charges +2pand −2p, respectively, through which they couple to a dy-namical gauge fieldAd. The dual photons ofAd mediatelong-range “electrodynamic” interactions between thebosons, which are just the familiar Biot-Savart interactionsbetween santidvortices. In addition, the particles interactthrough an assortment of short-range interactions containedin Hcore

d sA8d. Furthermore,Lnodal describes the interactionsgenerated by nodal Dirac-like fermions which will be in-cluded explicitly once we arrive at the dual representation ofEq. sA12d, as detailed in Sec. IV. Finally, the vortex potentialVsr d contains important information about the underlying lat-tice structure and the symmetry of the order parameter, asemphasized throughout the text.

We have derivedZvd as a continuum limit approximation

to the partition functionZXYd sA3d of the quantumXY-type

model sA1d. Actually, in real cuprates and in all other

physical systems, the opposite is true: It is the quantumXY-type representation that is an approximation toZv

d. Zvd

captures the general description of the quantum vortex-antivortex system, applicable to all superconductors and su-perfluids whose order parameter is a complex scalar. To ap-preciate this, imagine that for each given configuration of thephase, withsantidvortex positions fixed in Euclidean space-time, the microscopic action of a physical system is mini-mized with respect to the amplitude, after all other degrees offreedom have been integrated out. The subsequent summa-tion over all distinctsantidvortex positions leads to preciselyZv

d as the final resultsagain, we remind the reader that thecore dissipative terms will also generically appear inZv

d but,being small in underdoped cuprates, are neglected here asexplained in the main textd. In practice, this procedure isdifficult to carry out explicitly and the actual values of vari-ous terms that enterZv

d are hard to determine from “firstprinciples.” This is particularly true for core-core interactionterms appearing inHcore

d . For our purposes, it will suffice toapproximateVs2dsr ,r 8d→gdsr −r 8d, where g.0, and dropthe rest.

There is one crucial feature which distinguishesZvd from

the standard Feynman partition function: The vortex and an-tivortex quantum bosons arenot conserved. As particlesmake their way through imaginary time, vortices and anti-vortices canannihilateeach other; similarly, they can also becreatedat an instant in time; this is depicted in Fig. 3. Allsuch processes of creation and annihilation proceed inpairsof vortices and antivortices. Consequently, while the indi-vidual number of vortices and antivortices is not conserved,thevorticity, measured by dual chargeed= ±2p, is conservedand the gauge symmetry associated withAd is always main-tained sunless, of course, it is spontaneously broken by adual Higgs mechanism in dual superfluidd. In other words,these nonrelativisticsantidvortex bosons propagate throughspacetime permeated by a vortex-antivortex condensate, ofstrengthDv.

Feynman path integrals are beautiful but difficult to cal-culate with. Following the standard mapping48 we can ex-pressZv

d as a functional integral over complex fieldsCvsr ,tdand Casr ,td, which are the eigenvalues of vortex and anti-vortex annihilation operators, respectively, in the basis ofcoherent states,

Zvd →E DCvDCaE DAd expF−E d3xSCv

*s]t + iedAd0dCv

+ Ca*s]t − iedAd0dCa +

1

2Mus= + iedAd

s0d + iedAddCvu2

+1

2Mus=− iedAd

s0d − iedAddCau2 − mvsuCvu2 + uCau2d

+ Vsr dsuCvu2 + uCau2d + DvCv*Ca

* + Dv*CaCv

+g

2suCvu2 + uCau2d2 + Lnodal+

1

2Km

s] 3 Addm2DG , sA13d


214511-27

where the meaning of various terms is straightforward inlight of our earlier discussion; we have consolidated the no-tation so thated=2p, Km=sK0, J, Jd, the chemical potentialfor vortices ismv=ma, andg describes the short-range core-core repulsions. The “vortex-antivortex pairing” functionDvis crucial since it regulates the frequency of vortex-antivortexpair creation and annihilation processes.

The form ofZvd can be further simplified by exploiting the

vorticity conservation law. We observe that the action insA13d is invariant under gauge transformations:Cv→expsizdCv, Ca→exps−izdCa, Ad→Ad−s]zd /ed. This

prompts us to introduce bosonic “spinors”C=sCv* ,Cad

which carry a conserved dual chargeed and couple mini-mally to Ad,

Zvd →E DCDAdDs expF−E d3xSCLC + Lnodal+

1

2gs2

+1

2Km

s] 3 Addm2DG , sA14d

where

L = Fs]t + iedAd0d Dv

Dv* − s]t + iedAd0d G + S−

1

2Ms= + iedAd

s0d + iedAdd2 − mv + Vsr d + isD1, sA15d

and a Hubbard-Stratonovich scalar fields was deployed to decouple short-range repulsion. Now we setAd→0 and ignoreLnodal—they will be easily restored later—and note that the integration over vortex matter fieldsC gives

Zvd →E Ds expS−E d3x

1

2gs2D 3 det3]t −

1

2M=2 − mv + Vsr d + is Dv

Dv* − ]t −

1

2M=2 − mv + Vsr d + is4 . sA16d

The above partition function has a transition atmv=−uDvufwe are assuming that the minimum ofVsr d occurs at zero,with Ec

r , or Ecb as the case may be, having been absorbed into

mvg. For mv,−uDvu, the system ofsantidvortex bosons is inits “normal” state, withkCl=0. Formv.−uDvu, santidvortexbosons condense andkCl becomes finite. This is nothing butthe dual description of the superconducting transition dis-cussed in the main text. In the general vicinity of the transi-tion, it is useful to introducem2=mv

2− uDvu2, where nowm2.0 andm2,0 indicate dual normal and superfluid states,respectively. Focusing on distances longer thanÎDvM andenergies lower thanDv, the determinant insA16d can befurther reduced to

detF− ]t2 −

uDvu + Vsr d + is

2M=2 + m2 −

1

Mfs=2Vd + is=2sdg

+ 2uDvufVsr d + isg + fVsr d + isg2G + s¯d, sA17d

with additional termss¯d contributing unimportant deriva-tives.

By setting Vsr d and ssxd to zero, we observe that theabove expression assumes the form of the partition functiondeterminant for a system ofrelativistic quantum bosons ofmass m: detf−]t

2−c2=2+m2c4g, with the speed of “light”c=ÎuDvu /2M set to unity henceforth. The terms involvingVsr d andssxd describe the underlying potential and variousshort-range interactions of these relativistic bosons. Wetherefore can reexpress the determinantsA17d as a functional

integral over the relativistic boson fieldFsxd; this is a faith-ful representation of the originalsantidvortex partition func-tion at distances longer thanÎDvM and energies lower thanDv,

Zvd → Zd =E DFE DAdE Ds

3expF−E d3xFus] + i2pAddFu2 + m2sr duFu2

+ 2uFu2SuDvu + Vsr d −=2

2MDsisd +

uDFu2

Msisd

+ S 1

2g− uFu2Ds2 + Lnodal+

1

2Km

s] 3 Addm2GG ,

sA18d

where m2sr d=m2+2uDvuVsr d+Vsr d2−f=2Vsr d /Mg and wehave restoredLnodal and dual gauge fieldAd, through covari-ant derivatives ]→D=sD0,Dd=s]0+ i2edAd0, = + iedAd

s0d

+ iedAdd. The minimal coupling ofAd is mandated by dualcharge conservation:F→expsizdF, Ad→Ad−s]zd /ed.

The dual partition functionZd sA18d is the final result ofthis subsection. It describes the system of relativistic quan-tum bosons of massm and chargeed=2p in a magnetic fieldBd= = 3Ad

s0d. The virtual particle-antiparticle creation andannihilation processes in the vacuum of this theory are noth-ing but quantum vortex-antivortex pair excitations evolvingin imaginary time ssee Fig. 3d. In the “normal” vacuum,


214511-28

m2.0, the average size of such pairs is,m−1. This is justthe superconducting ground state of physical underdoped cu-prates. Form2,0, this “normal” vacuum is unstable to aHiggs phase, with a finite dual condensatekFl. The vortex-antivortex pairs unbind as infinite loops of virtual particle-antiparticle excitations ofF permeate this dual Higgsvacuum—this is the pseudogap state of cuprates. The inte-gration over the Hubbard-Stratonovich fields produces ashort-range repulsion12s4uDvu2dguFu4 followed by an assort-ment of other short-range interactions includingVsr d, powersof uFu higher than quartic and various derivatives. All theseadditional interactions are irrelevant in the sense of long-distance behavior but might play some quantitative role atintermediate length scales. For simplicity, we shall mostlyignore them in this paper. Finally, with the change of nota-tion 4uDvu2g→g andLnodal incorporated into the self-actionfor Ad as detailed in Sec. IV, the dual Lagrangian insA18dreduces toLd s36d. The arguments in the text can then befollowed to arrive atHd s37d and s38d.

2.Hd in the Villain approximation

A useful approximation to the ordinaryXY model is dueto Villain. In this approximation, the exponential of the co-sine function is replaced by an infinite sum of the exponen-tials of parabolas. We will illustrate this approximation firstfor the 2D case and follow up with thes2+1dD quantumXY-like model.

a. Classical (thermal) phase fluctuations

Here we apply the Villain approximation to our classicalXY model of a phase fluctuating two-dimensionald-wavesuperconductors14d. As we will see presently, the final Cou-lomb gas representation coincides with Eq.s23d—however,the advantage of the Villain approximation is that it willallow us to obtain explicit expressions for the core energiesEc

rsbd in terms of the coupling constants of the original Hamil-tonian s14d.

Denoting the sites of the blue lattice byr=sx,yd, the par-tition function of the model can be written as

Z =E Spr

dfrDexpor

hJfcoss=xfrd + coss=yfrdg + J12srd

3fcossfr+x+y − frd + cossfr+y − fr+xdgj, sA19d

where the lattice operator= is defined according to=dfsr d= f r+d− f r , and coefficientsJ12srd denoteJ1 sor J2d if r is inthe lower left corner of a redsor blackd plaquette.

In the Villain approximation, the exponent of a cosine isreplaced by a sum of Gaussian exponents that has the sameperiodicity 2p,

expfb cosgg < RVsbd on=−`

+`

expS−bVsbd

2sg − 2pnd2D .

sA20d

The fitting functionsbVsbd andRVsbd are determined by therequirement that the lowest Fourier coefficients of the two

functions coincide. In particular, the functionbVsbd has thefollowing asymptotic behavior for low and high tempera-tures:

bVsbd < 5b for b @ 1

S2 lnb

2D−1

for b ! 1.6 sA21d

The sum overn in sA20d can be transformed by decouplingthe quadratic term in the exponent via the Hubbard-Stratonovich transformation,

expfb cosgg ~ on=−`

`

expS−1

2bVsbdn2 + ignD . sA22d

By introducing integer fieldsuxsrd, uysrd, andw±srd to applysA22d to the cosine terms insA19d, we obtain

Z = oua

ow±

E pr

dfre−orhfua2srd/2J8g+fw+

2srd+w−2srdg/2J128 srdj

3 eiorfuasrd=afr+w+srdsfr+x+y−frd+w−srdsfr+y−fr+xdg,

sA23d

wherea=x,y, and coefficientsJ12srd andJ8 are defined as

J8 = bVsJd, sA24d

J12srd = bVfJ12srdg. sA25d

The integration over the anglesfr can be performed afterapplying the discrete analog of the integration by parts to thesums in the last exponent insA23d,

ox

fsrd=xgsrd = ox

fsrdfgsr + xd − gsrdg

= ox

ffsr − xd − fsrdggsrd

= − ox

= fsrdgsrd. sA26d

Note that we distinguish between the “right difference” op-

erator = and the “left difference” operator=. Integrationover the phasesfr yields the following expression for thepartition function:

oux,uy.w±

e−or„hfux2srd+uy

2srdg/2J8j+hfw+2srd+w−

2srdg/2J128 srdj…

3d„= ·vsrd + ¯ …, sA27d

where= ·vsrd denotes two-dimensional lattice divergence ofvsrd and dots denote

fw+srd − w+sr − x − ydg + fw−sr − xd − w−sr − ydg.

We rewrite the constraint appearing as the Kroneckerd func-tion in the sum as

= · sv + wfw+,w−gd = 0,

wherew=swx,wyd denote the following linear combinationsof integer fieldsw±srd:


214511-29

wx =w+srd + w+sr − yd

2−

w−srd + w−sr − yd2

, sA28d

wy =w+srd + w+sr − xd

2+

w−srd + w−sr − xd2

. sA29d

The constraint can then be resolved as

v = b − wfw+,w−g,

where

b = s=yL,− =xLd sA30d

and Lsrd has the meaning of the timelike component of avector potential. At this point it is useful to pause and estab-lish a simple geometrical interpretation of the various fieldswe have introduced. Variablesuxsrd anduysrd are coupled tothe phase differencesfr+x−fr and fr+y−fr and thereforethey reside on the links emanating fromr in the positivexand y directions, respectively. IntegersLsrd, on the otherhand, are related to link variableuxsrd through the differenceLsrd−Lsr− yd. Consequently, we must associateLsrd withthe centers of the blue plaquettes, which coincide with eitherred or black sites. Nevertheless, we will continue to use no-tation Lsrd tacitly implying that r refers to the lower leftcorner of thesred or blackd plaquette associated withL. Hav-ing resolved the constraint, we find that the partition functionZ can be written in terms of integer-valued fieldsw±srd andLsrd only,

oL,w±

exporF−

sbfLg − wfw±gd'2

2J8−

w+2srd + w−

2srd2J128 srd G .

sA31d

To obtain the description in terms of continuous rather thaninteger-valued fieldsLsrd, we use the Poisson summationformula,

oL=−`

`

fsLd =E−`

`

dL ol=−`

`

e2pilLfsLd. sA32d

The partition functionZ assumes the following form:

E−`

`

pr

dLsrdolsrd

expor

h2pil srdLsrdjexphFfbfLsrdggj,

sA33d

where we have defined a functional exphFfbsrdgj accordingto

ow±

expForS−

sb − wfw±gd'2

2J8−

w+2srd + w−

2srd2J128 srd

DG .

sA34d

In the limit when constantsJ1 and J2 are infinitesimallysmall, only the configurationsw±srd=0 contribute to theFfbg. This limit, which corresponds to the usual 2DXYmodel, is described by partition functionZ0 given by

E−`

`

pr

dLsrdolsrd

expFor

S2pil srdLsrd −s=aLd2

2J8DG .

sA35d

For finiteJ12, we must resort to approximate evaluation ofthe functionalF(bsrd),

exphFfbxsrd,bysrdgj = ow±

expF− orS sba − wad2

2J8

+w+

2srd + w−2srd

2J128 srdDG . sA36d

The quadratic terms containingb can be decoupled using theHubbard-Stratonovich transformation,

eFfbsrdg =E pr

dZxsrddZysrd 3 ow±

expForS−

J8Za2srd2

+ iZasrdfbasrd − wasrdg −w+

2srd + w−2srd

2J812srdDG ,

sA37d

wherea=x,y. Using explicit expressions forwa, we obtain

eFfbg =E pr

dZxsrddZysrdow±

3expForS−

J8Za2srd2

−w+

2srd + w−2srd

2J812srd

+ iZasrdbasrd −i

2w+srdfZxsrd + Zxsr + yd + Zysrd

+ Zysr + xdg −i

2w−srdf− Zxsrd − Zxsr + yd + Zysrd

+ Zysr + xdgDG . sA38d

The sums overw±srd can be performed by employing theVillain approximation sA20d backward. Note that the cou-pling constantsJ±8 are restored to the original values of cou-pling constantsJ±,

eFfbsrdg =E pr

dZxsrddZysrd

3expHorF−

J8Za2srd2

+ iZasrdbasrd + J12srd

3ScosZxsrd + Zxsr + yd + Zysrd + Zysr + xd

2

+ cos− Zxsrd − Zxsr + yd + Zysrd + Zysr + xd

2DGJ .

sA39d

To quadratic order, the expression in the exponent is


214511-30

orF−

J8Za2srd2

+ iZasrdbasrd − J12srd

3fZxsrd + Zxsr + ydg2 + fZysrd + Zysr + xdg2

4G .

sA40d

Note thatZx andZy components are completely decoupled atthe quadratic level. To proceed, one can double the unit cell,in which case the expression in the exponent becomes diag-onal in the momentum space. Alternatively, one can use anequivalent, but technically simpler procedure of keeping theoriginal unit cell. In this latter case, the momentum spaceproblem reduces to the diagonalization of a 232 matrix con-necting modes at wave vectorsq and q−g, whereg=psx+ yd.

It is convenient to representJ12srd as

J12srd =J1 + J2

2+ eig·rJ1 − J2

2; J + dJeig·r.

After Fourier transformation, the exponent insA40d becomes

iZxsqdbxs− qd − ZxsqdZxs− qdJ8 + Js1 + cosqyd

2

−idJ

2sinqyZxsg − qdZxsqd + sx ↔ yd. sA41d

Since Zasrd and basrd are real, their Fourier componentssatisfy

Zas− qd = Za* sqd, sA42d

bas− qd = ba* sqd. sA43d

In the last expression for the partition function, we found thatthe terms withZx andZy decouple and we thus can integrateover Zxsrd andZysrd separately.

The expressions in this and especially the next subsectioncan be significantly economized by using a check mark todenote two-component vectors,

bsqd = S bsqdbsq − gd

D .

Using this notation, the contribution due toZxsrd can bewritten as

i

2bx

Ts− qdZxsqd −1

4Zx

Ts− qds¯dZxsqd, sA44d

wheres¯d=J8+ J+ J cosqys3+dJ sinqys2 and superscriptTdenotes the transpose of a matrix. NowZa can be integratedout. Apart from the overall normalization constant,Ffbg isgiven by

Ffbsrdg = −1

4E dq

s2pd2fbxTs− qdGsqydbxsqd + sx ↔ ydg,

sA45d

where matrixG is defined as

G =1

DsqydsJ8 + J − J cosqysz − dJ sinqys2d sA46d

and the determinantD equals

Dsqyd = J8sJ8 + 2Jd + fJ2 + sdJd2gsin2 qy. sA47d

When J=dJ=0, we find

Ffbsqdg → −1

2J8basqdbas− qd,

which restores the limit of an ordinary 2DXY modelsA35d.Returning to the partition functionsA33d and using the

expressionsA45d for Ffbg we just found, we are now inposition to integrate out the gauge fieldLsrd and obtain theanalogue of the Coulomb gas representation for our model.Note thatsA30d implies

bxsqd = s1 − e−iqydLsqd, sA48d

bysqd = − s1 − e−iqxdLsqd. sA49d

The partition function now becomes

Z = olsrdE

−`

`

pr

dLsrd

3expFE dq

s2pd2fiplTs− qdLsqd − Ls− qdMLsqdgG ,

sA50d

where the 232 matrix M is given by

M =1

2Dsqydhs1 − s3 cosqydfJ8 + Js1 − s3 cosqydg

+ ss1dJ sin2 qydj + sx ↔ yd. sA51d

After integration overLsrd, we obtain

Z = olsrd

expF−p2

4E dq

s2pd2 lTs− qdMsqdlsqdG , sA52d

where matrixM is the inverse ofM. The elements of matrixM satisfy the following simple identities:

M11sqd = M22sq − gd, sA53d

M12sqd = M21sqd. sA54d

Consequently, the integrand in the exponent of partitionfunction sA52d can be written as

2fls− qdM11sqdlsqd + lsg − qdM12sqdlsqdg. sA55d

The explicit form ofM =M−1 is rather cumbersome. Fortu-nately, we will only need the leading- and subleading-orderterms in the long-wavelengthsq→0d expansion,

M11sqd =4sJ8 + 2Jd

qx2 + qy

2 +sJ8 − 4J + 18dJdsqx

4 + qy4d + 12dJqx

2qy2

3J8sqx2 + qy

2d2

+ Osq2d, sA56d


214511-31

M12sqd = − dJ + Osq2d. sA57d

The terms of orderOsq2d correspond toMsr−r8d decreasingat least as fast asur−r8u−4. Returning to the real-space rep-resentation, we obtain

olsrd

expH−p2

2 or,r8

flsrdM11sr − r8dlsr8d

+ eig·rlsrdM12sr − r8dlsr8dgJ , sA58d

where

Mabsr − r8d =E−p

p dqx

2pE

−p

p dqy

2peiq·sr−r8dMabsqd.

sA59d

The two terms in the exponent ofsA58d are easy to interpret.Recall that integerslsrd coupled toLsrd effectively reside atthe centers of the plaquettes of the blue lattice correspondingto either black or red sites in Fig. 2. The terms containingM11 clearly describe the average interaction between twoplaquettesirrespectiveof their “color,” while the terms withM12 reflect the difference between the red and black sites.For example, the strength of interaction between two blacksites separated by two lattice spacings withr=0 andr=2xwill be different from interaction between two red sites atr= x, r8=3x due to the factor expsig·rd that multipliesM12.

At large distances, the Fourier transform can be evaluatedby comparison to the standard lattice Green’s function in twodimensions. The difference

M11sqd − 4sJ8 + 2Jd1

4 − 2 cosqx − 2 cosqy

is finite at q=0, and therefore the Fourier transform of thisdifference vanishes at large distances. Thus

M11srd = 4sJ8 + 2Jd

3 E−p

p dqx

2pE

−p

p dqy

2p

eirq

4 − 2 cosqx − 2 cosqy+ ¯ .

sA60d

Using the well known asymptotic behavior of the lastintegral,37 we find

M11sr − r8d = M11s0d −4sJ8 + 2Jd

2pflnur − r8u + s d + ¯ g,

sA61d

where C1 can be related to the Euler-Mascheroni constantg<0.5772 asC1=g+lns2Î2d. Note that M11srd formallylogarithmically diverges becauseM11sqd is proportional toq−2 at small momenta. The difference,M11srd−M11s0d, how-ever, is finite. The overall infinite additive constant has asimple physical interpretation, just like for an ordinary two-dimensionalXY model, as will become clear in a moment.

The real-space expression forM12 can be easily calculateddirectly,

M12sr − r8d = − dJdrr8 + ¯ , sA62d

where ¯ denotes terms that decrease at least as fast asur−r8u−4. CombiningsA59d, sA61d, andsA62d, we obtain thefollowing expression, after separating off the terms withr=r8,

Z = olsrd

expF−p2

2 SM11s0dor

l2srd

+ orÞr8

M11sr − r8dlsrdlsr8d − dJor

eig·rl2srdDG .

sA63d

By applying the long-distance expansion ofM11srd in thesum containing terms withrÞr8, we find that the partitionfunction Z becomes

olsrd

expH−p2

2 FM11s0dSor

lsrdD2− dJo

r

eig·rl2srd

−4sJ8 + 2Jd

2po

rÞr8

lsrdlsr8dslnur − r8u + C1dGJ .

sA64d

We now return to the discussion of the formally divergentconstantM11s0d. This divergence is a reflection of the loga-rithmic dependence of a single vortex energy on the systemsize. If the number of the sitesN were finite, we would haveobtained a constant of order lnN for M11s0d instead of anoutright divergence. Although finite, this constant becomeslarge in the thermodynamic limitN→`, with the effect ofsuppressing all configurations of the integer-valued fieldlsrdexcept those that satisfy

or

lsrd = 0. sA65d

This is nothing but the charge neutrality condition in thepartition function of a 2D Coulomb plasma. Restricting our-selves only to such configurations, we obtain

Z = olsrd

expFp2

2 S4sJ8 + 2Jd2p

orÞr8

lsrdlsr8dslnur − r8u + C1d

+ dJor

eig·rl2srdDG . sA66d

A further simplification is achieved by noticing that

orÞr8

lsrdlsr8d = Sor

lsrdD2− o

r

l2srd.

Since the first term on the right-hand side vanishes by virtueof sA65d, the partition function equals


214511-32

Z = olsrd

expFp2

2 S4sJ8 + 2Jd2p

orÞr8

lsrdlsr8dlnur − r8u

−4C1sJ8 + 2Jd

2por

l2srd + dJor

eig·rl2srdDG . sA67d

This is the Coulomb gas representation of our model describ-ing chargeslsrd residing on black and red plaquettes andinteracting with long-ranged forces. ConditionsA65d, there-fore, is simply an expression of the overall neutrality of thesystem—only the configurations with the same number ofvortices and antivortices contribute to the partition function.

The Hamiltonian of the system can be finally recast as

Hvd = − psJ8 + J1 + J2d o

rÞr8

lsrdlsr8dlnur − r8u + Ecr o

rPRl2srd

+ Ecb o

rPBl2srd, sA68d

where the core energies of the vortices on redsRd and blacksBd plaquettes are expressed through the original parametersof the model as

Ecr = pC1sJ8 + J1 + J2d −

p2

4sJ1 − J2d, sA69d

Ecb = pC1sJ8 + J1 + J2d +

p2

4sJ1 − J2d. sA70d

The HamiltoniansA68d is of the form equivalent to Eq.s23dderived in the main text from the continuum formulation.Note that in the “low-temperature” limitJ@1, coefficientsJand J8=bVsJd coincide fsee Eq.sA21dg, and the agreementwith the continuum formulation is complete: the effectivestrength of the long-range interaction between vortices is

J = J + J1 + J2.

b. Quantum phase fluctuations

The derivation in 2+1 dimensions follows closely thesteps of the two-dimensional case considered in the previoussubsection. Denoting the imaginary time byt, the partitionfunction of the model is

Z =E pr

DfrstdexpF−E0

b

dtor

Lsr,tdG ,

where r is defined precisely like in the 2D case andr =sr ,td. The Lagrangian of our quantum model is definedas

− Lsr,td = −K0

2fr

2 + i f fr + Jfcoss=xfrd + coss=yfrdg

+ J12srdfcossfr+x+y − frd + cossfr+y − fr+xdg.

sA71d

The sign of the Berry phase is chosen to be positive for later

convenience; obviously the partition function is not affectedby the change. It is convenient to replace the integrals overcontinuous variablet by sums over discretetn sabbreviatedoften ast belowd separated by intervals of “length”e. Forbrevity, we will usefr+t to denotefsr ,t+ed.

The terms containing time derivatives can be transformedas follows:

expFE dtorS−

K0

2fr,t

2 + i f fr,tDG= expFo

r−

K0e

2Sfr+t − fr

eD2

+ or

i f efr+t − fr

e G .

sA72d

After completing the square, we have

exporF−

K0

2eSfr+t − fr − i

f

K0eD2

−e f2

2K0G .

Note that this expression can be formally replaced by a sum,

omsr d

exporF−

K0

2eS=tfr − i

f

K0e − 2pmsr dD2

−e f2

2K0G ,

since, clearly, only the termm=0 survives in the limit ofsmalle. The latter form is convenient because now the Pois-son identity

on=−`

`

expF−a

2n2 + infG =Î2p

ao

m=−`

`

expF−sf − 2pmd2

2aG

sA73d

can be applied. The result is

expFE dtorS−

K0

2fr,t

2 + i f fr,tDG~ o

utsr dexpHo

rF−

e

2K0utsr d2 + iutsr d

3Sfr+t − fr −i f e

K0D − e

f2

2K0GJ , sA74d

whereutsr d is an integer-valued field. Using the identity, thepartition functionZ can now be rewritten as


214511-33

ou

ow±

E pr

dfr expFior

fuisr d=ifr + w+sr dsfr+x+y − frd + w−sr dsfr+y − fr+xdgG3expF− o

rSua

2sr d2J8

+w+

2sr d + w−2sr d

2J128 srd+

esut − fd2

2K0DG . sA75d

The reader should bear in mind that throughout the Appen-dix, the Greek indices exclusively denote the spacelike com-ponentsx,y of a three-vector, while latin indices denote bothspacelike and timelike components, as the case may be. ThecoefficientsJ8 andJ128 srd are defined as

J8 = bVseJd, sA76d

J12srd = bVfJ12serdg. sA77d

We now proceed to transform the above expression by shift-ing the differences of the phasesfr onto the difference of thefields usr d and w±sr d by using discrete integration by partssA26d, just as was done in the two-dimensional case,

Z = ou

ow±

E pr

hdfrd„= ·usr d + fw+sr d − w+sr − x − ydg

+ fw−sr − xd − w−sr − ydg…j

3expF− orSua

2sr d2J8

+esut − fd2

2K0+

w+2sr d + w−

2sr d2J128 srd

DG ,

sA78d

where bold letters stand forthree-dimensionalvectors and

= ·u denotesthree-dimensionaldivergence.In the absence of the next-nearest coupling terms repre-

sented byw±, the d-function constraint in Eq.sA78d is re-

solved byu==3a, where the lattice curl is defined as

ui = ei jk= jLksr − ekd, e= sx,y,td.

In our case of ad-wave superconductor and finitew±, werewrite the constraint as

= · husr d + wfw+,w−gj = 0, sA79d

where w=swx,wy,0d and wxsyd are defined in the previoussubsection sectionsA28d. The solution is clearly

u = = 3 L − wfw+,w−g.

One can easily check that the choice ofL is not unique:for arbitrary scalar functionjsr d,

h= 3 fL + = jsr dgji − f= 3 Lgi = ei jk= j=kjsr − dkd

= ei jk= j=kjsr d = 0. sA80d

This gauge invariance implies that a gauge-fixing term mustbe introduced when replacing the sums over integer fielduby summation overL in order to avoid multiple counting.

The next step is most easily derived in the temporal gauge,

dsL3d ; pr

dL3sr d,0.

Afterwards, the results will be generalized to an arbitrarygauge-fixing condition. We proceed by rewriting the partitionfunction as

Z = oL

ow±

expF− orS s= 3 L − Ffw±gd'

2

2J8

+e

2K0fs= 3 Ld0 − fg2 +

w+2sr d + w−

2sr d2J128 srd

DGdL3sr d,0

sA81d

and apply the Poisson formula in order to obtain a theorydepending on continuous rather than integer-valued gaugefield L,

oLsr d

dL3,0f„L1sr d,L2sr d,L3sr d…

= prE dL1sr ddL2sr d o

l1sr d,l2sr d

3e2pior lasr dLasr df„L1sr d,L2sr d,0…

= prE

−`

`

dL1sr ddL2sr ddL3sr ddsL3dolsr d

d=·l

3e2pior l jsr dL jsr df„L1sr d,L2sr d,L3sr d…. sA82d

In performing the last step, we formally introducedl3L3 andan additional sum overl3sr d. The delta functiond(L3sr d)ensures that the exponent is not affected. All terms in thesum overl3 are therefore equal, and in order to avoid mul-tiple counting we need to impose a constraint, chosen as

= ·l =0, by assigning

l3 = − s=3d−1s=1l1 + =2l2d.

Note that the result of applying operators=3d−1 to an integerfield is another integer. The integer-valued fieldlsr d withzero divergence describes nonbacktracking closed loops onthe 2+1 space-time lattice. The fieldLsr d is now continuousand the temporal gauge condition


214511-34

QtfLg = pr

d„L3sr d…

can be replaced37 by an arbitrary gauge-fixing condition

QfLg; examples are= ·L=0 sLandau gauged or =' ·L'=0sradiation gauged,

oLsr d

dL3,0f„L1sr d,L2sr d,L3sr d…

=E−`

`

pr

dLsr dQfLsr dgolsr d

e2pior lsr d·Lsr dffLsr dgd=·l .

sA83d

This final identity allows us to rewrite our partition functionas

Z =E−`

`

pr

dLsr dQfLsr dgeotFfs= 3 Ld'g

3 olsr d

d=·leor h2pil sr d·Lsr d−se/2K0dfs= 3 Ld0 − fg2j, sA84d

whereFfs=3Ad'g has already been calculated insA45d andcan be used as is, provided that proper definitionssA77d ofJ8 andJ128 srd are replaced.

The remaining steps leading to the “Coulomb” gas repre-sentation of 3D vortex loops are conceptually similar to the2D case from the previous subsection. The algebra, however,is considerably more involved. We therefore will go slowlyand first wade through the derivation for the simple caseJ1=J2=0. This is just the ordinarys2+1dD XY model, ap-propriate for ours-wave pedagogical exercise from the maintext and the beginning of this appendixsA2d. Only theconfigurations w±sr d=0 contribute to the functional

Ffs=3Ad'g sA45d and we recover the usual13 anisotropic

3D XY model in a uniform magnetic fieldH = f t,

Z0 =E−`

`

pr

dLsr dQfLsr dgolsr d

d=·l

3 eor h2pil sr d·Lsr d−fs= 3 Ld'2 /2J8g−se/2K0dfs= 3 Ld0 − fg2j.

sA85d

To obtain the lattice loop gas representation, we need tointegrate out the gauge fieldL. The most transparent con-nection with the results for the 2D model is obtained by

using the radiation gauge=aLa=0,

or

s= 3 Ld02 = o

rf=xLysr − yd − =yLxsr − xdg2.

sA86d

Expanding the square and shifting the difference operatorsvia sA26d we have

or

s= 3 Ld02

= or

− Lysr − yd=x=xLysr − yd

− Lxsr − xd=y=yLxsr − xd − 2=yLysr − yd=xLxsr − xd

= or

− Lmsr d=n=nLmsr d − f=mLmsr dg2. sA87d

We introduce the following notation for the lattice ana-logues of wave vectorsqj appearing from discrete left-sidedor right-sided derivatives after Fourier transformation:

Qjsqd =eiqj − 1

i, sA88d

Qjsqd =1 − e−iqj

i, sA89d

Qjgsqd = Qjsq − gd =

− e−iqj − 1

i, sA90d

Qjgsqd = Qjsq − gd =

1 + e−iqj

i. sA91d

The arguments ofQj andQj is assumed to beq unless speci-fied otherwise.

We define the Fourier transformation as

fsr d =1

boq0

E dq'

s2pd2eiq·r fsqd,

where the sum over frequenciesq0 runs through q0=0,2p /b , . . . ,2p /e and the integrals overqx,qy extend from−p to p. Using the definition and properties that follow fromit,

fsqd = eoq0

E dq'

s2pd2e−iq·r fsqd, sA92d

f2sr d =1

beoq0

E dq'

s2pd2 fsqdfs− qd, sA93d

we obtain in the radiation gauge

or

s= 3 Ld02 = − o

rLmsr d=n=nLmsr d

= oq0

E dq'

bes2pd2Lms− qdQnQnLmsqd.

sA94d

Similarly,


214511-35

or

s= 3 Ld'2 = o

rf− L0sr d=m=mL0sr d − Lmsr d=0=0Lmsr d

− 2=0L0sr − td=aLasr − eadg, sA95d

where the last term vanishes due to our choice of the gauge.In the momentum space, we have

or

s= 3 Ld'2 = o

q0

E dq'

bes2pd2fL0s− qdL0sqdQmQm

+ Lms− qdLmsqdQ0Q0g. sA96d

The above definitions are generally valid but now we fo-cus again on the simple case ofJ1=J2=0. The partition func-tion Z0, given bysA85d, can be written as

Z0 = olsr d

d=·leor2pil sr d·Lfsr dE

−`

`

pr

dLsr d

3 df=aLageor h2pil sr d·Lsr d−fs= 3 Ld'2 /2J8g−se/2K0ds= 3 Ld0

2j.

sA97d

In arriving at the expression above, we performed a shift ofvariableL→L+L f, whereL f is a timeindependentvectorpotential corresponding to a constant and uniform magnetic

field f t. After Fourier transformation, the expression in theexponent can be written as

oq0

E dq'

bes2pd2F2pil sqd · Ls− qd −L0s− qdL0sqdQmQm

2J8

− Lms− qdLmsqdS 1

2J8Q0Q0 +

1

2K8QnQnDG . sA98d

Note that temporal and spacelike components are indepen-dent and can be integrated out separately. Integration overL0is trivial and yields

expF− 2p2J8oq0

E dqxdqy

bes2pd2

l0s− qdl0sqd

QmQm

G= expF− 2p2J8o

q0

E dqxdqy

bes2pd2

l0s− qdl0sqd4 − 2 cosqx − 2 cosqy

G .

sA99d

The remaining integral

E−`

`

pq

dLsqddfQaLagexpHorF2pil ns− qdLnsqd

− S 1

2J8Q0Q0 +

1

2K8QnQnDLms− qdLmsqdGJ

sA100d

can be computed by switching to 2D transverse and longitu-dinal components ofL, which we define on the lattice as

LLsqd = iQxLxsqd + QyLysqd

Q'

, sA101d

LTsqd = i− QyLxsqd + QxLysqd

Q'

, sA102d

where

Q' = ÎQaQa = ÎQxQx + QyQy.

TheLxsr d andLysr d can be expressed through the transverseand longitudinal components of the gauge filedL as

Lxsqd = − iQxLLsqd − QyLTsqd

Q'

, sA103d

Lysqd = − iQyLLsqd + QxLTsqd

Q'

. sA104d

Now observe that the Jacobian of the transformationsLx,Lyd→ sLL ,LTd is unity; 2D divergence ofL is propor-tional to LL as expected,

QaLa = − iLLQ'

and

las− qdLasqd = lTs− qdLTsqd + lLsqdLLsqd.

The integralsA10d can be written as

E−`

`

pq

dLTsqdexpHoq0

E dqxdqy

bes2pd2F2pil Ts− qdLTsqd

− S 1

2J8Q0Q0 +

1

2K8QaQaDLTs− qdLTsqdGJ ,

sA105d

which finally gives

Z0 ~ olsr d

d=·l exp3− p2oq0

E dqxdqy

ebs2pd212J8l0s− qdl0sqd

QaQa

+lTs− qdlTsqd

1

2J8Q0Q0 +

1

2K8QaQa24 . sA106d

This is the desired vortex loop gas representation of ourmodel. Rather than integrating out the gauge fieldL insA85d, one can partially perform the sum over the integerslsr d and arrive at yet anothersduald representation of parti-

tion function Z. The constraint= ·lsr d=0 in the partitionfunction Z0 sA85d is rewritten using auxiliary variablesasr das


214511-36

olsr d

d=·l,0 expF−or

l2sr d

2bVsb8d+ 2pio

rlsr d · Lsr dG

= olsr d

prE

0

2p

dasr dexpF−1

2bVsb8dorl2sr d

+ 2pior

lsr d · Lsr d + i o asr d= · lsr dG= o

lsr dprE

0

2p

dasr dexpF−1

2bVsb8dorl2sr d

+ ior

l is2pLisr d − =iasr ddG< p

rE

0

2p

dasr dexpFor ,i

b8cosf=iasr d − 2pLigG .

sA107d

The last equation describes a lattice superconductor. Notethat the partition functionsA85d does not contain quadratictermsl2sr d. Instead, one introduces a vortex core energy term

−1

2bVsb8dorl2sr d

by hand, and then, in the final lattice superconductor repre-sentation, a limitb→` is taken. Such a system is called afrozen superconductor. Alternatively, terms proportional tol2

can be kept finite. Such an “unfrozen lattice superconductor”is equivalent to an ensemble of vortex loops, namely anXYmodel augmented with an additional core energy that makesformation of vortices more difficult. Thus, applyingsA107dandsA85d, we obtain the partition functionZ0 describing the

lattice superconductor in a fieldf coupled to fluctuatinggauge fieldL,

prE

0

2p

dasr dE−`

`

dLsr dQfLsr dg

3expFor ,i

b8 cosf=iasr d − 2pLig

− orS 1

2J8s= 3 Ld'

2 +1

2K8fs= 3 Ld0 − fg2DG .

sA108d

The last step of our derivation is the standard37 Ginzburg-Landau expansion of the action, and for completeness wereproduce here the derivation following Kleinert.37

First, we introduce a complex fieldUr =expfiasr dg and

define covariant derivative operatorsDi ,Di according to

DxFsr d = Fsr + xde−2piLx − Fsr d, sA109d

DxFsr d = Fsr d − Fsr − xde2piLxsr−xd. sA110d

The following identity, which expresses the cosine insA108dthroughUr , can be proved easily:

or

cosf=xasr d − 2pLxg = or

Ur*S1 +

1

2DxDxDUr .

Thus, for a given fixed configuration of dual gauge fieldL insA108d, the sum over all configurations of angular variablesasr d is

ZXYfLg =E Dasr deor ,ib8 cosf=iasr d−2pLig

and can be transformed into

ZXYfLg =E Da expF3b8or

Ur*DUrG , sA111d

where D=s1+ 16DiDi

d. OperatorD is Hermitian, and there-

fore allows decompositionD=K2. Let us show thatZXY isproportional to

E Dfa,F,F*ge−s1/12b8dor uFr u2+1/2or ,r 8sFr*Krr 8Ur 8+Ur

*Krr 8Fr 8d,

sA112d

where the notation

E DfF,F*g ¯ = prE

−`

`

d ReFrd Im Fr¯

is used. To establish the equivalence, we group the terms inthe exponent as

−1

12b8SFr* − 6b8o

r8

Ur8* Kr8rDSFr − 6b8o

r8

Kr8rUr8D+ 3b8o

ror8r9

Ur9* Kr9rKrr 8Ur8. sA113d

After a shift of variables and integrating out the auxiliaryfieldsFr , the result coincides withsA111d up to an unimpor-tant proportionality factor. To obtain the equivalent descrip-tion in terms of fieldF, we now integrate over the angular

variablesar . To simplify notation, we definex1=KF /2 and

x2=KTF* /2, or more explicitly,

x1sr d =1

2or8

Krr 8Fr8,

x2sr d =1

2or8

Kr8rFr8* .

Integration over the phases insA111d now amounts to calcu-lation of disentangled integrals at separater ,

E dasr deeiasr dx2sr d+e−iasr dx1sr d = 2pI0fÎ4x1sr dx2sr dg,

where I0 denotes the modified Bessel function. Thus, omit-ting nonessential overall prefactors, the expression forZXYfLg assumes the following form:


214511-37

E DfF,F*ge−s1/12b8dor uFr u2+or ln I0fÎ4x1sr dx2sr dg.

Finally, after applying the Taylor expansion

ln I0sxd =x2

4−

x4

64+ ¯ ,

and retaining only the leading terms, we obtain

Z0 =E pr

dFsr ddF*sr ddLsr dQfLsr dg

3expH− orF 1

24sDiFd*sDiFd +

1

4S 1

3b8− 1DuFsr du2

+uFsr du4

64+

1

2J8s= 3 Ld'

2 +1

2K8fs= 3 Ld0 − fg2GJ .

sA114d

The partition functionsA114d is the desired dual represen-tation of our initial anisotropicXY model with the Berryphase, in the simple caseJ1=J2=0.

Armed with the experience from the above derivation, wenow return to the partition functionZ of the full-fledgedmodel sA84d containingJ1 andJ2. As we will demonstrate,the effect of the next-nearest-neighbor interactions will berather modest: to the leading order, only the term propor-tional to uFsr du2 will be modified. The prefactors of this termwill be modulated, having different values on the red and theblack plaquettes.

To arrive at the dual representation ofZ, we seek a gaugethat will ensure the decoupling of the temporal and spatialcomponents ofLsqd, similarly to the radiation gauge in thesimple example above. The bilinear terms inLm appearing inthe exponent can be classified as follows: first, there is a

contribution from the terms=3Ad02 which in an arbitrary

syet unknownd gauge has been already calculated insA87d,

−1

2K8fLms− qdLmsqdQnQn − QnQmLms− qdLnsqdg.

sA115d

To facilitate the bookkeeping of various terms resulting from

Ffs=3Ld'g, we use the following set of identities:

bxTs− qd = − ieiq0L0

Ts− qdSQy 0

0 QygD + ieiqyQ0Lx

Ts− qds3,

sA116d

bxsqd = − ieiq0SQy 0

0 QygDL0sqd − ie−iqyQ0s3Lxsqd.

sA117d

The expressions forbysqd can be obtained by replacingx↔yand the overall change of sign. Using these identities, thebilinear form

bxTs− qdSG11sqyd G12sqyd

G21sqyd G22sqydDbxsqd + sx ↔ yd

can be written as a sum of two groups: the diagonal terms are

L0Ts− qdSQyQyG11sqyd Qy

gQyG12sqyd

QygQy

gG21sqyd QygQy

gG22sqydDL0sqd + Ly

Ts− qd

3S Q0Q0G11sqyd − Q0Q0G12sqyd

− Q0Q0G21sqyd Q0Q0G22sqydDLysqd + sx ↔ yd.

sA118d

In addition, we obtain cross-terms that couple the spatial and

temporal components ofLi,

Q0L0Ts− qdfPsqydLysqd + PsqxdLxsqdg + c.c.,

sA119d

where c.c. denotes complex conjugation and matricesPsqadare defined as

Psqad = S− QaG11sqad QaG12sqad

QagG21sqad − Qa

gG22sqadD .

Note that the terms diagonal inL0 are exactly what we en-countered insA51d when we considered the 2D example.The off-diagonal terms can be eliminated altogether bychoosing a gauge defined by the following relation betweenLxsqd andLysqd:

PsqydLysqd + PsqxdLxsqd = 0. sA120d

The matrix equation can be resolved byLysqd=GsqdLxsqd,where matrixG is defined asGsqd=−P−1sqydPsqxd. The spa-tial part of the action in momentum space becomes

oq0

E dqxdqy

bes2pd2hipfl xTs− qd + l y

Ts− qdGsqdgLxsqd

− Lxs− qdXsqdLxsqdj, sA121d

where the 232 matrix X is defined as

1

4K8FSQ'

2 0

0 sQ'g d2D + GTs− qdSQ'

2 0

0 sQ'g d2DGsqdG

+1

4FS G11sqxd − G12sqxd

− G21sqxd G22sqxdD

+ GTs− qdS G11sqxd − G12sqxd− G21sqxd G22sqxd

DGsqdG . sA122d

Note that we omitted the second term insA115d as it is onlyof the orderQ'

6 and can be safely neglected for extractingthe long-distance behavior. After integrating outLx, we ob-tain


214511-38

expF−p2

4 oq0

E dqxdqy

bes2pd2fl xTs− qd + l y

Ts− qdGsqdg

3Xfl xsqd + GTs− qdl ysqdgG , sA123d

where X=X−1. Expanding this expression in the region ofsmall momenta as insA55d and retaining only the leading-order terms, as in the two-dimensional example, we find

− p2oq0

E dqxdqy

bes2pd2

1

1

2Jq0

2 +1

2K8q'2

3fqylxs− qd − qxlys− qdgfqylxsqd − qxlysqdg

q'2 .

sA124d

Observe that there are no cross-terms that couple modes atwave vectorsq and q−g to the order ofq0 and q−2. Thespatial part of the action after integrating out the gauge fieldsis equivalent to the resultsA106d obtained in the frameworkof simple modelZ0 where J8 is replaced by the effective

coupling constantJ=J8+J1+J2.CombiningsA124d andsA68d, the final form of the action

Z in terms of closed vortex loopslsr d can be written as

Z = olsr d

d=·l exp3orf2pi lsr d ·A fsr d − Ec8srdl0

2sr dg

− p2oq0

E dqxdqy

bes2pd212Jl0s− qdl0sqd

q'2

+1

1

2Jq0

2 +1

2K8q'2

lTsqdlTs− qd24 , sA125d

whereE8csrd= ±p2sJ1−J2d /4 depending on whetherr cor-responds to a black or a red plaquette.

We had intentionally used momentum representation forthe last two terms in the exponent. It is important to recog-nize that these terms are precisely what one would have ob-tained for the usual 3DXY model with no next-nearest-neighbor interaction and the effective nearest-neighbor

coupling constant equal toJ=J8+J1+J2. Thus, we may in-troduce a dual gauge fieldAsr d and present the partitionfunction as

Z = olsr dE

−`

`

pr

dAsr dQfAsr dgd=·l

3 expForS2pi lsr d · fAsr d + A fsr dg − Ec8srdl2sr d

−s= 3 Ad'

2

2J−

s= 3 Ad02

2K8DG . sA126d

By shifting A →A −A f back, we obtain

Z =E−`

`

pr

dAsr dQfAsr dgexpForS2pi lsr d ·Asr d

− Ec8srdl2sr d −s= 3 Ad'

2

2J−

fs= 3 Ad0 − fg2

2K8DG .

sA127d

Note that in the absence of the next-nearest-neighbor inter-actions, a similar expressionsA85d contained no quadraticterms l2sr d and the core energy −f1/bVsb8dgl2sr d was intro-duces by hand. In the present case, the differenceDEc8 of thecore energies on the black and red sites isfinite due to theanisotropic next-nearest-neighbor interactions. However, theaverage magnitude is still zero within our model, and herewe also need to introduce a constant average core energyterm f1/bVsb08dgl

2sr d. Thereby, we replaceEc8srd in sA127dby

1

bVfb8srdg→ 1

bVsb08d+ Ec8srd,

where the functionb8srd is implicitly defined by this equa-tion.

The remaining steps repeat the derivation leading fromsA107d–sA114d with the replacementb8→b8srd and resultin the Ginzburg-Landau expansion of our dual theory,

Z =E pr

dFsr ddF*sr ddLsr dQfLsr dg

3expF− orH 1

24sDiFd*sDiFd +

1

4S 1

3b8srd− 1DuFsr du2

+1

64uFsr du4 +

1

2J8s= 3 Ld'

2 +1

2K8fs= 3 Ld0 − fg2JG .

sA128d

This is our final result—the partition functionsA128d rep-resents the Ginzburg-Landau functional of a dual type-II su-perconductor appropriate for our model and subjected to a

constant dual magnetic fieldf.


214511-39

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6Z. Tešanović, Phys. Rev. B59, 6449s1999d.7I. F. Herbut, Phys. Rev. Lett.88, 047006s2002d; B. H. Seradjeh

and I. F. Herbut, Phys. Rev. B66, 184507s2002d.8M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani,

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10C. Howald, H. Eisaki, N. Kaneko, M. Greven, and A. Kapitulnik,Phys. Rev. B67, 014533s2003d; M. A. Steiner and A. Kapit-ulnik, e-print cond-mat/0406227.

11H. D. Chen, O. Vafek, A. Yazdani, and S. C. Zhang, Phys. Rev.Lett. 93, 187002s2004d.

12Z. Tešanović, Phys. Rev. Lett.93, 217004s2004d.13M. P. A. Fisher and D. H. Lee, Phys. Rev. B39, 2756s1989d.14H. D. Chen, S. Capponi, F. Alet, and S. C. Zhang, e-print cond-

mat/0312660; H. D. Chen, C. Wu, and S. C. Zhang, Phys. Rev.Lett. 92, 107002s2004d.

15E. Altman and A. Auerbach, Phys. Rev. B65, 104508s2002d.16While the Cooper and the real-space pairs correspond to two dis-

tinct limiting behaviors, there is still a sense in which they aretwo sides of the same coin: in both cases, charge modulationarises primarily from the particle-particle channel. Similarly, arecent e-print, P. W. Anderson, e-print cond-mat/0406038, alsoexamines the effect on the LDOS of the modulations in theparticle-particle channel. This puts these works in the categorydifferent from theories which focus on nonuniformities in theparticle-hole channel, for example H. C. Fu, J. C. Davis, and D.H. Lee, e-print cond-mat/0403001. Other prominent examples ofsuch theories are J. Zaanen and O. Gunnarson, Phys. Rev. B40,7391 s1989d; K. Machida, Physica C158, 192 s1989d; S. A.Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tran-quada, A. Kapitulnik, and C. Howald, Rev. Mod. Phys.75, 1201s2003d; S. Chakravarty, R. B. Laughlin, D. K. Morr, and C.Nayak, Phys. Rev. B63, 094503s2001d; Y. Zhang, E. Demler,and S. Sachdev,ibid. 66, 094501s2002d. A useful general studyof a nonuniform superconductor is found in D. Podolsky, E.Demler, K. Damle, and B. I. Halperin,ibid. 67, 094514s2003d.

17A Hofstadter-type problem is also at the root of various inhomo-geneous phases of quantum spin-dimer models; see L. Balents,L. Bartosch, A. Burkov, S. Sachdev, and K. Sengupta, e-printcond-mat/0408329 and references therein. For early discussionsof valence-bond solids in such quantum spin-dimer models, seeS. Sachdev and N. Read, Int. J. Mod. Phys. B5, 219s1991d and

M. Vojta and S. Sachdev, Phys. Rev. Lett.83, 3916s1999d.18O. Vafek, A. Melikyan, M. Franz, and Z. Tešanović, Phys. Rev. B

63, 134509s2001d, and references therein.19M. Sutherland, D. G. Hawthorn, R. W. Hill, F. Ronning, S.

Wakimoto, H. Zhang, C. Proust, E. Boaknin, C. Lupien, L.Taillefer, R. Liang, D. A. Bonn, W. N. Hardy, R. Gagnon, N. E.Hussey, T. Kimura, M. Nohara, and H. Takagi, e-print cond-mat/0301105.

20There is a large and growing field-theory literature on noncom-pact, parity-preserving QED3, which is the low-energy limit ofEq. s9d. Those interested will find T. Appelquist and L. C. R.Wijewardhana, e-print hep-ph/0403250, C. S. Fischer, R.Alkofer, T. Dahm, and P. Maris, e-print hep-ph/0407104, S. J.Hands, J. B. Kogut, L. Scorzato, and C. G. Strouthos, e-printhep-lat/0404013, and J. Alexandre, K. Farakos, and N. E.Mavromatos, e-print hep-ph/0407265 to provide a good over-view of frontier issues and a useful source of additional refer-ences.

21Representingam as a Us1d gauge field constitutes the “natural”gauge choice for this problem due to the following fundamentalfeature of a spin-singlet superconductor: consider a system madeup of twodistinctspecies of fermions, “up”sud and “down” sdd.The normal part has the form consisting entirely of bilinearsu†uandd†d so thatu andd flavors areseparatelyconserved. Notethat we are not at all concerned here with the symmetry withrespect to rotations betweenu and d—such symmetry may ormay not be present and the relevant symmetry of the normal partis just the global Uus1d3Uds1d. The inclusion of pairing termsof the form ud and d†u† sbut not uu or dd and their complexconjugatesd breaks this symmetry by violating the conservationlaw for the total fermion number, the sum of “up” and “down”flavors. There remains, however, an intactcontinuousUu−ds1dsymmetry associated with therelative fermion number, the dif-ference between “up” and “down” flavors. This continuous sym-metry signals the remaining conservation lawsspin conservationin spin-singlet superconductorsd. Actually, our latticed-wave su-perconductor models13d is a simple illustration of general fea-ture: consider screening the bond phase factor ofDi j by sitephase factors arising from the gauge transformed electron fieldscis: there are 2N bond phase factors expsiui j d versus onlyN sitephases exps−iwid. The most natural solution is to introduce 2Nsite phase factors exps−iwi↑d and exps−iwi↓d and attach them viagauge transformation toci↑ and ci↓. In this way, one can com-pletely eliminate thecenter of massexpsiui j d18 from Eq.s13d bya judicious choice of expsiwisd: expsiui j dexps−iwi↑− iw j↓d→exps−iaij d; expsiui j dexps−iwi↓− iw j↑d→expsiaij d, whereaij isa bond phase antisymmetric under↑↔↓ exchange. This is noth-ing but the tight-binding lattice version of the FT transformationand leads directly to the Us1d representation of the Berry gaugefield a↔aij . Here exps2iaij d is determined by expsiwi↑− iw j↑− iwi↓+ iw j↓d, where expsiwisd are found in terms ofcenter ofmass ui j ’s from expsiwi↑+ iw j↑+ iwi↓+ iw j↓d↔exps2iui j d. Notealso that the hopping term in the Hamiltonian acquires a gaugefield factor expsiwi↑s↓d− iw j↑s↓dd for spin ↑ s↓d fermions, withwi↑s↓d−w j↑s↓d being the lattice equivalents of the gauge fieldsvAsBd featured in the continuum FT transformation. Sinceaij isultimately given by theshalf ofd phase differencesswi↑−w j↑d−swi↓−w j↓d—expressed in terms ofui j ’s—its configurations arenoncompact, i.e., monopole-free, by construction. Note thatthese arguments do not generally apply to superconductors


214511-40

which are not spin-singlet. For example, if we have a singleflavor of spinless fermionsf, the normal part made up off†fbilinears has only a single Us1d symmetry—pairing terms of theform f†f† and f f sin the odd angular momentum channeld breakthis Us1d symmetry down todiscreteZ2.

22In short, the gauge theorys1,9d is noncompact by construction.Recently, there has been much interest in quantum spin systemswhere the underlying effective gauge theory is compact but itsmonopolesinstantond configurations are dynamically irrelevantat a critical point or in a critical phase. In such cases, one isagain back to a noncompact QED3 with massless bosons or fer-mions; see M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N.Nagaosa, and X.-G. Wen, e-print cond-mat/0404751 and refer-ences therein.

23There are two basic conditions in play. The first is that the startingfermions of the QED3 theory are electrons which, althoughstrongly renormalized by Mott-Hubbard correlations, still carryboth spin and charge, i.e., the true spin-charge separationsforexample, into RVB-style spinons and holonsd is absent. The sec-ond condition is that these fermions are paired by ad-wave,spin-singlet, anomaloussi.e., particle-particled gap functionwhose amplitude is robust. The topological phase fluctuations ofthis gap function are consequently ordinaryhc/2e vortices/antivortices, carrying standard charge 2e supercurrentsor super-flowd. The Nernst effect experiments could be interpreted as theliteral confirmation of the second condition; below the Nernstdome in Fig. 1, the first condition—the spin-charge locking intosrenormalizedd electrons—is then an almost certain implication.

24This chirally symmetric “algebraic Fermi liquid” phase has un-usual thermodynamic and transport properties; see O. Vafek andZ. Tešanović, Phys. Rev. Lett.91, 237001s2003d. For example,its specific heat,T2, just as in a neighboring superconductingstatessee Fig. 1d. This translates into heat transport similar tothat of a nodal d-wave superconductor, i.e., the criticalpseudogap state is a “thermal metal.” On the other hand, sincevortex-antivortex pairs are unbound, the same state is also a“charge insulator,” as emphasized in Ref. 5 and detailed in thismanuscript. This implies breakdown of the Wiedemann-Franzlaw in the pseudogap state. TheT2 specific heat and heat trans-port are due to spin excitations carried by nodal BdG fermions.Remarkably, the pseudogap is also a “spin dielectric” in thesense that Pauli spin susceptibility vanishes asx,q2, in contrastto x,q in a superconductor. Consequently, the Wilson ratio alsovanishes as temperature goes to zero.

25In general, the Hamiltonians14d will also contain terms which donot involve the standardXY phase differences of bond phasesbut are instead due toDi j itself being a “hopping term” in ad-wave superconductorsunlike the case of a simples-wave su-perconductord. The leading such term isK cossu12−u23+u34

−u41d around a plaquette of the CuO2 lattice. Such terms aredown by a factor,D2/ t*2 relative to theXY terms kept in Eq.s14d and one can neglect them in cuprates, whereD remainssignificantly smaller thant* for most of the underdoped regime,judged by the ratio ofvD to vF.19 However, such terms mightbecome important, along with many other longer-range termsexcluded from Eqs.s14d and s36d, in the calculation of vortexcore energies later in the text sinceK is not necessarily smallerthanJ1−J2.

26D. R. Nelson, inPhase Transitions and Critical Phenomena, ed-ited by C. Domb and J. L. LebowitzsAcademic Press, London,

1983d, Vol. 7, p. 1.27J. V. Jose, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys.

Rev. B 16, 1217s1977d.28The vortex core massM can be computed in a specific micro-

scopic model, say ad-wave BCS model of vortex quantum tun-neling fA. Mizel sunpublisheddg; also, see J. H. Han, J. S. Kim,M. J. Kim, and P. Ao, e-print cond-mat/0407156 and referencestherein. Similarly, one can also envision computing the masswith a Gutzwiller correlated BCS wave function.

29Ch. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile, and Ø.Fischer, Phys. Rev. Lett.80, 3606s1998d; E. W. Hudson, S. H.Pan, A. K. Gupta, K.-W. Ng, and J. C. Davis, Science285, 88s1999d; S. H. Pan, E. W. Hudson, A. K. Gupta, K.-W. Ng, H.Eisaki, S. Uchida, and J. C. Davis, Phys. Rev. Lett.85, 1536s2000d.

30M. Franz and Z. Tešanović, Phys. Rev. B63, 064516s2001d.31H. Tsuchiura, M. Ogata, Y. Tanaka, and S. Kashiwaya, Phys. Rev.

B 68, 012509s2003d and references therein.32Q.-H. Wang, J. H. Han, and D.-H. Lee, Phys. Rev. Lett.87,

167004s2001d.33P. A. Lee and X.-G. Wen, Phys. Rev. B63, 224517s2001d.34L. B. Ioffe and A. J. Millis, Phys. Rev. B66, 094513s2002d.35Viewing cuprates as entirely analogous to superfluid4He, how-

ever, should be studiously avoided as emphasized elsewhere inthis paper—the essential role of gapless fermions, the Galileaninvariance being broken by the CuO2 lattice, and the conversionof the Goldstone mode to gapped plasmon in superconductorsare but a few examples of the calamities that will be visited uponthose who take the similarities too far.

36In the limit of a superconductor with very tight vortex cores, thesmall number of “normal” excitations within the core impliesboth small mass and small viscosity. An important point here isthat, in such ans-wave superconductor, those few vortex corestates have a large gap and rapidly adjust to even rather fastvortex motion. Consequently, such vortex motion is decidedlyquantum, characterized by small mass of order of only a fewelectron masses but effectively vanishing core viscosity. We as-sume the same is true in HTS, where vortex core states areknown to be significantly gapped from STM experiments. Ofcourse, the presence of gapless nodal quasiparticles far awayfrom the cores will generate dissipation during vortex motion,but that effect is treated separately in the main text. Since thereis no clean way of separating gapped core states from gaplessand nearly gapless states just “outside” the core, in transportcalculations a small empirical Bardeen-Stephen core dissipationshould be included for better quantitative accuracy.

37H. Kleinert,Gauge Fields in Condensed MattersWorld Scientific,Singapore, 1989d.

38A. A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 s1957d; fSov.Phys. JETP5, 1174s1957dg.

39Since we are discussing ans-wave case here, the pseudogapD inthe fermionic action would have to be changed from adx2−y2- toan s-wave form.

40J. M. Singer, M. H. Pedersen, T. Schneider, H. Beck, and H.-G.Matuttis, Phys. Rev. B54, 1286s1996d, and references therein.

41Although dDi j ↔dni +dnj remains a useful approximation andwill be used occasionally in this paper.

42A more explicit argumentation for thedBdsr d↔dDi j correspon-dence goes as follows: Note thatdDi j as defined in Eq.s33d is

nothing but the Hubbard-Stratonovich field decoupling theui j2


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term in the action. By the reasoning of the previous “pedagogi-cal” subsection, this immediately makesdDi j equivalent to themodulation of the dual inductionBd. But why should this modu-lation be translated into the modulation of the pairing gap am-plitude Di j as our notation seems to imply? Consider the spatialregion whereBd is largerssmallerd than its average. This regionattractssrepelsd dual vortices, i.e., vortices in the dual fieldF.Consequently, the amplitude ofF is reducedsenhancedd in thisregion, which translates into thesantidvortices inui j itself stay-ing away fromsbeing drawn tod the same region. This, by theanalyticity of the complex gap functionDi j near ansantidvortexposition, finally implies that its amplitude must be largerssmallerd than its average in the regions whereBd is largerssmallerd than its average. Consequently,dBdsr d↔dDi j to theleading order, wheredDi j indeed assumes the meaning of themodulation in thed-wave pairing amplitude, up to an overallfactor which can only be determined from a fully microscopictheory. In the present paper, this overall factor should be treatedas an adjustable parameter. Finally, note that the above reason-ing does not imply thatuDi j u is the operator canonically conju-gate to ui j . Rather, it simply relates the modulation in the

ground-state expectation value of such ansunknownd operator todDi j .

43N. R. Cooper, S. Komineas, and N. Read, e-print cond-mat/0404112.

44M. Franz, I. Affleck, and M. H. S. Amin, Phys. Rev. Lett.79,1555 s1997d.

45Again, we should remind the reader that this dual Lagrangiancontains the contribution from the charge channel but ignoresthe spin. The coupling to the spin channel, associated with nodalfermions, enters through the Berry gauge fieldam and its contri-bution toLd is important near a critical point but not otherwise.In the parlance of effective field theory,Ld describes the “high-energy” physics of a charge sector relative to the “low-energy”physics of spin. Of course, the effect ofa remainsessentialforthe low-energy fermiology throughout the pseudogap state.

46E. Brown, Phys. Rev.133, A1038 s1964d.47J. Zak, Phys. Rev.134, A1602 s1964d; 134, A1607 s1964d.48J. W. Negele and H. Orland,Quantum Many-Particle Systems

sAddison-Wesley, New York, 1988d.49Only five of the six parameters are independent:dDi j are related

to dual fluxesdBd, whose average over a unit cell is zero.


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-wave superconductor: Application to the Cooper-pair charge-density wave in underdoped cuprates

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Transcript of -wave superconductor: Application to the Cooper-pair charge-density wave in underdoped cuprates