Duality and the vibrational modes of a Cooper-pair Wigner crystal

T. Pereg-Barnea1 and M. Franz2

1Department of Physics, The University of Texas at Austin, Austin, Texas 78712-1081, USA2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

�Received 27 April 2006; revised manuscript received 5 June 2006; published 28 July 2006�

When quantum fluctuations in the phase of the superconducting order parameter destroy the off-diagonallong-range order, duality arguments predict the formation of a Cooper-pair crystal. This effect is thought to beresponsible for the static checkerboard patterns observed recently in various underdoped cuprate superconduct-ors by means of scanning tunneling spectroscopy. Breaking of the translational symmetry in such a Cooper-pairWigner crystal may, under certain conditions, lead to the emergence of low-lying transverse vibrational modeswhich could then contribute to thermodynamic and transport properties at low temperatures. We investigatethese vibrational modes using a continuum version of the standard vortex-boson duality, calculate the speed ofsound in the Cooper-pair Wigner crystal, and deduce the associated specific heat and thermal conductivity. Wethen suggest that these modes could be responsible for the mysterious bosonic contribution to the thermalconductivity recently observed in strongly underdoped ultraclean single crystals of YBa2Cu3O6+x tuned acrossthe superconductor-insulator transition.

DOI: 10.1103/PhysRevB.74.014518 PACS number�s�: 74.72.�h, 74.20.Mn, 74.25.Fy, 74.25.Dw

I. INTRODUCTION

Ground states of superfluids and superconductors arecharacterized by a sharply defined macroscopic phase and anuncertain total number of particles. The mathematical ex-pression of this phenomenon is the number-phase duality: thephase � of the superconducting �or superfluid� order param-eter is an operator that is quantum mechanically conjugate to

the particle number operator N, leading to the uncertaintyrelation �N ���1. A similar duality exists locally. Con-sider, for simplicity, a lattice model of a superconductor �or asuperfluid� with the number and phase operators on site idenoted by ni and �i, respectively. These operators satisfy�ni , � j�= i�ij which yields the local version of the above un-certainty relation,1

�ni �� j � �ij . �1�

The uncertainty relation �1� has an interesting implication fora state of matter obtained by phase disordering a supercon-ductor or a superfluid. �By phase disordering we mean de-struction of the off-diagonal long-range order by phase fluc-tuations in such a way that the amplitude of the orderparameter remains finite and large.� In such a phase-disordered state the local phase is maximally uncertain,which, according to �1�, allows the particle number to belocally sharp. Cooper pairs �or bosons� may thus minimizetheir interaction energy by forming a crystal.

A precise mathematical description of this phenomenon isgiven in terms of vortex-boson duality2–4 which maps thesystem of Cooper pairs �bosons� near a superfluid–Mott-insulator transition onto a fictitious dual superconductor inan applied external magnetic field. The Cooper-pair �boson�crystal emerges as the Abrikosov vortex lattice of the dualsuperconductor.

Over the past several years it has become increasinglyclear that the theoretical ideas summarized above may havefound a concrete physical realization in the underdoped re-gime of high-Tc cuprate superconductors. According to one

school of thought the ubiquitous pseudogap phenomenon5

can be thought of as a manifestation of phase-disordered su-perconductivity in a d-wave channel.6–10 This view is sup-ported by a number of experiments11–14 which, in essence,probe the phase fluctuations directly. Given this evidence andthe dual relationship between the number and phase one iscompelled to ask whether the expected pair crystallizationcan be observed in underdoped cuprates. The answer to thisquestion appears to be a tentative “yes” and we shall elabo-rate on this point below.

The experimental technique of choice to probe forCooper-pair crystallization is scanning tunneling microscopy�STM�. Recent advances in STM provided a number ofatomic resolution measurements of the local density of states�LDOS� in a host of cuprate materials. These reveal an intri-cate interplay between quasiparticles, vortices, order and dis-order under the umbrella of high-Tc superconductivity.15–20

The STM scans have been performed on large areas offreshly cleaved samples of Bi2Sr2CaCu2O8+� �BiSCCO� andCa2−xNaxCuO2Cl2 �Na-CCOC� cuprate superconductors.Real-space maps of LDOS show clear modulations whichcan be systematically studied in Fourier space. Results of thevarious measurements can be summarized as follows. Super-conducting samples exhibit energy dispersing features at en-ergies that are small compared to the maximum supercon-ducting gap. When the samples are underdoped toward thepseudogap phase new nondispersive features appear. Thesefeatures have periodicity close to four lattice constants butare not always commensurate with the underlying ionic lat-tice. At low energies they coexist with the dispersingfeatures18 and are present up to energies of the order of thegap maximum, where the dispersing features no longer ap-pear.

The appearance of dispersive features in the supercon-ducting phase can be understood in the context of impurityscattering of the quasiparticles.21,22 The peaklike structuresin the Fourier-transformed STM maps represent the momen-tum transfer in these scattering processes. The latter are sen-sitive to the quasiparticle dispersion and the coherence

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factors23 and thus allow mapping of the underlying Fermisurface, the gap function, and reflect on the nature of theordered state. The dispersive features also appear in thepseudogap phase,18 indicating that the nodal quasiparticlessurvive the transition. Furthermore, the similarity betweenthe observed peaks in the superconducting and pseudogapphases suggests that the anomalous �off-diagonal� order pa-rameter is responsible for the pseudogap phenomenon.23 Theexistence of a residual superconducting order parameter inthe insulating phase is consistent with the picture of phasedisordered superconductivity.

The nondispersive features that are seen in underdopedsamples indicate the formation of additional charge orderclose to and in the pseudogap phase. A crucial characteristicof these modulations is that they leave the low-energy phys-ics intact. This is manifested by the coexistence of the staticcharge modulations with the low-energy dispersing featuresand by the V-shaped gap in the density of states. It is there-fore difficult to imagine that an ordinary charge density wavein the particle-hole channel is responsible for the nondisper-sive modulations. A conventional charge density wave wouldgap out the nodal quasiparticles and would therefore changethe low-energy properties of the system, in contrast to vari-ous experimental results.24,25

An interpretation of the static modulations in terms of thepair Wigner crystal �PWC� was first suggested by Chen etal.26 who also proposed a test designed to distinguish be-tween PWC and ordinary charge density waves. A detailedtheory of the PWC in the context of the phase fluctuationscenario was developed by Tešanović,27 who applied the du-ality map to the case of a d-wave superconductor and dem-onstrated that scrambling of the order parameter phase byvortex-antivortex fluctuations indeed leads to Wigner crys-tallization of Cooper pairs. It was further argued that theobserved patterns are consistent with a detailed account ofthe phase-fluctuating d-wave order parameter on the latticewith the relevant amount of charge carriers in the Cu-Oplanes.28 Using different variations on this theme,Anderson29 and Balents et al.30 arrived at similar conclu-sions.

In this article we propose and investigate an additionalmanifestation of the existence of a PWC. Cooper-pair crys-tallization should be accompanied by emergence of trans-verse vibrational modes, absent in the superfluid. These canbe though of as transverse “magnetophonons” of the dualAbrikosov lattice. Both the superfluid and PWC phases sup-port longitudinal modes �phase mode or second sound in theformer, longitudinal phonon in the latter�. For charged sys-tems these are gapped due to the long-range nature of theCoulomb interaction and thus cannot be excited at low tem-peratures. Transverse modes, on the other hand, correspondto shear deformations of the PWC and thus have the charac-ter of gapless acoustic modes. This is strictly true if one canignore the coupling of the PWC to the underlying ionic lat-tice. If such a coupling is present, then the new modes ac-quire a gap proportional to the strength of this coupling. Attemperatures smaller than the gap scale such vibrationalmodes would not contribute to thermodynamics or transport.We shall assume in the following that this coupling to thelattice is negligible although it is not a priori clear why this

should be so. The key hint comes from the fact that at least insome cases the PWC has been reported to be incommensu-rate with the underlying ionic lattice.17,19 This points to veryweak coupling, for if the coupling to the lattice were strong,PWC would always be commensurate. We shall return to thisimportant point in Sec. VIII.

Using the formalism of the vortex-boson duality we cal-culate the normal modes of the pair Wigner crystal and notethat such modes are accessible through the measurement ofheat transport. Indeed, Doiron-Leyraud et al.31 have reportedthe appearance of a new bosonic mode, with a T3 contribu-tion to the thermal conductivity in YBCO. The onset coin-cides with the transition to the pseudogap state at the criticaldoping xc. Our calculations below show that this additionalthermal conductivity may be a result of the PWC lattice vi-brations.

This article is organized as follows. In the next section webriefly review a simplified version of the duality transforma-tion that maps a phase-fluctuating superconductor onto the“dual” superconductor in an applied magnetic field. In Sec.III we show that vortices in the dual model correspond toCooper pairs and derive the interaction potential betweenthem. In Sec. IV we consider an Abrikosov vortex lattice inthe dual superconductor and calculate its normal modes. InSec. V we evaluate the inertial mass of Cooper pairs �themass that determines their kinetic energy� which is needed tocomplete the description of the normal modes. In the finalsections we use the foregoing results to estimate the contri-bution of the PWC vibrational mode to the specific heat andthe thermal conductivity, summarize our results and discusstheir implications for the physics of cuprates.

II. DUALITY AND WIGNER CRYSTALLIZATION OFCOOPER PAIRS

The route to a detailed understanding of the charge modu-lations in a pair Wigner crystal goes through a duality trans-formation. The need for the dual description has to do withthe fact that Cooper pairs are highly nonlocal objects, mostnaturally described in momentum space when they form acoherent superconducting state. The duality transformationtrades a description of a superconductor with fluctuatingphase for a model in which the vortex field plays the primaryrole. The full theory, relevant to cuprates, must take intoaccount the dynamics of vortices at the lattice scale in thepresence of both Cooper pairs and nodal quasiparticles. Itsdetailed implementation is quite complicated and is given inRefs. 27 and 28. Much of the complication arises preciselybecause of the need to describe phenomena on relativelyshort length scales, of the order of the period of the PWC,which is typically close to four lattice spacings. Since weshall be concerned primarily with the low-energy, long-wavelength properties of the PWC, we will contend our-selves with a simpler continuum version of the dualitytransformation32 to which we input the correct lattice struc-ture by hand. As usual, the long-wavelength physics willdepend only on the symmetry and dimensionality of theproblem.

We thus consider an effective theory for superconductiv-ity in a single Cu-O plane defined by the partition function

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Z=�D�� ,�*�exp�−�0��d�d2xL /�� where � is a complex

scalar order parameter and the Lagrangian density is

L =1

2K���� −

2ie

cA���2

+ U��2� . �2�

The Greek index =0,1 ,2 labels the temporal and spatialcomponents of �2+1�-dimensional vectors, �= ��c ,�x ,�y�,and c is the speed of light in vacuum. The parameters K

= �K0 , K1 , K1� are related to the compressibility and phasestiffness, respectively. They could be calculated, in principle,from an underlying microscopic theory via the standard gra-dient expansion. In the present work we shall determinethese parameters by matching to the relevant experimental

quantities: K0 will be expressed through the Thomas-Fermi

screening length and K1 through the London penetrationdepth.

The electromagnetic vector potential A is explicitly dis-played in order to track the charge content of various fields.In a more general model one could allow A to fluctuate inspace and time and thus implement the long-range Coulombinteraction; however, we shall not pursue this here. U is apotential function that sets the value of the order parameter� in the superconducting state in the absence of fluctuations.

Let us now proceed by disordering the phase of the field�= �ei�. We fix its amplitude �=�0 at the minimum ofthe potential U��2� and retain the phase � as a dynamicalvariable,

L L�c

=1

2K��� −

2�

0A�2

, �3�

where K= ��K /c��02 and 0=hc /2e is the superconduct-

ing flux quantum. To explicitly display the relativistic invari-ance of the above theory it is expedient to rescale

A0 →�K1

K0A0 �4�

and measure time in natural units related to length as x0=cd where

cd =�K1

K0c �5�

is the speed of the phase mode in the superconductor. Thepartition function becomes Z=�D�� ,�*�exp�−�d3xL� with

L =1

2K��� −

2�

0A�2

�6�

and K=�K0K1.Next we consider fluctuations in the phase. These can be

decomposed into smooth fluctuations and singular �vortex�fluctuations

�� = ��s + ��v, �7�

� � ��s = 0. �8�

The smooth fluctuations are curl free and the singular fluc-tuations are related to the density of vortices through theircurl in the temporal direction,

�� � ���0 = 2��i

qi�„r − ri��… . �9�

Here ri�� are the locations of vortex centers and qi are therespective vortex charges �+1 for vortices and −1 for anti-vortices�.

In order to shift our point of view from the condensate tothe vortices we decouple the quadratic term by an auxiliaryfield, W, using the familiar Hubbard-Stratonovich transfor-mation. The resulting Lagrangian is

L =1

2KW

2 + iW���v −2�

0A� + iW���s� . �10�

We may replace the third term by −i�W�s �and a vanishingsurface term� and integrate over the smooth fluctuations �s.This results in the constraint �W=0. In order to enforcethis constraint we replace the field W by a�2+1�-dimensional curl, W= ���Ad�, and rewrite Eq. �10�as

L =1

2K�� � Ad�

2 + i�� � Ad� · ���v −2�

0A� . �11�

Integrating by parts in the last term and introducing the vor-tex three-current

jv =1

2�� � ��v, �12�

we obtain

L =1

2K�� � Ad�

2 + 2�iAd · jv −2�i

0�� � Ad� · A , �13�

where the centerdot represents the scalar product in 2+1dimensions. The vortex current jv is minimally coupled tothe gauge field Ad. This coupling implies that vortices act aselectric charges for the dual gauge field.

In order to complete our description of the system interms of vortex degrees of freedoms we introduce a vortexfield � that is related to the vortex current through

jv = i��*�� − ���*��� . �14�

We elaborate the theory by adding kinetic and potential en-ergy terms for the field � and write32,33

L =1

2�� − 2�iAd

��2 + V��� +1

2K�� � Ad�

2

−2�i

0A · �� � Ad� + LEM�A� , �15�

where we have added, for the sake of completeness, the elec-tromagnetic Maxwell term for A. The inclusion of the poten-tial V restores the physics of short-range interactions be-tween the vortex cores, which has been neglected when wemade the London approximation in Eq. �3�.

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We have now reached our goal of describing a phase-fluctuating superconductor in terms of a vortex field �coupled to a dual gauge field Ad. If we ignore the coupling toelectromagnetism, contained in the second line, the dual La-grangian formally describes a fictitious superconductorcoupled to a fluctuating gauge field. As we shall see shortlythe temporal component of the dual magnetic field, Bd=��Ad, is related to the charge density and, thus, the dualsuperconductor is generally subjected to nonzero magneticfield.

At the mean-field level this theory exhibits two distinctphases separated by a second-order transition. In a dual “nor-mal” phase ��=0, vortex fluctuations are bounded and formfinite loops in space-time. This is a phase-coherent supercon-ductor in the direct picture. In the dual “superconducting”phase the vortices condense, ���0, and vortex fluctuationsproliferate. This is the pseudogap phase in the direct picturewhere phase fluctuations destroy the long-range supercon-ducting order.

Beyond the mean field a third possibility exists in whichthe dual superconductor has a phase-incoherent condensate.In this case ���0 but the dual phase coherence is de-stroyed by quantum melting of the vortex lattice. This phaseis akin to the “vortex liquid” phases known from the studiesof vortex matter.4 A special case is a situation when most ofthe dual vortices remain in a crystal but some small fractionmelts and delocalizes through the sample. This is known tohappen in a thermally fluctuating lattice superconductor34

when the magnetic flux per unit cell of the lattice is close toa major fraction—e.g., � 1

2 + pq

�, with q� p. In the direct pic-ture this situation corresponds to the supersolid phase wherethe PWC coexists with superconductivity. Such coexistenceseems to occur in samples of Na-CCOC.20,25 Whether such adual supersolid phase occurs in the class of quantum modelsconsidered in this paper is a very interesting open questionwhich we shall not attempt to answer here.

III. INTERACTION BETWEEN COOPER PAIRS

The dual model describes a fictitious superconductor. Weshall assume that this is a type-II superconductor and there-fore vortices of the dual field can be present. Vortices in thedual model represent Cooper pairs in the direct picture. Inorder to see this we note that the dual flux is quantized inunits of 1: compare the term ��−2�iAd

�� in the dualmodel to ��−2�i 0

−1A�� in the direct model. Next let usobtain the relation between the real electromagnetic chargeand the dual magnetic field. The coupling 2�i 0

−1A · ���Ad� implies that the electromagnetic current is given byje /�c=2� 0

−1���Ad� or

je = 2e�� � Ad�. �16�

Specifically, the flux in the temporal direction is related tothe charge density, �=2e���Ad�0. This means that a vortexin � carries the electric charge

Qv =� ��r�d2r = 2e� �� � Ad�0d2r = 2e , �17�

where the integration is over a surface containing the vortex.

With nonzero density of Cooper pairs the dual supercon-ductor is subject to a dual magnetic field, Bd0= ���Ad�0. Ifwe assume that we are in the dual type-II limit, a dual Abri-kosov vortex lattice is formed. From now on we shall dropthe adjective “dual” whenever there is no potential for con-fusion. In the vortex lattice the magnetic field exhibits a pe-riodic modulation with one flux quantum per unit cell. In thedirect picture this corresponds to a charge density modula-tion with charge 2e per unit cell. This is the pair Wignercrystal. Figure 1 illustrates the spatial variation of the orderparameter � and the field Bd in such a dual Abrikosov lattice.

We are interested in the interaction between vorticeswhich we regard as point particles located at the phase sin-gularities associated with each vortex. We consider a staticconfiguration of vortices located at points ri and neglect fluc-tuations in both � and Ad. This corresponds to a dual mean-field approximation. We emphasize that this is a highly non-trivial mean-field theory since it describes the originalsuperconductor in the presence of strong quantum fluctua-tions. From Eq. �15� the energy of a collection of static vor-tices becomes

EMF =�c

2� d2r���− 2�iAd��2 + K0

−1�� � Ad�2� ,

�18�

where, for simplicity of notation, we regard Ad as a three-dimensional vector with z component always zero. We havealso passed back to conventional units reversing the scalingintroduced in connection with Eq. �4�.

The interaction energy is easiest to evaluate in the dualLondon approximation, where we assume a constant ampli-tude of the order parameter ��r�=�0, but arbitrary variationin its phase �. As in the case of ordinary superconductors theLondon approximation is valid when the intervortex distancea��d, the dual coherence length; i.e., the vortex cores do notoverlap. �d is difficult to estimate reliably since it depends onV in Eq. �15� which has been essentially added by hand toaccount for the short-distance behavior of �real� vortices. Weshall assume that �d is of the order of ionic lattice spacing

FIG. 1. �Color online� A schematic plot of the spatial variationof the order parameter � and the field Bd in the dual Abrikosovvortex lattice. �d and �d denote the dual coherence length and dualpenetration depth, respectively. The dashed line represents the mag-netic field bd�r� associated with a single isolated vortex; Bd in adense lattice is a superposition of such field profiles centered atindividual vortices.

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deep in the dual superconducting phase. Close to the transi-tion it diverges �as ��0

−1 in the mean-field theory� and thuswe expect the London approximation to break down in thislimit.

To find the minimum of the energy we regard EMF as afunctional of Ad and � and vary it with respect to the gaugefield

�EMF

�Ad= 0, �19�

with the result

2��02��� − 2�Ad� = K0

−1 � � Bd. �20�

Now we can rewrite the energy as

EMF =�c

2K0� d2r��d

2�� � Bd�2 + Bd2� , �21�

where

�d2 =

1

4�2�02K0

. �22�

�d is the dual penetration depth which in the context of theAbrikosov lattice characterizes the magnetic size of a vortex.In the direct picture it represents the size of the charge cloudassociated with a single Cooper pair.

If we apply the curl operation to Eq. �20�, we obtain thedual London equation

− �d2�2Bd + Bd = zn�r� , �23�

where

n�r� = �i

��r − ri� �24�

is the vortex density. The � functions appear due to the mul-tiple valuedness of the phase in the presence of vortices; �����r�=2�z�i��r−ri�. It is clear that Bd= zBd. The solu-tion of the London equation �23� can be written as

Bd�r� =� d2r�G�r − r��n�r�� , �25�

where G�r� is a Green’s function subject to

�1 − �d2�2�G�r� = ��r� . �26�

The above equation has a simple solution in the Fourierspace,

G�k� =1

1 + �d2k2 . �27�

Using a vector identity ���Bd�2=Bd · �����Bd�+� · �Bd���Bd� and discarding the vanishing surface termwe rewrite the vortex energy �21� as

EMF =�c

2K0� d2rBd · �− �d

2�2Bd + Bd� . �28�

With the help of Eqs. �23� and �25�, this can be recast as adensity-density interaction

EMF =1

2� d2r� d2r�n�r�V�r − r��n�r�� , �29�

where we have introduced the intervortex potential V�r�= ��c /K0�G�r�. In view of Eq. �24� one can also write this asan interaction between pointlike particles located at ri,

EMF =1

2�ij

V�ri − r j� . �30�

For future reference it is useful to give the intervortexpotential in terms of physically accessible quantities. Specifi-cally, we can trade the compressibility K0 /�c for theThomas-Fermi screening length �TF. In conventional termsthe latter is given by35

1

�TF2 = 4�e21

d

�n0

�, �31�

where n0 is the two-dimensional density of particles and wehave added the factor of 1 /d to account for the layer thick-ness. We thus have �c /K0=4�e2�TF

2 /d. This allows us towrite an explicit expression for the intervortex potential inthe Fourier space as

V�k� =er

2

k2 + �d−2 , �32�

with the effective charge

er2 =

4�e2

d��TF

�d�2

. �33�

Two remarks are in order. First, despite its appearanceV�r� is not electrostatic in origin; e2 appears simply becausewe chose to express the compressibility in terms of theThomas-Fermi screening length. The interaction is mediatedby phase fluctuations in the physical superconductor. Second,it is useful to consider the interaction in real space. Oneobtains V�r�= �er

2 /2��k0�r /�d� where k0�x� is the Hankelfunction of zeroth order. Of interest is its asymptotic behav-ior

V�r� =er

2

2����

2

r

�d�1/2

e−r/�d, r � �d,

ln� r

�d� + 0.12, r � �d.� �34�

Thus the interaction has a finite range, of the order of thedual penetration depth. Approaching the transition, �d di-verges. The long-distance behavior of V indicates that in thislimit the interaction becomes long ranged but, at the sametime, its strength vanishes since er→0.

IV. NORMAL MODES IN A COOPER-PAIR WIGNERCRYSTAL

The problem of normal modes in a lattice with long-rangeCoulomb interaction has been studied previously in the con-text of electronic Wigner crystals.36 In two dimensions, vi-brations of a square lattice with centrally symmetric interac-

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tions between electrons contain modes with imaginaryfrequencies, indicating that the lattice is, in general, unstable.In the dual picture this corresponds to the well-known factthat while the triangular Abrikosov vortex lattice is stable,the square lattice represents an energy maximum and is,therefore, unstable.37 It is also known that the square vortexlattice can be stabilized if the interactions possess fourfoldanisotropy.38 Such anisotropies can arise from the d-wavesymmetry of the order parameter39 or from band structureeffects associated with the underlying ionic lattice.40 In thefollowing we shall make an assumption that similar termswith fourfold anisotropy exist in our dual superconductor.These terms will stabilize the square vortex lattice but willnot affect our discussion of the long-wavelength vibrationalmodes.

We employ the standard formalism for lattice vibrations:we assume that vortices are located at points ri=Ri+uiwhere Ri denotes vectors of a Bravais lattice and ui are smalldisplacements. The interaction potential �30� is expanded tosecond order in the displacements,

V�Rij + uij� � V�Rij� + uij�Dij

��uij� , �35�

where Rij =Ri−R j, uij =ui−u j, and

Dij�� = � �2V�Rij + u�

�u��u� �u=0

= − er2� d2k

�2��2

k�k�

k2 + �d−2eik·Rij

�36�

is the dynamical matrix. Aside from a constant V0=�ijV�Rij�, the elastic energy can be written as

�E = 2er2�I1 − I2� , �37�

with

I1 = �i

ui�ui

��j

Dij��, �38�

I2 = �ij

ui�uj

�Dij��. �39�

Note that the intervortex potential V is defined every-where in space �not only on the lattice sites� and therefore kis not restricted to the first Brillouin zone. In order to keep allmomentum integration variables within the Brillouin zonewe replace k→q+Q and

� d2k → �Q�

BZd2q , �40�

where Q is a reciprocal lattice vector �eQ·R=1 for any latticevector R�.

We now analyze the terms I1 and I2 by Fourier transform-ing the variables ui,

ui = a2�BZ

d2k

�2��2eik·Riuk. �41�

The momentum integral is over the first Brillouin zone, and ais the vortex lattice constant which we include in order for ukto maintain the dimension of length. Combining Eqs.�36�–�41� we find

In = −� �dk��dk���dq�uk�uk�

� �Q

�q� + Q���q� + Q���q + Q�2 + �d

−2

� a4�ij�e−i�k+k��·Ri+q·�Ri−Rj�, n = 1,

e−ik·Ri−ik�·Rj+q·�Ri−Rj�, n = 2,� �42�

where �dk�=d2k / �2��2 and the integration extends over thefirst Brillouin zone. Performing the real-space sums leads to

�E = 2er2� �dk�uk

�u−k� Z���k� , �43�

with

Z���k� = �Q� �k� + Q���k� + Q��

�k + Q�2 + �d−2 −

Q�Q�

Q2 + �d−2� . �44�

The normal modes of the system are related to the eigen-values zi�k� of the matrix Z���k�. Explicitly mv�i

2=2er2zi�k�

where mv is the dual-vortex mass. We thus need to evaluatethe reciprocal lattice sums indicated in Eq. �44�. These sumsare slowly convergent and great care must be taken in evalu-ating them; specifically, we need to employ the Ewald sum-mation technique. The following derivation is similar to thatgiven by Fetter41 which was done in the context of a trian-gular Abrikosov vortex lattice.

In order to proceed analytically we split Z���k�=Z���1��k�

+Z���2��k� with

Z���1��k� =

k� k�

k2 + �d−2 �45�

and Z���2��k� containing the sum over all Q�0. Evaluation of

the latter is greatly simplified if we assume that �d−1 is much

smaller than the smallest reciprocal lattice vector, Q0�2� /a. Our estimate of �d below will justify this assump-tion. Thus,

Z���2��k� � �

Q�0� �k� + Q���k� + Q��

�k + Q�2 −Q�Q�

Q2 � . �46�

We note that all the dependence on �d is contained in Z���1�

��k� which is, in addition, independent of the lattice struc-ture. Z��

�2��k�, on the other hand, depends on the lattice struc-ture but is independent of �d.

The detailed calculation of Z���2��k� is given in the Appen-

dix. The result can be summarized as

Z���k� =k�k�

k2 + �d−2 +

�

2a2����k2 − 2k�k�� , �47�

where for the square lattice �=0.066. Equation �47� is validfor long wavelengths, k�2� /a.

The normal modes are readily found through the eigen-values of the matrix Z���k�,

�1�k� = vsk , �48�

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�2�k� = vsk� 2

�a2

1

k2 + �d−2 − 1, �49�

where

vs2 = �

er2

mv. �50�

The first, acoustic, mode is transverse and its sound velocityis vs. The second mode is longitudinal. At long wavelengths,k��d, the longitudinal mode is also acoustic with velocityvs��d /a��2/�. As the intervortex interaction becomes longranged ��d→�� the latter speed diverges and the mode be-comes gapped, which is expected on general grounds. Withthe estimated �d of about ten vortex lattice constants thelongitudinal mode is unimportant �as we shall see it is theinverse of the sound velocity that enters the expressions forthe specific heat and thermal conductivity�. Also, had weretained the long-range Coulomb interaction mediated by theelectromagnetic gauge field fluctuations, the longitudinalmode would be gapped for any value of �d. A schematic plotof the two modes is given in Fig. 2.

V. DUAL VORTEX MASS

In order to evaluate the sound velocity of the transverseacoustic mode it is essential to estimate the mass of the dualvortex. Naively this mass is simply the Cooper-pair mass2me. However, this is not the parameter mv that enters thesound velocity expression. The difference between the twooriginates from our scheme of calculating the normal modes.We have modeled the PWC as a system of point masses andsprings. This analogy is useful but not quite suitable for thecase of Cooper pairs that are delocalized over many latticesites. The finite size of the Cooper pairs causes significantoverlap between their wave functions, and this leads to asituation that is quite different from the familiar case of vi-brations of pointlike ions in a solid.

To make these considerations more concrete let us nowestimate the effective Cooper-pair size given by the dual pen-etration depth �d. According to the STM experiments thecharge modulation amounts to �=1% –5% of the totalcharge density.42 In a dual vortex lattice this corresponds to

the rms amplitude variation of the dual magnetic field �

� �Bd− Bd�2�1/2 / Bd, where Bd is the average field and theangular brackets denote the spatial average. The above rmsvariation depends on the ratio of �d to intervortex spacinga.43 Namely, a2 /�d

2��, which leads to the estimate

�d � �5 − 10�a . �51�

Near half filling we can estimate a��2a0�5 Å, where theionic lattice constant a0=3.8 Å in YBCO. Thus, even deep inthe dual superconducting phase, the charge and mass of theCooper pair are distributed over many lattice sites.

In the present case it is intuitively clear that only thefraction of the Cooper-pair mass associated with the pairdensity wave should contribute to the vibrational degrees offreedom. In particular, as �d→� the charge distribution be-comes homogeneous, the system is superfluid and cannotsupport any transverse modes. We thus seek a mass param-eter associated with the kinetic energy of a moving dual vor-tex. The appropriate parameter is the inertial mass of a dualvortex. In order to determine the latter we note that, if wediscard the coupling to electromagnetic field, the vortex dy-namics are given by a relativistic theory Eq. �15�. Corre-spondingly, the rest energy of a dual vortex is given by

Erest = mvcd2. �52�

Here, mv is the inertial mass of the dual vortex that we seekand cd is the dual speed of light—i.e., the phase velocity ofthe dual gauge field defined in Eq. �5�. We may thereforeestimate the energy of a vortex line, Erest, in the standardway44 and deduce its inertial mass through Eq. �52�.

We consider a single vortex located at the origin. Its en-ergy can be expressed by combining Eq. �28� with the dualLondon equation �23� and n�r�=��r� as

Erest =�c

2K0bd�0� . �53�

Henceforth we shall denote magnetic field associated with asingle isolated vortex by bd�r�. The energy of a vortex de-pends linearly on the magnetic field at its center. For a cir-cularly symmetric vortex the London equation can be solvedby the Hankel function

bd�r� =1

2��d2k0� r

�d� �

1

2��d2�ln��d

r� + 0.12� , �54�

where the last equality holds for r��d. The divergence asr→0 is unphysical and occurs due to our neglect of the orderparameter amplitude variation in the vortex core. A reliableestimate is obtained by evaluating

bd�0� � bd��d� �1

2��d2 ln��d� , �55�

where �d=�d /�d is the dual Ginzburg-Landau parameter. �dis assumed larger than unity �dual type-II regime�, but sinceit appears inside the logarithm, its exact value is unimpor-tant.

The dual vortex mass is thus given by

FIG. 2. A schematic plot of the acoustic transverse mode�dashed line� and the longitudinal mode �solid line�. The functionsare plotted with �d=10a where a is the vortex lattice constant and��k� is presented in units of vs.

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mv =�

4�c�d2K1

ln��d� =4e2

dc2� �

�d�2

ln��d� . �56�

We have used the London penetration depth,

�−2 =4�e2ns

mec2 �57�

with the superfluid density ns=2�02 /d to eliminate K1

=��02 /2mec. In the above estimate one should take the

mean-field value of �—i.e., the value it would have in theabsence of phase fluctuations. In YBCO, we thus take ��1000 Å, the value at optimum doping. Taking �d=10 andd=12 Å gives mv�1.1�10−5�� /�d�2�2me�, where me is theelectron mass.

A more instructive way of expressing mv is to estimate themean-field superfluid density close to half filling by ns�1/ �2a0

2d� and obtain

mv = 2me� a0

�d�2 ln��d�

2�. �58�

As expected, when Cooper pairs are localized and �d ap-proaches the lattice constant a0 the inertial mass of the dualvortex approaches that of the Cooper pair. When, on theother hand, �d�a0, the Cooper pair is delocalized over manylattice spacings and the dual vortex mass becomes small.

VI. INTERLAYER TUNNELING AND DUALMONOPOLES

A linear dispersion such as the one we found for the trans-verse mode, combined with Bose-Einstein distribution forphonons, leads to specific heat Cv�Td where d is the dimen-sionality of the system. The thermal conductivity, within thesimple Boltzmann approach, is �= 1

3Cvvs� where � denotesthe phonon mean free path. Assuming the latter is T indepen-dent, as is the case for phonons scattered by the sampleboundaries, we have ��Td. We thus arrive at a conclusionthat, in order to agree with experiment, the PWC phononsmust propagate in all three dimensions.

Our theory has thus far focused on the purely two-dimensional �2D� physics of the Cu-O layers. However, it isclear that vibrations of PWC in the adjacent layers will becoupled. There are two main sources of this coupling: �i� thepair tunneling between the layers represented by the inter-layer Josephson term and �ii� the Coulomb interaction be-tween the induced charge modulations. Inclusion of the in-terlayer coupling will cause the PWC phonons to propagatebetween the layers. To complete our calculation we mustdetermine the associated sound velocity in the z direction.

Ideally, we would like to extend our duality map to asystem of weakly coupled layers and repeat the calculationof the normal modes. Unfortunately, there is no straightfor-ward generalization of the vortex-boson duality to a systemin 3+1 dimensions; �2+1�-dimensional systems are specialin this respect. The reason for this can be seen most clearlyby returning back to Eq. �10�. In �3+1�D it is not possible toenforce the zero-divergence constraint on W by expressingit as a curl of a gauge field; the curl operation is only mean-

ingful in three dimensions. On a more fundamental level in�3+1�D vortices cannot be regarded as point particles;rather, they should be thought of as strings. Thus, rather thana dual superconductor, in �3+1�D the duality map produces astring theory. On physical grounds we still expect the PWCto form in a �3+1�D phase-disordered superconductor butthe underlying mathematical structure of the dual theory ap-pears to be more complicated and beyond the scope of thispaper.45,46

We thus continue describing the physics of the individuallayers by the �2+1�D duality and consider the effect of weakinterlayer coupling on the resulting quantum state. We focusfirst on the Josephson coupling, which mediates tunneling ofCooper pairs between layers. Formally we imagine general-izing our starting Lagrangian �2� into a Lawrence-Doniachmodel44,47 for a multilayer system by attaching a layer indexm to the scalar field � and adding a term

J0

2a02 �m − �m+12 �59�

to couple the layers. The cross terms describe tunneling ofphysical Cooper pairs between the layers. The amplitude forthis process, J0, is related to the c-axis London penetrationdepth �c by

J0 =a0

2

d

�2c2

16�e2�c2 . �60�

From the point of view of an individual layer removal ofa Cooper pair represents a monopole event. A tunneling eventoccurring at a particular instant of imaginary time adds orremoves a Cooper pair from the plane. In the dual descrip-tion, this event corresponds to the appearance or disappear-ance of a vortex; the magnetic flux lines associated with itoriginate or terminate at the same point, as illustrated in Fig.3�a�. A point in space-time which represents a drain �source�of a magnetic flux is known as a magnetic monopole �anti-monopole�. The vortex must reappear in the adjacent layer,and this corresponds to an antimonopole. In ordinary super-conductors vorticity is strictly conserved which can be re-garded as a consequence of the absence of monopoles in thereal world. In the dual superconductor vorticity is conservedonly if we consider a system with a fixed number of Cooperpairs. Once we introduce the Josephson tunneling, vorticityis conserved only globally �i.e., the total number of vorticesin all layers is constant� and we must permit monopole-antimonopole pairs to occur in the adjacent layers.

FIG. 3. �Color online� �a� A schematic illustration of the fluxlines in the presence of a monopole. �b� Josephson tunneling withlateral displacement of the dual vortex. The shaded regions repre-sent charge densities associated with a Cooper pair in the PWC.

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To model the interlayer tunneling we should thus regardthe dual gauge field Ad as compact and append to the dualLagrangian �15�, a term describing monopole-antimonopoleevents occurring in the adjacent layers. Since, ultimately, weonly need the mean field-description of the intervortex inter-action, we shall adopt a simpler and physically more intui-tive approach. In essence, all we need to complete our de-scription of PWC vibrations is an effective potential, akin toV�ri−r j�, that will describe the coupling between vortices inadjacent layers induced by the Josephson tunneling. Takingour clue from Eq. �30� we describe the system with manylayers by a Hamiltonian H=H0+HJ with

H0 =1

2�m�

r�

r��m

† �r��m† �r��V�r − r���m�r���m�r� ,

HJ = �m�

r�

r�J�r − r����m

† �r��m+1�r�� + H.c.� . �61�

The operator �m† �r� creates a vortex at point r in the mth

layer and �r=�d2r. H0 describes interactions between vorti-ces within a layer, and HJ generates the Josephson coupling;J is the amplitude for interlayer tunneling to be discussedshortly. For now we regard vortices as infinitely heavy �nokinetic energy in the plane�.

Our strategy will be to treat HJ as a small perturbation onthe eigenstates of H0. This is justified as long as J is verysmall. In the limit of infinite mass the unperturbed eigen-states are labeled by the positions �rim� of N individual vor-tices within each layer. The energy of this state is simplygiven by Eq. �30�. Clearly, there will be no correction to theenergy to first order in HJ. The leading contribution appearsin second order and describes a virtual hop of a Cooper pairfrom one layer to the next and then back. For simplicity weconsider a case of two layers, m=1,2, and evaluate thesecond-order correction to the energy from such processes:

�E�2� = ��=±1

�s�

s�

N,NHJN + �s,N − �s��2

E�N,N� − E�N + �s,N − �s��. �62�

In an obvious notation N ,N� denotes a state with N pairs ineach layer, located at �ri1� and �ri2�, while N+�s ,N−�s��represents a state with one pair added �removed� at s in layer1 and one pair removed �added� at s� in layer 2, for�=1�−1�.

The matrix element in the numerator is easily seen to bejust J�s−s��2 but the energy denominator requires somecareful thought. An obvious contribution to the energy dif-ference comes from the interaction between Cooper pairsexpressed through Eq. �30�. For �= +1 this is simply theinteraction energy of a vortex added at s in layer 1 and an-tivortex added at s� in layer 2,

�U = − �i

�V�s − ri1� − V�s� − ri2�� . �63�

For �=−1 the overall sign is reversed. However, this cannotbe the whole story since Eq. �63� does not account for thefact that the state with N vortices per layer corresponds to theabsolute minimum of the total energy of the unperturbed

system. At the level of Eq. �30� this requirement is imple-mented as a constraint. This is adequate as long as N is heldconstant. Once we allow for number fluctuations, however,we must consider the energy cost U0 of removing �adding� aCooper pair from �to� a layer. Ultimately, this cost is relatedto the electric charge neutrality: removing or adding a pairfrom or to a neutral layer costs Coulomb energy. Thus, weestimate U0 as the electrostatic energy of the charge distri-bution corresponding to an extra pair at s in layer 1 and amissing pair at s� in layer 2. The missing pair is modeledsimply as an effective positive charge distribution on theotherwise neutral background. We have

U0 =1

2��

R�

R�

��R���R��R − R�

, �64�

with R= �r ,z� a three-dimensional vector. � is the dielectricconstant which reflects the polarizability of the insulatingmedium between the Cu-O layers. Its value is around 10 inYBCO.48 Taking

��R� = �1�r − s���z� − �1�r − s����z − d� , �65�

where �1�r�= �2e�bd�r� is the planar charge density associ-ated with a single Cooper pair in PWC, we obtain

U0 =�2e�2

��

r�

r�bd�r�bd�r��� 1

r − r�−

1��r − r� − l�2 + d2� ,

�66�

with l=s−s�. For general values of l and d the above integralmust be evaluated numerically. However, it turns out that weshall only require its leading behavior in the limit d, l��d—i.e., the situation when the distance between the twocharge clouds is small compared to their lateral size given by�d. In this limit we can expand

U0 ��2e�2

�d�� d

2�d+

�

8

l2

�d2� . �67�

The energy correction �62� thus becomes

�E�2� = − �s�

s�� J2

U0 + �U+

J2

U0 − �U� , �68�

where we have suppressed various arguments in order todisplay the structure of the result. We notice that �E�2� de-pends on the positions �ri1� and �ri2� through �U. Upon in-tegration over s, s� this will produce interactions betweenvortices in different layers, as expected. The origin of thisinteraction lies in the fact that the energy of the virtual inter-mediate state depends on the position of the extra Cooperpair relative to the pairs already present in that layer. Sincethe positions of the former before and after the tunnelingevent are strongly correlated this induces interaction betweenCooper pairs in the adjacent layers.

We can make the form of the interlayer interaction moreexplicit in the limit �U�U0 by expanding

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�E�2� � − �s�

s�

2J2

U0�1 + ��U

U0�2

+ O��U

U0�4� . �69�

The first term is a constant, but the second term, when ex-panded with the help of Eq. �63�, contains the expression

Vz�ri1 − r j2� = 4�s�

s�V�ri1 − s�

J�s − s��2

U03�s − s��

V�s� − r j2� ,

�70�

which provides an explicit interaction potential between thepairs located in the adjacent layers. Although we derived Eq.�70� assuming just two layers, it is obvious that it generalizesto a multilayer system.

In order to complete our computation of Vz we must de-termine the form of the tunneling matrix element J�s−s��. Inthe direct picture, modeled by a Lawrence-Doniach-typeHamiltonian,47 Cooper pairs are assumed to tunnel “straightup” �or down�—i.e., from a point s in one layer to the points in the adjacent layer. A subtlety in the dual picture arisesfrom the fact that HJ describes tunneling of dual vortex coresand not Cooper pairs directly. We recall that even when theposition of the dual vortex is sharply localized at a point s,the associated Cooper-pair charge �and number� density isdelocalized over the length scale �d around s with the prob-ability density given by bd�r−s�. Thus, we can think of aCooper pair as being described by a wave function whoseenvelope varies as �bd�r−s�. As illustrated in Fig. 3�b� astraight-up tunneling of a Cooper pair in general may lead todual vortex core tunneling with nonzero lateral displacement.The associated amplitude to tunnel from point s to s� will begiven by the overlap of the two pair wave functions,��r�bd�s−r�bd�r−s��. This last integral is somewhat diffi-cult to evaluate because of the square root, but the resultingfunction is clearly close to bd�s−s��. In the following wethus use

J�r� � J0bd�r� , �71�

which turns out to be properly normalized since �rbd�r�=1.An important implication of the above result �71� is that

tunneling over lateral distances larger than �d is exponen-tially suppressed and we may indeed use the approximation�67� for U0 when evaluating Vz from Eq. �70�. In fact we willuse only the first term in �67� which gives the leading con-tribution of the form

Vz�r� =4J0

2er4

U03 ��d

4�s�

s�bd�r − s�bd�s − s��2bd�s��� ,

�72�

where we have used the relation V�r�=er2�d

2bd�r� in order toexpress everything in terms of bd. It is worth noting that inEq. �72� the prefactor

EJ =4J0

2er4

U03 �73�

characterizes the energy scale for the interlayer couplingwhile the expression in the square brackets represents a di-

mensionless function of spatial variable r whose range is setby �d. The interaction is repulsive.

We could now repeat the calculation of the normal modesfor the layered system. This calculation,49 however, islengthy and does not yield any new physical insights to theproblem at hand. All we need, in fact, is the result for thespeed of sound in the z direction. This can be readily esti-mated in analogy with Eq. �50� which states that up to anumerical constant the speed of sound squared equals theenergy scale of the interaction divided by mass. This ofcourse is a well-known result for ordinary systems of springsand masses; what Eq. �50� confirms is that this result remainsvalid even in the case of interactions with finite range. Wethus obtain

vz2 �

EJ

mv=

4J02er

4

U03mv

. �74�

The above is a perturbative result and remains valid onlyas long as J0 /U0�1. With the help of Eqs. �67� and �60� thiscondition becomes

1 �J0

U0= �a

d�2 �

16��2��d

�c�2

� 4.1 � 102��d

�c�2

, �75�

where �=e2 /�c�1/137 is the fine-structure constant. Thelatter appears because the Josephson coupling connects two“dual worlds” through the real world coupling. Since �c istypically very large in cuprates �104–105 Å, Eq. �75�should be well satisfied except very close to the transition.

We now briefly discuss the direct Coulomb interaction,which will turn out to be negligible in most cases. If weassume that only the interaction between neighboring layersis important, then the standard treatment leads to acousticdispersion for the transverse mode along the z direction withthe sound velocity �EC /mv where EC is of the order of theCoulomb potential at distance d,

EC =�2�e�2

�d. �76�

Here 2�e�2ea2 /�d2 is the fraction of Cooper-pair charge that

is modulated. As mentioned above ���1–5�% in the STMexperiments.

For the typical values of various parameters listed belowEq. �82� and �d=20 Šwe find EJ /EC�30; the Josephsonterm dominates. It is also important to note that EJ and ECexhibit very different scaling with �d. In particular Eq. �73�implies that EJ��d

2 while EC��d−4. Thus, even if far away

from the transition the direct Coulomb contribution is sig-nificant, as the transition is approached and �d becomeslarge, the Josephson coupling between the layers alwaystakes over. In the following we shall thus focus exclusivelyon the latter.

VII. SPECIFIC HEAT AND THERMAL CONDUCTIVITY

Having found the eigenmodes of the PWC as well as theeffective mass of dual vortices, we may now combine theseresults and calculate the specific heat and the thermal con-

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ductivity associated with the transverse vibrational modes.As mentioned above the longitudinal modes either havemuch higher speed or are gapped and thus will not contributeat low temperatures.

Let us first estimate the transverse in-plane sound velocity�50� in terms of physically meaningful parameters. Combin-ing Eqs. �33� and �56� this becomes

vs = c� ��

ln��d���TF

�� . �77�

We observe that the dependence on �d has dropped out, ex-cept through �d. We expect Eq. �77� to be valid only deep inthe PWC state; as one approaches the transition point thedual London approximation ceases to apply.

Thomas-Fermi screening length is normally of the orderof inverse Fermi wave vector kF, or several Å. The ratio ofthe two length scales is thus about 10−2–10−3. For �d�10the square root is 0.30. Equation �77� thus gives vs, about anorder of magnitude smaller than the Fermi velocity in cu-prates �the latter is �10−2c�. This makes sense physicallysince the PWC vibrations are a purely electronic phenom-enon, and, thus, on dimensional grounds one expects vs andvF to be of similar order of magnitude.

Similarly we can express the interplane sound velocity vzassociated with the Josephson coupling. Evaluating Eq. �74�with the help of Eqs. �33�, �56�, �60�, and �67� we obtain

vz = c�3/2

8�2 ln��d��a0

d�2��TF

2

d��� �d

��c�2

. �78�

Thus, vz grows with �d. According to the criterion �75� theperturbation theory breaks down when ��d /�c�� becomes ofthe order of 10. A glance at Eqs. �77� and �78� confirms thatthis makes sense physically for in this limit vz approaches vs.The sound velocity becomes isotropic, and we may no longertreat the interlayer tunneling as a perturbation.

Repeating the standard calculation of the phonon specificheat for the case of sound velocity with uniaxial anisotropyleads to

Cv =2�kB

4

�2�3vs2vz

T3, �79�

where �=�0�dxx3 / �ex−1�=�4 /15. The associated thermal

conductivity in the Cu-O plane becomes

� =1

3Cvvs� = �T3, �80�

with

� =2�kB

4�

3�2�3vsvz. �81�

Combining Eqs. �77� and �78� with Eq. �81� we obtain

� = �0.007–3.5��10 Å

�d�2 mW

K4 cm, �82�

where the lower bound obtains if we take �=1030 Å and�c=6.2� as found for optimally doped YBCO,50 while the

upper bound obtains for �=2020 Å and �c=37.7�, valuesrelevant to the 56-K “ortho-II” phase. In both cases we take�TF�a0=3.8 Å, d=12 Å, �=10, and �d=10. We assume thatthe phonon mean free path ��0.05 mm, which is the geo-metric average of the width ��500 m� and the thickness��10 m� of samples used in the experiment.31

The experimental result31 for the point farthest from thetransition gives ��0.5 mW/ �K4 cm�. With the value of �d

=25–50 Å, estimated from the STM data via Eq. �51�, ourresult above is broadly consistent with the experimental data,although it has to be noted that it depends strongly on theassumed values of the input parameters, most notably on theab-plane and c-axis penetration depths. These are known inYBCO with high accuracy50 but they also vary strongly withdoping, and it is not entirely clear which values one shouldadopt in the estimate of �. We have argued above that oneshould take the underlying mean-field, noninteracting values,which are presumably most accurately approximated by thevalues measured near the optimal doping. Within the class ofphase fluctuation models considered here, any reduction ofsuperfluid density upon underdoping is attributed to an inter-action effect beyond mean-field theory. In reality, part of thechange may be associated with the change of the underlyingmean-field ground state but it is impossible to make a precisestatement about this. It is also possible that the measuredvalues at optimal doping already reflect a fair amount offluctuations and the underlying mean-field superfluid densityshould be higher. The range of values displayed in Eq. �82� ismeant to be indicative of these various uncertainties.

VIII. SUMMARY AND OPEN ISSUES

We have studied the normal modes of a pair Wigner crys-tal employing a duality transformation from a phase fluctu-ating superconductor to a fictitious dual type-II supercon-ductor in applied magnetic filed. Vortices in the dualsuperconductor represent Cooper pairs in the original modeland the vibrational modes of the PWC can be calculated asmagnetophonons of the dual Abrikosov vortex lattice. As-suming that the pinning of the dual vortex lattice to the un-derlying ionic lattice is negligible, as suggested by the factthat PWC is incommensurate in some cases, the transversemagnetophonon is acoustic and will thus contribute to thelow-energy thermal and transport properties of the system.For charged systems longitudinal modes will be gapped. Ourmain result is the estimate of the sound velocity of the trans-verse mode which determines the magnitude of its contribu-tion to the specific heat and thermal conductivity. The in-plane velocity vs is found to be about an order of magnitudelower than the Fermi velocity vF. Together with our estimateof the interplane sound velocity vz, which we assumed to bedetermined primarily by the Josephson coupling between theneighboring layers, our considerations yield a T3 contributionto the thermal conductivity with a prefactor consistent withthe recently reported bosonic mode in strongly underdopedsingle crystals of YBCO.31

An important length scale in this problem is the dual pen-etration depth �d, which has the physical meaning of the sizeof the Cooper pair in the PWC. STM experiments indicate

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that �d is much larger than the distance between the Cooperpairs; the latter are extended objects with strong zero-pointmotion and overlapping wave functions. This is the mainreason why duality is a useful concept in this problem: itprovides a convenient tool for the description of a stronglyquantum fluctuating system of Cooper pairs in terms of dualvortices that can be treated as point particles. The latter arelocal objects, and their physics can be accurately described inthe mean-field approximation.

A key assumption, underlying all our preceding consider-ations and results, is that the pair Wigner crystal is essen-tially decoupled from the ionic lattice. As mentioned in theIntroduction transverse modes of a PWC pinned to the ioniclattice would be gapped and thus irrelevant at low tempera-tures. The duality transformation reviewed in Sec. II showsthat PWC indeed can exist in continuum, independently ofany underlying crystalline lattice. There is, therefore, nological contradiction implied by the above assumption. In thecontext of cuprates one must ask to what extent does thecontinuum model reflect the physics of Cooper pairs movingin the copper-oxygen planes. The key issue here is whetherPWC is commensurate or incommensurate with the underly-ing ionic lattice, since incommensurate PWC cannot bepinned, except by disorder. Pinning by disorder affects themagnetophonon mean free path but in general does not opena gap in the phonon spectrum.

The problem of commensurability is a difficult one toanalyze theoretically as it involves the details of PWC ener-getics, band structure, and electron-ion interaction. Our argu-ment in favor of the incommensurate PWC is thereforelargely phenomenological and is based on the followingthree observations. First, the checkerboard patterns in at leastsome experiments17,19 have been reported to have periodclearly different from 4 ionic lattice constants �the valuesrange between 4.2a0 and 4.7a0�, implying incommensuratePWC. The mere existence of such an incommensurate PWCindicates that the coupling to the lattice must be extremelyweak even in the case when the PWC is commensurate. Sec-ond, the PWC appears to exist for a relatively wide range ofdopings. If only commensurate PWC were allowed, then onewould expect dramatic changes in PWC structure uponvariation of the pair density; in particular, the unit-cell sizewould vary significantly as the PWC adjusted to differentdoping levels. No such dramatic variations are observed. Itappears, instead, that PWC structure, where it exists, islargely independent of doping. Finally, we found that Cooperpairs in a crystal proximate to a superconductor have wavefunctions delocalized over many lattice constants. We mayexpect that a potential with ionic lattice periodicity should berelatively ineffective in pinning such delocalized objects.

An issue which might require further consideration is thedependence of the sound velocity of the PWC on the dopinglevel of the system. Experimentally the strength of the T3

mode goes to zero continuously as the transition to the su-perconducting state is approached. In our theory the transi-tion is marked by the divergence of the dual penetrationdepth �d. We found the in-plane sound velocity to be inde-pendent of �d. This is due to the exact cancellation betweenthe strength of the intervortex interaction er and the dualvortex mass mv: both vanish as �d→�. The c-axis sound

velocity is proportional to �d2 which ensures that the thermal

conductivity of the bosonic modes decreases as the transitionis approached, in agreement with the experiment. It is impor-tant to emphasize, however, that there is no reason to expectthat this result will remain valid very close to the transition.First, vz is obtained in a perturbation theory which is onlyvalid as long as �d /�c�1. Outside of this regime one musttreat the full 3D system. Second, the dual London approxi-mation also breaks down near the transition since the vortexcores begin to overlap. To address the physics of magne-tophonons near the transition one would have to treat vortexvibrations in the full Ginzburg-Landau theory, Eq. �18�.Qualitatively we expect that the amplitude fluctuations willmake the intervortex interaction stronger, leading to an in-crease in vs and thus a reduction of thermal conductivity asthe transition is approached, in accordance with experiment.Detailed calculations, however, present a daunting challengeand are left for future investigation.

Another open issue is the inclusion of the detailed struc-ture of the PWC. We have assumed a simple square Bravaislattice of Cooper pairs. Experiments18–20 and detailed theo-retical considerations9,28 indicate a lattice with square sym-metry, but with a more complicated internal structure. Withinour approach this could be modeled as a square vortex latticewith a basis. The vibrational spectrum of such a lattice willbe more complicated but will retain the acoustic mode de-rived above which reflects the center-of-mass motion of theunit cell.

STM studies also indicate the presence of domain wallsand other defects in the PWC. Such defects would scatterPWC phonons and cause a short mean free path �. Our esti-mate, on the other hand, suggests that in YBCO � should beof the order of the sample size. This implies that the PWC inYBCO is much more homogeneous than that in BiSCCO andNa-CCOC. Given the extreme purity of the YBCO crystalsused in the thermal conductivity measurement31 this is per-haps not surprising. Unfortunately YBCO is not amenable tohigh-precision STM studies due to its lack of a natural cleav-age plane. It would be interesting to see if the bosonic modecould be observed in the thermal conductivity of Na-CCOC.Based on our model we would expect it to be much weakerdue to much shorter mean free path �.

ACKNOWLEDGMENTS

The authors are indebted to E. Altman, J. C. Davis, T. P.Davis, N. Doiron-Leyraud, M. P. A. Fisher, K. LeHur, L.Taillefer, and Z. Tešanović for stimulating discussions andcorrespondence. The work has been supported in part byNSERC, CIAR, A.P. Sloan Foundation, and the Aspen Cen-ter for Physics.

APPENDIX: EVALUATION OF Zxy��

This appendix follows the calculation of Fetter,41 adjustedto the case of the square lattice. The matrix Z��

�2� is a symmet-ric rank-2 tensor. In terms of the vector k it can therefore bewritten as

T. PEREG-BARNEA AND M. FRANZ PHYSICAL REVIEW B 74, 014518 �2006�

014518-12

Z���2��k� = A�k2���� + B�k2�

k�k�

k2 , �A1�

where A and B are scalar functions of k2. The trace of Z�2�

vanishes to first order in 1/ ��d2Q0

2�; thus,

2A�k2� + B�k2� = 0. �A2�

We evaluate the off-diagonal elements of the matrix to deter-mine B�k2�.

Let us define

S�k� = 2�nv �Q�0

� �Qx + kx��Qy + ky��Q + k�2 −

QxQy

Q2 � , �A3�

where nv=1/a2 is the density of vortices. The Ewald sum-mation technique splits the above sum into two parts, S=Sd�k�+Sr�k�, in such a way that Sd converges rapidly inreal space while Sr converges rapidly in reciprocal space.One obtains, for any Bravais lattice,41

Sd = 2�j

�1 − eik·Rj�XjY j

Rj4 �1 + �nvRj

2�e−�nvRj2,

Sr = 2�nv �Q�0

�Qx + kx��Qy + ky��Q + k�2 e−�Q + k�2/4�nv

− 2�nvkxky

k2 �1 − e−k2/4�nv� , �A4�

where R j = �Xj ,Y j� is a lattice vector. The above sums can beeasily evaluated numerically for arbitrary k. For small wave

vector k we may expand Sd and Sr to second order,

Sd � 2kxky�j

�Xj

2Y j2

Rj4 �1 + �nvRj

2�e−�nvRj2,

Sr � −1

2kxky��

Q

��1 −Q2

8�nv�e−Q2/4�nv + 1� , �A5�

where �� denotes a summation that excludes the zero vector.In order to perform the sums for the square lattice we sub-stitute

R = a�l,m�, Q =2�

a�l,m� , �A6�

and arrive at

Sd = kxky�l,m� 2l2m2

�l2 + m2�2 �1 + ��l2 + m2��e−��l2+m2�,

Sr =1

2kxky��l,m

� �1 −�

2�l2 + m2��e−��l2+m2� + 1� .

The sums over m and l are rapidly convergent and can beevaluated numerically. This gives the off-diagonal part ofZ�2�,

Zxy�2��k� = − 0.415

a2

2�kxky − �a2kxky , �A7�

with �=0.066. Thus, B�k2�=−�a2k2.

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