Weak magnetism and non-Fermi liquids near heavy-fermion critical points

PHYSICAL REVIEW B 69, 035111 ~2004!

Weak magnetism and non-Fermi liquids near heavy-fermion critical points

T. SenthilDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Matthias VojtaInstitut fur Theorie der Kondensierten Materie, Universita¨t Karlsruhe, Postfach 6980, 76128 Karlsruhe, Germany

Subir SachdevDepartment of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA~Received 19 June 2003; revised manuscript received 13 October 2003; published 23 January 2004!

This paper is concerned with the weak-moment magnetism in heavy-fermion materials and its relation to thenon-Fermi-liquid physics observed near the transition to the Fermi liquid. We explore the hypothesis that theprimary fluctuations responsible for the non-Fermi-liquid physics are those associated with the destruction ofthe large Fermi surface of the Fermi liquid. Magnetism is suggested to be a low-energy instability of theresulting small-Fermi-surface state. A concrete realization of this picture is provided by a fractionalized Fermi-liquid state which has a small Fermi surface of conduction electrons, but also has other exotic excitations withinteractions described by a gauge theory in its deconfined phase. Of particular interest is a three-dimensionalfractionalized Fermi liquid with a spinon Fermi surface and a U~1! gauge structure. A direct second-ordertransition from this state to the conventional Fermi liquid is possible and involves a jump in the electronFermi-surface volume. The critical point displays non-Fermi-liquid behavior. A magnetic phase may developfrom a spin-density-wave instability of the spinon Fermi surface. This exotic magnetic metal may have a weakordered moment, although the local moments do not participate in the Fermi surface. Experimental signaturesof this phase and implications for heavy-fermion systems are discussed.

DOI: 10.1103/PhysRevB.69.035111 PACS number~s!: 71.10.Li

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I. INTRODUCTION

The competition between the Kondo effect and intermment exchange determines the physics of a large clasmaterials which have localized magnetic moments coupto a separate set of conduction electron.1 When the Kondoeffect dominates, the low-energy physics is well describedFermi-liquid theory~albeit with heavily renormalized quasparticle masses!. In contrast when the intermoment exchandominates, ordered magnetism typically results.

A remarkable experimental property of such magnestates is that the magnetism is often very weak—the ordemoment per site is much smaller than the microscopic lomoment that actually occupies each site. The traditionalplanation of this feature is that the magnetism arises ouimperfectly Kondo-screened local moments. In other worthe magnetism is to be viewed as a spin-density wavedevelops out of the parent heavy Fermi-liquid state. We whenceforth denote such a state as SDW. Clearly a SDW smay be a small moment magnet.

A different kind of magnetic metallic state is also possibin heavy-fermion materials where the moments order at rtively large energy scales, and simply do not participatethe Fermi surface of the metal. In such a situation, the sration moment in the ordered state would naively be lari.e., of order the atomic moment.

Often the distinction between these two kinds of magnestates can be made sharply: the two Fermi surfaces in thestates may have different topologies~albeit, the same volumemodulo the volume of the Brillouin zone of the orderestate!, so that they cannot be smoothly connected to oanother.

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In recent years, a number of experiments have unearsome fascinating phenomena near the zero-temperature~T!quantum transition between the heavy-fermion liquid andmagnetic metal. In particular, many experiments do noteasily2–5 into a description in terms of an effective Gaussitheory for the spin-density-wave fluctuations, renormalizself-consistently by quartic interactions.6–10 This theorymakes certain predictions on deviations from Fermi-liqubehavior as the heavy Fermi-liquid state approaches mnetic ordering induced by the condensation of the spdensity-wave mode; those predictions are, however, inagreement with experimental findings. This conflict raisthe possibility that the magnetic state being accessed is nthe first category discussed above: a SDW emerging froheavy Fermi liquid. Rather, it may be the second kindmagnetic metal where the local moments do not participat all in the Fermi surface. In other words, the experimesuggest that the Kondo effect~crucial in forming the Fermi-liquid state! is itself suppressed on approaching the magnstate.

This proposal clearly raises several serious puzzles. Hdo we correctly describe the non-Fermi-liquid physics nthe transition? If this non-Fermi-liquid behavior is accompnied by the suppression of the Kondo effect, how doreconcile it with the weak moments found in the magnestate? The traditional explanation for the weak magnetismapparently in conflict with the picture that the Kondo effeand the resultant heavy Fermi-liquid state are destroyedapproaching the magnetic state. In other words, the naexpectation of a large saturation moment in a magnetic mwhere the local moments do not participate in the Fesurface must be revisited.

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T. SENTHIL, MATTHIAS VOJTA, AND SUBIR SACHDEV PHYSICAL REVIEW B69, 035111 ~2004!

The weakness of the ordered moment in the magnstate may be reconciled with the apparent suppression oKondo effect if we assume that there are strong quanfluctuations of the spins that reduce their moment. Sstrong quantum effects may appear to be unusual in thdimensional systems, but may be facilitated by the coupto the conduction electrons~even if there is no actual Kondscreening!. In this paper we study specific states where suquantum fluctuations have significantly reduced the ordemoment~or even caused it to vanish!, and the evolution ofsuch states to the heavy Fermi liquid.

We begin with several general pertinent observatioFirst, consider the heavy Fermi-liquid state. This Fermliquid behavior is accompanied by a Fermi surface whiremarkably, satisfies Luttinger’s theorem only if the locmoments are included as part of the electron count.~Such aFermi surface is often referred to as the ‘‘large Fermi sface,’’ and we will henceforth refer to such a phase as F!.The absorption of the local moments into the Fermi voluis the lattice manifestation of the Kondo screening ofmoments. We take as our starting point the assumptionthe Kondo effect becomes suppressed on approachingmagnetic state. What then happens to the large Fermiface?

In thinking about the resulting state theoretically, it is important to realize that once magnetic order sets in, there isharp distinction between a large Fermi volume whichcludes the local moments, and a Fermi volume that excluthe local moments—the latter is often loosely referred to‘‘small.’’ This is because the Fermi volumes can onlydefined modulo the volume of the Brillouin zone, and tonset of magnetic order at least doubles the unit cellhence at least halves the Brillouin-zone volume.~There can,however, be a distinction between the Fermi-surface topgies in the two situations.!

In this paper we will take the point of view that the prmary transition involves the destruction of the large Fersurface, and that the resulting small-Fermi-surface state hdistinct physical meaning even in the absence of magnorder. The magnetic order will be viewed as a low-eneinstability of the resulting state in which the local momenare not to be included in the Fermi volume.

Evidence in support of this point of view exists. In thexperiments the non-Fermi-liquid behavior extends to teperatures well above the Neel ordering temperature evenaway from the critical point. This suggests that the fluctutions responsible for the non-Fermi-liquid behavior havery little to do with the fluctuations of the magnetic ordparameter. Some further support is provided by the resultinelastic neutron-scattering experiments that apparentlycritical behavior at a range of wave vectors including~butnot restricted to! the one associated with magnetic orderiin the magnetic metal.5 Finally, there even exist materials iwhich the non-Fermi-liquid features persist into the magncally ordered side—this is difficult to understand if the noFermi-liquid physics is attributed to critical fluctuationsthe magnetic order parameter.

Conceptually, as we asserted above, it pays to allowthe possibility of a nonmagnetic state in which the suppr

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sion of the Kondo effect removes the local moments fromFermi volume, resulting in a ‘‘small Fermi surface,’’ evethough such a state may not actually be a ground state insystem of interest. In our previous work11 we argued thatsuch states do exist as ground states of Kondo lattice moin regular d-dimensional lattices, and that the violationLuttinger’s theorem in such a state was intimately linkedthe presence of neutralS51/2 andS50 excitations inducedby topological order~see also Appendix A!: we dubbed suchground states FL* .

Clearly, it is worthwhile to explore metallic magnetistates that develop out of such FL* states~just as the usuaSDW state develops out of the Fermi liquid!. Such states,which we will denote SDW* , represent a third class of metallic magnetic states distinct from both the conventionSDW and the conventional local-moment metal describabove. As we will see, in such magnetic states the local mments do not participate in the Fermi surface. Neverthethey may have a weak ordered moment. Thus these soffer an opportunity for resolution of the puzzles mentionabove. The properties and the evolution of such states,their parent FL* states, to the Fermi liquid will be the subjeof this paper. The SDW* states inherit neutral spinS51/2spinon excitations andS50 ‘‘gauge’’ excitations from theFL* states, which will be described more precisely belothese excitations coexist with the magnetism and the metbehavior. The experimental distinction between the SDand SDW* states is however subtle, and will also be dscribed in this paper.~The FL and FL* states can be easildistinguished by the volumes of the Fermi surfaces, butdistinction does not extend to the SDW and SDW* states.!

We emphasize that a wide variety of heavy-fermion mterials display non-Fermi-liquid physics in the vicinity of thonset of magnetism that is, to a considerable extent, unisal. However, the detailed behavior at very low temperatappears to vary across different systems. In particularsome materials a direct transition to the magnetic statevery low temperature does not occur~due for instance tointervention of a superconducting state!. In other materials,such a direct transition does seem to occur at currentlycessible temperatures. In view of this, we will not attemptpredict the detailed phase diagram at ultralow temperatuWe focus instead on understanding the universal non-Feliquid physics not too close to the transition and its relatito the magnetic state.

A. Summary of results

Our analysis is based upon nonmagnetic translatiinvariant states that have a small Fermi surface (FL* ), andthe related transitions to the heavy Fermi liquid~FL!. As weshowed previously,11 the FL* state has a Fermi surface olong-lived electronlike quasiparticles whose volume doescount the local moments. The local moments are insteadstate adiabatically connected to a spin-liquid state with emgent gauge excitations. Such spin liquids can be classifiedthe gauge group determining the quantum numbers carby the neutralS51/2 spinon excitations and the gauge extations, and previous work12,13 has shown that the mos

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WEAK MAGNETISM AND NON-FERMI LIQUIDS NEAR . . . PHYSICAL REVIEW B 69, 035111 ~2004!

prominent examples areZ2 and U~1! spin liquids. TheZ2

spin liquids are stable in all spatial dimensionsd>2, whilethe U~1! spin liquids exist only ind>3 @the latter correspondto the existence of a Coulomb phase in a compact U~1!gauge theory ind>3, as discussed in Ref. 14#. Correspond-ingly, we also have the metallicZ2 FL* and U~1! FL* states.Our previous work11 considered primarily theZ2 FL* state,whereas here we focus on the U~1! FL* state.

As we have already discussed, these nonmagnetic smay lead to magnetic order at low energies, or in proximstates in a generalized phase diagram. In this manner thstate leads to the SDW state, while the FL* states lead to theZ2 SDW* and the U~1! SDW* states. The relation betweethe metallic SDW and SDW* states has a parallel to thabetween the insulating Ne´el state and the AF* state of Refs.13 and 15.

We will also discuss the evolution from the U~1! SDW*state to the conventional Fermi liquid. As explained earlthe underlying transition is that between FL and FL* stateswhich controls the nature of the Fermi surface. In Ref.we argued that the spinon pairing in theZ2 FL* state impliedthat there must be a superconducting state in between thandZ2 FL* states. There is no such pairing in the U~1! FL*state, and hence there is the possibility of a direct transibetween the FL and U~1! FL* states: this transition and thnature of the states flanking it are the foci of our paper. Nthat the volume of the Fermi surfacejumpsat this transition.Nevertheless the transition may be second order. Thimade possible by the vanishing of the quasiparticle resion an entire portion of the Fermi surface~a ‘‘hot’’ Fermisurface! on approaching the transition from the FL sidNon-Fermi-liquid physics is clearly to be expected at sucsecond-order Fermi-volume changing transition. We reitethat the U~1! FL* state is only believed to exist ind.2.

We study the FL and U~1! FL* states by the ‘‘slave’’boson method, introduced in the context of the singmoment Kondo problem.16 In this method, the condensatioof the slave boson marks the onset of Kondo coherencecharacterizes the FL phase. In contrast the slave boson icondensed in the FL* phase. Fluctuations about this meafield description lead to the critical theory of the transitiinvolving a propagating boson coupled to a compact U~1!gauge field, in the presence of damping from fermioniccitations.

We note that earlier studies17,18 of single-impurity prob-lems found a temperature-induced mean-field transitiontween a state in which the slave boson is condensed~andhence the local moment is Kondo screened! and a state inwhich the boson has no condensate: however, it wasrectly argued that this transition is an artifact of the mefield theory, and no sharp transition exists in the singmoment Kondo problem atT.0. If we now naivelygeneralize this single-impurity model to the lattice, we wfind that theT50 ground state always has Kondo screeniIt is only upon including frustrating intermoment exchaninteractions—equivalent to having ‘‘dispersing’’ spinonsthat it is possible to break down Kondo screening and rea state in which the slave boson is not condensed. This t

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Our analysis of the aboved53 U~1! gauge theory leadsto the schematic crossover phase diagram as a function oKondo exchangeJK andT shown in Fig. 1.

The crossover phase diagram in Fig. 1 is similar to thaa dilute Bose gas as a function of chemical potential atemperature.19,20 Here the bosons are coupled to a U~1!gauge field, and this is important for many of the criticproperties to be described in the body of the paper. Notain Fig. 1 the density of bosons isnot fixed, and varies as afunction of T, JK , and other couplings in the HamiltonianIndeed, the contours of constant boson density have a cplicated structure, which are similar to those in Ref. 20. Tvariation in the boson density is a crucial distinction froearlier analyses22,23 of boson models coupled to dampeU~1! gauge fields: in these earlier works, the boson denwas fixed at aT-independent value. As we will see, allowinthe boson density to vary changes the critical properties,has significant consequences for the structure of the crover phase diagram and for theT dependence of observable

FIG. 1. Crossover phase diagram for the vicinity of thed53quantum transition involving breakdown of Kondo screening.JK isthe Kondo exchange in the Hamiltonian introduced in Sec. III. Tonly true phase transition above is that at theT50 quantum criticalpoint at JK5JKc between the FL and FL* phases. The ‘‘slave’’bosonb measures the mixing between the local moments andconduction electrons and is also described in Sec. III. The crovers are similar to those of a dilute Bose gas as a functionchemical potential and temperature, as discussed in Refs. 1920—the horizontal axis is a measure of the boson chemical potial mb . The boson is coupled to a compact U~1! gauge field; atT50 this gauge field is in the Higgs/confining phase in the FL staand in the deconfining/Coulomb phase in the FL* state. There is nophase transition atT.0 between a phase withb&Þ0 and a phasewith ^b&50 because such a transition is absent in a theory witcompactU~1! gauge field ind53 ~Ref. 21! ~the mean-field theoriesof Secs. III and IV C do show such transitions, but these will tuinto crossovers upon including gauge fluctuations!. The compact-ness of the gauge field therefore plays a role in the crossovers in‘‘renormalized classical’’ regime above the FL state~this has notbeen worked out in any detail here!. However, the compactnessnot expected to be crucial in the quantum-critical regime. The croover line displayed between the FL and quantum-critical regimcan be associated with the ‘‘coherence’’ temperature of the heFermi liquid. At low T, as discussed in the text, there are likelybe additional phases associated with magnetic order~the SDW andSDW* phases!, and these are not shown above but are shownFig. 2; they also appear in the mean-field phase diagram in Fig

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We will show that non-Fermi-liquid physics obtains in thquantum-critical region of this transition. Furthermore, wargue that fluctuation effects may lead to a spin-density wdeveloping out of the spinon Fermi surface of the U~1! FL*phase, thereby obtaining the U~1! SDW* phase. The ex-pected phase diagram and crossovers for the evolution fthe U~1! SDW* phase to the FL phase are shown in Fig.We examine few different kinds of such U~1! SDW* phasesdepending on the details of the spinon Fermi surface.also describe a number of specific experimental signaturethe U~1! SDW* phase which may help to distinguish it frommore conventional magnetic metals.

B. Relation to earlier work

We have already mentioned a number of precursors toideas in our discussion so far. Here, we complete thisnoting some other related developments in the literature

Early on, Andrei and Coleman24 and Kaganet al.25 dis-cussed the possibility of the decoupling of local momeand conduction electrons in Kondo lattice models. Andand Coleman had the local moments in a spin-liquid swhich is unstable to U~1! gauge fluctuations, and did nonotice violation of Luttinger’s theorem. The possibility osmall electronic Fermi surfaces was noted by Kaganet al.,but no connection was made to the requirement this impoon emergent gauge excitations.11

More recently, Burdinet al.described many aspects of thphysics we are interested in here in a dynamical mean-fitheory of a random Kondo lattice.26 In this work, we ob-tained a state in which local moments formed a spin liqand stayed essentially decoupled from the conduction etrons. They emphasized that the transition between sustate~which is the analog of our FL* states! and a conven-tional heavy Fermi liquid~the FL state! should be understoodas a Fermi-volume changing transition. However questiof emergent gauge structure were not addressed by them

FIG. 2. Expected phase diagram and crossovers for the evtion from the U~1! SDW* phase to the conventional FL. Two different transitions aregenericallypossible at zero temperature: Upomoving from the SDW* towards the Fermi liquid, the fractionalization is lost first followed by the disappearance of magnetic orNevertheless the higher-temperature behavior in the region ma‘‘quantum critical’’ is non-Fermi-liquid-like, and controlled by thFermi-volume changing transition from FL to FL* . This may beloosely associated with the breakdown of Kondo screening.

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Demleret al.27 discussed fractionalized phases of Konlattice models. However, they did not consider any stawith long-lived electronlike quasiparticles, as are presenthe FL* phase.

Recently Essler and Tsvelik28 discussed the fate of onedimensional Mott insulator under a particular long-rangeterchain hopping. At intermediate temperatures, they obtastate with a small Fermi surface, in that the Fermi-surfavolume does not count the local moments.29 However, theirconstruction does not lead to a state with emergent gaexcitations in higher dimensions, and as they conclude, tstate is unstable to magnetic order at low temperatures.believe this low-T state is an ordinary SDW state, and arealizations of small Fermi surfaces at intermediate temptures are remnants of one-dimensional physics. In contrall our constructions are genuinely higher dimensional, aonly work for d>2.

The physics of the destruction of the large Fermi surfaby the vanishing of Kondo screening has been addresseinteresting recent works30–32 using an ‘‘extended dynamicamean-field theory.’’ We have argued in our discussion abothat vanishing of Kondo screening is conceptually quitedifferent transition from the onset of magnetic order; constent with this expectation, Sun and Kotliar31 found two dis-tinct points associated with these transitions. It is our conttion that the critical theory of the FL to U~1! FL* transition~discussed in the present paper! is thed53 realization of thelarge-dimensional critical point with vanishing Kondscreening found by Sun and Kotliar.

C. Outline

The rest of the paper is organized as follows: In Sec.we briefly review the properties of various fractionalizeFermi liquids (FL* ). A specific U~1! FL* state where thespinons form a Fermi surface is considered. In Sec. III,construct a mean-field description of this state and its tration to the heavy Fermi liquid. This transition involvesjump in the Fermi-surface volume but is nevertheless shoto be second order within the mean-field theory. This is mapossible by the vanishing of the quasiparticle residueZ on anentire Fermi surface~a hot Fermi surface! as one moves fromthe heavy Fermi liquid to the fractionalized Fermi liquiFluctuations about this mean-field description are then csidered. In Sec. IV, we first consider fluctuation effects onphases—in particular the FL* phase. We argue that thspecific-heat coefficientg diverges logarithmically once theleading-order fluctuations are included. Furthermore, flucttions also make possible a spin-density-wave instabilitythe spinon Fermi surface, leading to a U~1! SDW* state. Toillustrate possible phases, we will discuss an improved mefield theory which includes the SDW order parameter, apresent phase diagrams showing the influence of temperaand magnetic field. We then examine fluctuation effectsthe critical point of the transition between FL* and FL inSec. V. We argue that the logarithmic divergence ofspecific-heat coefficient persists in the quantum-criticalgion, and also that non-Fermi-liquid transport obtains theIn Sec. VI, we discuss the properties of the U~1! SDW*

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phase in greater detail with particular attention to its idenfication in experiments. A discussion of the implications fvarious experiments in Sec. VII will conclude the paper.

II. FRACTIONALIZED FERMI LIQUIDS

The existence of nonmagnetic translation-invariant smFermi-surface states was shown in a recent paper by11

with a focus on two-dimensional Kondo lattices. Such stawere obtained when the local-moment system settles infractionalized spin liquid~rather than a magnetically orderestate! due to intermoment interactions. A weak Kondo copling to conduction electrons does not disrupt the structurthe spin liquid but leaves a sharp Fermi surface of quasiticles whose volume counts the conduction-electron denalone ~a small Fermi surface!. Thus these states have frationalized excitations that coexist with conventional Fermliquid-like quasiparticle excitations. We dubbed these staFL* ~to distinguish them from the conventional Fermi liquFL!. We also pointed out an intimate connection betweendisappearance of the large Fermi surface and fractionation, and this is discussed further in Appendix A.

The FL* phase can be further classified by the naturethe spin liquid formed by the local moments. Recent yehave seen considerable progress in our understanding oftionalized spin liquids. An important feature of spin-liqustates ind>2 is that they possess emergent gauge structPut simply, this means that the distinct excitations in suphases interact with each other through long-ranged intetions which can be mathematically encapsulated as gainteractions. In other words, the effective-field theory of tstate is a gauge theory in its deconfined phase. The two nral possibilities are that the emergent gauge group is eiZ2 or U~1!. The former is allowed in any dimensiond>2while the latter is only allowed ind53 ~or higher!.

The Z2 states have been discussed at length in the litture and in the present context in our earlier work.11 In con-trast, the U~1! states have not been discussed much, thotheir possible occurrence~in d53) and their universal properties have been appreciated by many workers in the fiWe therefore provide a quick discussion: The distinct exctions in thed53 U~1! spin-liquid phases are neutral spin-1spinons, a gapless~emergent! gauge photon, and a gappepoint defect~the ‘‘monopole’’!. The spinons are minimallycoupled to the photon and hence interact through emerlong-ranged interactions. For simple microscopic modelsrealize such phases, see Refs. 14 and 33. A crucial distincbetween theZ2 spin liquids is that the spinons in this phaare not generically paired, i.e., the spinon numberconserved.34

Several classes of spin liquids are theoretically posswith the same gauge structure. These may be characteby the statistics of the spinons, their band structure, etc.the rest of this paper, we will focus on a particular thredimensional U~1! spin-liquid state with fermionic spinonthat form a Fermi surface. A specific toy model which dplays this phase is presented in Appendix B.

As with the Z2 spin liquids discussed in Ref. 11, thgauge structure in the U~1! spin-liquid state is also stable t

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a weak Kondo coupling to conduction electrons.35 The re-sulting U~1! FL* state consists of a spinon Fermi surfacoexisting with a separate Fermi surface of conduction etrons. There will also be gapless photon and gapped mopole excitations. The physical electron Fermi surface~asmeasured by de Haas–van Alphen experiments for insta!will have a small volume that is determined by the condution electrons alone.

In our previous work, we pointed out that the transitiofrom a Z2 FL* phase to the heavy FL will generically bpreempted by superconductivity. This is due to the pairingspinons in theZ2 phase. In contrast, we expect that dueconservation of spinon number a direct transition betwethe U~1! FL* and heavy FL phases should be possible.

III. MEAN-FIELD THEORY

A simple mean-field theory allows a description both oU~1! FL* phase and its transition to the heavy FL. Consida three-dimensional Kondo-Heisenberg model, for concreness on a cubic lattice:

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Proceeding as usual, we consider a decoupling of bothKondo and Heisenberg exchange using two auxiliary fiein the particle-hole channel. Treating the fluctuations of thauxiliary fields by a saddle-point approximation@formallyjustified for a large-N SU(N) generalization#, we obtain themean-field Hamiltonian

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phases. First, there is the usual Fermi-liquid~FL! phase when

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b0 , x0 , m f are all nonzero.~Note thatb0Þ0 implies thatx0Þ0.) This phase is readily seen to have a large Fesurface as expected. Second, there is a phase (FL* ) wherethe Kondo hybridizationb050 but x0Þ0. ~In this phasem f50.! This mean-field state represents a situation whthe conduction electrons are decoupled from the local mments and form asmall Fermi surface. The local-momensystem is described as a spin fluid with a Fermi surfaceneutral spinons. We expect thatx0;JH .

The transition between these two different states canbe examined within the mean-field theory. Interestingly,transition is second order~despite the jump in Fermi volume!and is described byb0→0 on approaching it from the Fermliquid side. How can a second-order transition be associawith a jump in the volume of the electron Fermi surfacThis can be understood by examining the Fermi surfaclosely in this mean-field theory.

The mean-field Hamiltonian is diagonalized by the traformation

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Ek12ek, uk

21vk251. ~10!

Consider first the FL* phase whereb0505m f , but x0Þ0. The electron Fermi surface is determined byconduction-electron dispersionek and is small. The spinonFermi surface encloses one spinon per site and has vohalf that of the Brillouin zone. For concreteness, we wconsider the situation where the electron Fermi surface dnot intersect the spinon Fermi surface. We will also assuthat the conduction-electron filling is less than half.

Now consider the FL phase near the transition~smallb0).In this case, there are two bands corresponding toEk6 : onederives from thec electrons~with weakf character! while theother derives from thef particles~with weakc character!. Wewill call the former thec band and the latter thef band. Forsmall b0, both bands intersect the Fermi energy so thatFermi surface consists of two sheets~see Fig. 3!. The totalvolume is large, i.e., includes both local moments and cduction electrons. Upon moving toward the transition to F*(b0 decreasing to zero!, thec-Fermi surface expands in siz

03511

i

e-

f

soe

ed

s

-

e

elese

e

-

to match onto the small Fermi surface of FL* . On the otherhand, thef-Fermi surface shrinks to match onto thespinonFermi surface of FL* .

Upon increasingb0 in the FL state and depending on thband structure, another transition is possible, where thcband becomes completely empty. Then, the Fermi-surftopology changes from two sheets to a single sheet—sutransition between two conventional Fermi liquids is knowas Lifshitz transition and will not be further considered he

The quasiparticle weightZ close to the FL-FL* transitionis readily calculated in the present mean-field theory. Forelectron Green’s function we find

G~k,ivn!5uk

2

ivn2Ek11

vk2

ivn2Ek2. ~11!

Therefore at the Fermi surface of thec band ~which hasdispersionEk1 , the quasiparticle residueZ5uk

2). At thisFermi surface,Ek1'ek'0 so that

Ek1'ek1b0

2

ek2ek f⇒uk'2

JH

b0vk . ~12!

Using Eqs.~10!, we then findZ'1 on thec-Fermi surface.At the Fermi surface of thef band on the other hand,Z

5vk2 . Also near this Fermi surface,uek2ek fu't wheret is

the conduction-electron bandwidth. We have assumed areasonable thatt@JH . Thus for thef-Fermi surface,

Ek1'ek1b0

2

ek2ek f⇒uk'2

t

b0vk . ~13!

This then gives

Z5vk2'S b0

t D 2

. ~14!

Thus the quasiparticle residue stays nonzero on thec-Fermisurface while it decreases continuously to zero onf-Fermi surface on moving from FL to FL* . ~The f-Fermisurface is ‘‘hot’’ while thec-Fermi surface is ‘‘cold.’’!

Clearly the critical point is not a Fermi liquid.Z vanishesthroughout the hot Fermi surface at the transition, and n

FIG. 3. Fermi-surface evolution from FL to FL* : close to thetransition, the FL phase features two Fermi-surface sheets~the coldc and the hotf sheet, see text!. Upon approaching the transition, thquasiparticle residueZ on the hotf sheet vanishes. On the FL* side,the f sheet becomes the spinon Fermi surface, whereas thec sheet issimply the small conduction-electron Fermi surface.

1-6

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Fermi-liquid behavior results. It is interesting to contrast tresult with the spin-fluctuation model~Hertz-Moriya-Milliscriticality! where the non-Fermi-liquid behavior is only asociated with some hot lines in the Fermi surface, and csequently plays a subdominant role.

Despite the vanishing quasiparticle weightZ, the effectivemassm* of the large-Fermi-surface state does not divergethe transition in this mean-field calculation, because the etron self-energy is momentum dependent. Physically,quasiparticle at the hot Fermi surface is essentially madeof the f particle for smallb; even whenb goes to zero thefparticle ~the spinon! continues to disperse due to the nonvnishing x0 term. Indeed the low-temperature specific heC;gT with g nonzero in both phases. As we argue belothis is an artifact of the mean-field approximation and willmodified by fluctuations.

The detailed shape of the spinon Fermi surface in the F*phase~or the hot Fermi surface which derives from it in thFL phase! depends on the details of the lattice and the foof the local-moment interactions. For the particular modiscussed above, the spinon Fermi surface is perfenested. In more general situations, a non-nested spFermi surface will obtain. In all cases, however, the voluof the spinon Fermi surface will correspond to one spinper site.

IV. FLUCTUATIONS: MAGNETISM AND SINGULARSPECIFIC HEAT

Fluctuation effects modify the picture obtained in tmean-field theory in several important ways. We first discfluctuation effects in the two phases. The heavy Fermi-liqphase is of course stable to fluctuations—their main efbeing to endow thef particle with a physical electric chargthereby making it an electron.36,37 Fluctuation effects aremore interesting in the FL* state, and are described byU~1! gauge theory minimally coupled to the spinon Fersurface~which continues to be essentially decoupled frothe conduction-electron small Fermi surface!. This may bemade explicit by parametrizing the fluctuations in the actin the FL* phase as follows:

x rr 8~t!5eiarr 8(t)x0rr 8 . ~15!

The action then becomes

S5Sc1Sf1Sf c1Sb ,

Sc5E dt(k

ck~]t2ek!ck ,

Sf5E dt(r

f r~]t2 ia0! f r2 (^rr 8&

x0~eiarr 8 f r f r 81H.c.!,

Sc f52E dt(r

~brcr f r1H.c.!,

Sb5E dt(r

4ubr u2

JK. ~16!

03511

s

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tc-ep

-t,

llyonen

sdct

i

n

As usual, the fielda0 is introduced to impose the constraithat there is one spinon per site and may be interpreted atime component of the gauge field. By assumptionbr is notcondensed. It is useful to start by completely ignoringcoupling betweenc and f fermions. The action for thef par-ticles describes a Fermi surface of spinons coupled to a cpact U~1! gauge field.

An important simplification for the three-dimensional sytems of interest~as compared tod52) is that the U~1! gaugetheory admits a deconfined phase where the spinons potially survive as good excitations of the phase. In what flows we will assume that the system is in such a deconfiphase.~This is formally justified in the same large-N limit asthe one for the mean-field approximation.! This deconfinedphase has a Fermi surface of spinons coupled minimallygapless ‘‘photon’’@U~1! gauge field#. ~Due to the compact-ness of the underlying gauge theory, there is also a gapmonopole excitation.! Thus two static spinons interact witeach other through anemergent long-range1/r Coulomb in-teraction. Putting back a small coupling between thec and fparticles will not change the deconfined nature of this pha~In particular the monopole gap will be preserved.! This isthe advocated U~1! FL* phase.

A. Specific heat

The coupling of the massless gauge photon to the spiFermi surface leads to several interesting modificationsthe mean-field results. First, consider the effect of the spacomponents of the gauge field. It is useful to work in tgauge¹W •aW 50 so that the vector potential is purely tranverse. Unless otherwise stated, we assume a generic spFermi surface~without flat portions! henceforth. Integratingout the spinons and expanding the resulting action to qdratic order gives the following well-known form for thpropagator for these transverse gauge fluctuations:

Di j ~kW ,ivn![^ai~kW ,ivn!aj~2kW ,2 ivn!&5d i j 2kikj /k2

Guvnu/k1x fk2

.

~17!

HereG, x f are positive constants that are determined bydetails of the spinon dispersion, andvn is an imaginary Mat-subara frequency. Note that the gauge fluctuations are odamped in the small-q limit. As was first shown in a differentcontext by Holsteinet al.38 ~and reviewed in Appendix D!,this form of the gauge-field action leads to aT ln 1/T singu-larity in the low-temperature specific heat. Thus the speciheat coefficientg5C/T diverges logarithmically at low temperature in the U~1! FL* phase.

We also briefly mention the effect of the longitudin~time component! of the gauge field. This couples to thlocal f fermion density, and so its influence is very much lia repulsive density-density interaction. The longitudingauge-field propagator has a structure very similar to thaa standard random-phase approximation density-fluctuapropagator, and so does not lead to any non-Fermi-liqbehavior.

1-7

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B. Magnetic instability

The repulsive interaction mediated by the longitudinpart of the gauge interaction can lead to various instabiliof the spinon Fermi surface. In particular, it is interestingconsider a SDW instability of the spinon Fermi surface. Tresulting state will have magnetic long-range order tcould potentially have a weak moment as it is a SDW stthat is formed out of the spinon Fermi surface. Howevercontrast to the traditional view of the weak magnetism, hthe SDW instability isnot that of the large-Fermi-surfacheavy Fermi liquid. Despite the occurrence of magnelong-range order, this magnetic state is far from convtional. Because the SDW order parameter is gauge neuthe presence or absence of a SDW condensate has littlestantive effect on the structure of the gauge fluctuations.deed, the latter remain as in the U~1! FL* state, even afterthe magnetic order has appeared in the descendant~1!SDW* state. The spinons continue to be deconfined andcoupled to a gapless U~1! gauge field. Further, the monoposurvives as a gapped excitation—this yields a sharp disttion with more conventional magnetic phases. These gaexcitations coexist with the gapless magnons associatedbroken spin rotation invariance and with a Fermi surfacethe conduction electrons. However, due to the broken tralational symmetry in this state, there is no sharp distinctbetween small and large Fermi surfaces. So to reiterateexotic magnetic metal, dubbed U~1! SDW* , emerges as alow-energy instability of the spinon Fermi surface of the pent U~1! FL* state.

Different possibilities emerge for the formation of thspin-density wave out of the parent U~1! FL* phase, depending on the details of the spinon Fermi surface andstrength of the interactions driving the SDW instability. Wenumerate some of them below.

~a! Perfectly nested spinon Fermi surface: In this caarbitrarily weak interactions will drive a SDW instability. Ithe resulting state, the spinons are gapped. So upon inteing out the spinons, the effective action for the gauge fican be expanded safely in spatial and temporal gradiewith no long-range couplings. Gauge invariance nowmands that these terms in the gauge-field action havestandard Maxwell form. Consequently, the photon becomsharp propagating mode at low energies~below the spinongap! with linear dispersion. Despite clearly being a distinphase from conventional spin-density-wave metals, theperimental distinction is subtle.

~b! Generic spinon Fermi surface, weak interaction: Fogeneric spinon Fermi surface, the leading spin-density-winstability ~which will require an interaction strength beyonsome threshold value! will be at a wave vector that matcheone of the ‘‘2kF’’ wave vectors of the spinon Fermi surfacIn the resulting state, a portion of the spinon Fermi surfa~away from points connected by the ordering wave vec!survives intact. The damping of the gapless U~1! gauge fluc-tuations due to coupling to gapless spinons is preserConsequently the low-temperature specific heat will continto behave asC(T);T ln(1/T). Thus for this particular U~1!SDW* state its non-Fermi-liquid nature is readily manifest

03511

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by specific-heat measurements, providing a concreteample of a weak moment SDW metal with non-Fermi-liquthermodynamics at low temperature.

~c! Generic spinon Fermi surface, strong interaction:the interactions are strong enough, even for a non-nespinon Fermi surface, the spinons can develop a full gap wno portion of their Fermi surface remaining intact. The rsulting phase is the same as that obtained in~a!, and has asharp propagating linear dispersing photon at low energi

In Sec. VI we discuss experimental probes that can hdistinguish these U~1! SDW* phase from the conventionaspin-density-wave metals.

C. Mean-field theory with magnetism

In view of the possible occurrence of SDW phaseswill now consider a modified mean-field theory which catures the magnetic instability at the mean-field level, bdoes no longer correspond to a large-N saddle point. We willdiscuss the fully self-consistent solution of the mean-fiequations for arbitrary temperature and external magnfield.

The mean-field Hamiltonian, written down explicitly foSU~2! symmetry, takes the following form:

Hmf5(k

ekcka† cka2 (

^rr 8&~x rr 8

* f ra† f r 8a1H.c.!

1(r

m f ,r f ra† f ra2(

rbr~cra

† f ra1H.c.!

11

2 (r

~HW eff,r1HW ext!• f ra† sW aa8 f ra8

11

2HW ext•(

rcra

† sW aa8cra81Econst, ~18!

whereHW ext is the external field, and we have allowed forspatial dependence of the mean-field parametersm f ,r , x rr 8 ,br , HW eff,r . They have to be determined from the followinequations:

15^ f ra† f ra&, ~19!

2br5JK^cra† f ra&, ~20!

2x rr 85~12x!JH^ f ra† f r 8a&, ~21!

HW eff,r5xJH(r 8

MW r 8 , MW r51

2^ f ra

† sW aa8 f ra8&, ~22!

where the last sum runs over the nearest neighborsr 8 of siter. We have introduced a parameterx which allows to controlthe balance between ordered local-moment magnetismspin-liquid behavior of thef electrons. A valuex51/2 wouldcorrespond to an unrestricted Hartree-Fock treatment oforiginal Heisenberg interaction; we will employ valuesx

1-8

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ean

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,1/2 in order to model aweakmagnetic instability of thespinon Fermi-surface state. The constant piece of the Hatonian reads

Econst52(r

m f ,r1(r

2br2

JK1(

rr 8

2ux rr 8u2

~12x!JH

21

2 (r

HW eff,r•MW r . ~23!

For simplicity, we consider a simple cubic lattice, aassume a tight-binding dispersion for the conduction etrons, ek522t(a51,2,3cos(ka)2mc , where mc controls theconduction-band filling. The mean-field equations canself-consistently solved using a large unit cell, allowing fspatially inhomogeneous phases.39 In this section we restricour attention to mean-field solutions where thex rr 85x0fields are real~time-reversal invariant! and obey the full lat-tice symmetries, andbr5b0 is site independent. We emploa 231 unit cell, then antiferromagnetism is characterizedMW r• x5Msexp(iQ•r ) where Q5(p,p,p) is the antiferro-magnetic wave vector, andx is the magnetization axis~which is arbitrary in zero external field!.

In Fig. 4 we show a phase diagram obtained from sconsistently solving Eq.~18! together with the above meanfield equations at zero external magnetic field. A U~1! FL*phase withb050 andx0Þ0 is realized at intermediate temperatures. As expected, it is unstable to magnetic ordelow T, resulting in a U~1! SDW* ground state for smalJK—this phase has in additionMsÞ0. For the present parameter values, the spinon Fermi surface is gapped out inSDW* phase. IncreasingJK drives the system into the FLphase withb0Þ0, x0Þ0, andMs50; at low temperatures a

FIG. 4. Mean-field phase diagram ofHmf ~18! on the cubiclattice, as function of Kondo couplingJK and temperatureT. Pa-rameter values are electron hoppingt51, Heisenberg interactionJH50.1, decoupling parameterx50.2, and conduction-band fillingnc50.7. Thin~thick! lines are second-~first-! order transitions. The‘‘decoupled’’ phase is an artifact of the mean-field theory, andcorresponding transitions will become crossovers upon includfluctuations, as will the transition between the FL and U~1! FL*phases; the transitions surrounding the SDW and SDW* phases willof course survive beyond mean-field theory.

03511

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-

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conventional SDW phase intervenes where allb0 , x0 , Msare nonzero. Note that the transition between FL and SDWweakly first order at low temperatures. At high temperatuthe mean-field theory only has a ‘‘decoupled’’ solution wib05x05Ms50—this decoupling is a well-known meanfield artifact and reflects the presence of incoherent exctions.

In the FL phase, the above-mentioned Lifshitz transitioccurs atJK'1.7 in the low-temperature limit, i.e., forJK.1.7 only a single Fermi-surface sheet remains. Notethis transition does not lead to strong singularities inmean-field parameters.

The staggered magnetizations of the SDW and SDW*states as determined from the mean-field solution are shin Fig. 5; we can expect that fluctuation corrections will sinificantly reduce these mean-field values. We have also sied different values of the decoupling parameterx; in particu-lar smaller values ofx lead to a suppression of orderemagnetism in favor of the nonmagnetic FL* state, i.e., theSDW instability of FL* is shifted to lower temperatures~andbecomes completely suppressed at smallx); similarly, theordered moment in the SDW phases is decreased withcreasingx.

Interesting physics obtains when an external magnfield is turned on, and the corresponding mean-field phdiagram is discussed in Appendix C.

V. FLUCTUATIONS NEAR THE FERMI-VOLUMECHANGING TRANSITION

We now turn to the effects of fluctuations beyond tmean-field theory at the phase transition between the FLU~1! FL* phases. In mean-field theory, this transition occthrough the condensation of the slave boson fieldb. Such acondensation survives as a sharp transition beyond mfield only whenT50.

We begin by observing that in the mean-field theorythe important changes near the transition occur at theFermi surface. The cold Fermi surface~essentially made upof c particles! plays a spectator role. We therefore integra

eg

FIG. 5. Staggered magnetization determined from the mefield solutionHmf ~18!. Parameters are as in Fig. 4, the two curvcorrespond to two horizontal cuts of the phase diagram in Fig. 4T50, the first-order character of the SDW-FL transition is cleaseen. Note that smaller values of the decoupling parameterx yieldsmaller values of the magnetization in the SDW and SDW* phases.

1-9

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out the c fields completely from the action in Eq.~16! toobtain an effective action involving theb, f , and gaugefields alone. We also partially integrate outf excitations wellaway from the hot Fermi surface: this changes theb effectiveaction from the simple local term in Eq.~16!, and endows itwith frequency and momentum dependence. In this manwe obtain the following effective action at long distance atime scales:

S5Sb1Sf , ~24!

Sb5E dtd3r F bS ]t2mb2 ia02~¹Wr2 iaW !2

2mbD b

1u

2ubu41•••G , ~25!

and Sf has the same form as in Eq.~16!. Notice that thebfield has become a propagating boson, with the same tein the action as a microscopic canonical boson: here thterms arise from a (b, f ) fermion polarization loop integratewell away from thef-Fermi surface. The parametersmb ,mbmay be interpreted as the chemical potential and mass obosons, respectively. The (b, f ) fermion loop will also lead tohigher time and spatial gradient terms as well as a densdensity coupling betweenb andf in Eq. ~24!, but all these areformally irrelevant near the quantum-critical point of inteest.

A key feature of Eq.~24!, induced by taking the spatiaand temporal continuum limit, is that we have lost informtion on the compactness of the U~1! gauge fielda, i.e., thecontinuum action is now no longer periodic underarr 8→arr 812p, as was the case for the lattice action~16!. TheU~1! gauge field is now effectively noncompact, and conquently monopole excitations have been suppressed.monopole gap is finite in the U~1! FL* phase~which is theanalog of the ‘‘Coulomb’’ phase of the compact gautheory!.40 In the FL phase, the monopoles do not exist—thare confined to each other. This occurs due to the condetion of the boson field. However, the monopole gap isexpected to close at the transition,42 and so neglecting thecompactness of the gauge field is permissible. Indeed,continuum action~24! provides a satisfactory description othe critical properties of the FL to U~1! FL* transition atT50. However, as we noted in the caption of Fig. 1, tcompactness of the gauge field is crucial in understandthe absence of aT.0 phase transition above the FL phase21

The action in Eq.~24! above is similar to that popular ingauge theory descriptions22,23 of the normal state of opti-mally doped cuprates but with some crucial differences. Hthe chemical potential of the bosons is fixed while in Re22 and 23 the boson density was fixed; as we will see bethis significantly modifies the physical implications of thcritical theory, and the nature of the non-Fermi-liquid criticsingularities asT.0. Furthermore, we are interested speccally in d53, as opposed to thed52 case considered inRefs. 22 and 23.

The phase diagram of the action~24! was sketched in Fig1. The horizontal axis, represented in Fig. 1 byJK , is now

03511

er

sse

he

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-

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ysa-t

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accessed by varyingmb . Without any additional~formallyirrelevant! second-order time derivative terms forb in theaction, the quantum-critical point between the FL and U~1!FL* phases occurs precisely atmb50, T50. We will nowdiscuss the physical properties in the vicinity of this criticpoint first atT50, and then atT.0, followed by an analy-sis of transport properties using the quantum Boltzmaequation in Sec. V C. Section V D will comment on the efect of the SDW or SDW* phases that may appear at velow temperatures~these are not shown in Fig. 1, but sketchin Fig. 2!.

A. Zero temperature

In a mean-field analysis of Eq.~24!, we see that the FL*phase~the Coulomb phase of the gauge theory! obtains formb,0 with ^b&50, while the FL phase~the ‘‘Higgs’’ phaseof the gauge theory! obtains formb.0.

Consider fluctuations formb,0 in the FL* phase. Here,there are no bosons in the ground state, and all self-encorrections associated with the quartic couplingu vanish.41

The gauge-field propagator is given by Eq.~17!, and thisdoes contribute a nonzero boson self-energy. At small mmentap and imaginary frequenciese, the boson self-energyhas the structure~determined from a single gauge-boson echange process, as in Refs. 22 and 23!

Sb~k,i e!;k2~11c1ueu ln~1/ueu!1••• !, ~26!

wherec1 is some constant. Apart from terms which renomalize the boson massmb , these self-energy corrections aless relevant than the bare terms in the action, and so casafely neglected near the critical point. Notice also thSb(0,0)50, and so the quantum-critical point remainsmb50.

The critical exponents can now be determined as in R20 and 41, and are simply those of the mean-field theoryEq. ~24!:

n51/2, z52, h50. ~27!

As in Eq. ~26! we can also determine the fate of the bosquasiparticle pole as influenced by the gauge fluctuationsobtain

ImSbS k,e5k2

2mbD;sgn~e!e2ln~1/ueu!. ~28!

The boson lifetime is clearly longer than its energy, and tpole remains well defined. Finally, we recall our statemenSec. IV A that the gauge fluctuations lead to aT ln(1/T) spe-cific heat in the FL* phase, with a divergingg coefficient.This behavior remains all the way up to, and including, tcritical point. Parenthetically, we note that the same calcution in d52 dimensions will yieldC}T2/3.

We turn next tomb.0, in the FL phase. Here the bosonare condensed, and Eq.~26! or explicit calculations showthat

^b&[b0;~mb!1/2;~JK2JKc!1/2, ~29!

1-10

eSndsopa

thifi

D,

thi-

a

-

arb

nd

tioofe

in

aity

leti

uaa

lln-

ee

m-f

theta-ws

n

ptiche

s

y

thea-44thedif-s of

asnc-


whereJKc is the position of the critical point in Fig. 1. Thtransverse gauge-field propagator may be obtained as inIV A by integrating out both the bosons and fermions aexpanding the resulting action to quadratic order; the bocondensate leads to a ‘‘Meissner’’ term in the gauge progator so that Eq.~17! is replaced by

Di j ~kW ,ivn![^ai~kW ,ivn!aj~2kW ,2vn!&

5d i j 2kikj /k2

Guvnu/k1x fk21rs

. ~30!

Here rs is the boson ‘‘superfluid density,’’ and we havers

;b02. The presence of such a Meissner term cuts off

singular gauge fluctuations. The divergence of the specheat coefficientg(T) as a function of temperatureat thecritical point implies that it diverges atT50 on approachingthe transition from the FL side. As shown in Appendixthis is indeed the case, and we find thatg diverges asg; ln(1/b0). In experiments, such a divergingg is sometimesinterpreted as a diverging effective mass. Importantly,divergence ofg is unrelated to the singularity in the quasparticle residue on the hot Fermi sheet,Z, which obeysZ;b0

2 as shown in Eq.~14!, and so vanishes linearly asfunction of JK2JKc .

B. Nonzero temperatures

A crucial change atT.0 is that it is now no longer truethat Sb(0,0)50 in a region with^b&50. Instead, as in earlier studies of the dilute Bose gas,19,20 we have

Sb~0,0!52uE ddk

~2p!d

1

exp@k2/~2mbT!#21

5uz~3/2!

4p3/2~2mbT!3/2 in d53. ~31!

This behavior determines the crossover phase boundshown in Fig. 1. The physical properties are determinedthe larger of the two ‘‘mass’’ terms in theb Green’s function,umbu or Sb(0,0)—consequently, the crossover phase bouaries in Fig. 1 lie atT;umbu2/3;uJK2JKcu2/3. These bound-aries separate the U~1! FL* region at lowT andmb,0, andthe FL region at lowT and mb.0, from the intermediatequantum-critical region. Note that there is no phase transiin the FL region atT.0: this is due to the compactnessthe underlying U~1! gauge theory, and the fact that th‘‘Higgs’’ and ‘‘confining’’ phases are smoothly connecteda compact U~1! gauge theory in three total dimensions.21

We now briefly comment on the nature of the electrictransport in the three regions of Fig. 1. The behavior is qucomplicated, and we will first highlight the main results bsimple estimates in the present section. A more comppresentation based upon the quantum Boltzmann equaappears in Sec. V C.

The conventional FL region is the simplest, with the usT2 dependence of the resistivity—the gauge fluctuationsquenched by the ‘‘Meissner effect.’’

03511

ec.

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ec-

e

iesy

-

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teon

lre

In the U~1! FL* region, there is an exponentially smadensity of thermally excitedb quanta, and so the boson coductivity sb is also exponentially small. As in earlier work,43

the resistances of theb andf quanta add in series, and so thtotal b and f conductivity remains exponentially small. Thphysical conductivity is therefore dominated by that of thecfermions, which again has a conventionalT2 dependence.

Finally, we comment on the transport in the quantucritical region. This we will estimate following the method oRef. 22, with a more complete calculation appearing infollowing section. A standard Fermi’s golden rule compution of scattering off low-energy gauge fluctuations shothat a boson of energye has a transport scattering rate

1

tbtr~e!;TAe ~32!

for energiese!T2/3. From this, we may obtain the bosoconductivity by inserting in the expression

sb;E d3k tbtr~ebk!k2S 2

]n~ebk!

]ebkD , ~33!

where n(e)51/(ee/T21) is the Bose function, andebk5k2/(2mb)1Sb(0,0)5k2/(2mb)1c2T3/2 for some constantc2. Estimating the integral in Eq.~33! we find that there is anincipient logarithmic divergence at smallk which is cutoff bySb(0,0);T3/2, and sosb diverges logarithmically withT:

sb; ln~1/T!. ~34!

There are no changes to the estimate of thef conductivityfrom earlier work,22,23 and we haves f;T25/3. Using againthe composition rule of Ref. 43, we see that the asymptolow-temperature physical conductivity is dominated by tbehavior in Eq.~34!.

As an aside, we note that for the theory~24! in two spatialdimensions the result of Eq.~34! continues to hold, whereathe fermion part becomess f;T24/3. This implies that theasymptoptic low-T physical conductivity is dominated bEq. ~34! in d52 as well.

C. Quantum Boltzmann equation

We now address electrical transport properties oftheory~24! in more detail, using a quantum Boltzmann eqution. The analysis is in the same spirit as the work of Ref.but, as we have discussed in Sec. I A, the variation inboson density as a function of temperature leads to veryferent physical properties, and requires a distinct analysithe transport equation.

We saw in Sec. V B that the electrical conductivity wdominated by theb boson contribution, and so we focus othe time ~t! dependence described by the distribution funtion

f ~kW ,t !5^bk†~ t !bk~ t !&. ~35!

In the absence of an external~physical! electric fieldEW , wehave the steady-state valuef (kW ,t)5 f 0(k) with

1-11

edetin

qaarealfld

he.

c-

-

an

qu

hee

g

e

reic

up-In thelawade,om-theonw-d

u-


f 0~k![1

exp$@k2/~2mb!2mb1Sb~0,0!#/T%21, ~36!

with Sb(0,0) given in Eq.~31!. The transport equation in thpresence of a nonzeroEW (t) can be derived by standarmeans, and most simply by an application of Fermi’s goldrule. The bosons are assumed to scatter off a fluctuagauge field with a propagator given by Eq.~17! or ~30!, andthis yields the equation

] f ~kW ,t !

]t1EW ~ t !•

] f ~kW ,t !

]kW

52E2`

` dV

p E ddq

~2p!dImF kiDi j ~qW ,V!kj

mb2 G ~2p!

3dS k2

2mb2

~kW1qW !2

2mb2V D @ f ~kW ,t !@11 f ~kW1qW ,t !#

3@11n~V!#2 f ~kW1qW ,t !@11 f ~kW ,t !#n~V!#, ~37!

where n(V) is the Bose function at a temperatureT asabove.

We will now present a complete numerical solution of E~37! for the case of a weak, static electric field, to lineorder inEW . The analysis near the quantum-critical point pallels that of Ref. 45, with the main change being that instof the critical scattering appearing from the boson seinteractionu, the dominant scattering is from the gauge-fiefluctuations@note, however, that it is essential to include tinteraction u to first order in the self-energy shift in Eq~36!#. We write

f ~kW ,t !5 f 0~k!1kW•EW f 1~k!, ~38!

where we notice thatf 1 depends only on the modulus ofkand is independent oft. We now have to insert Eq.~38! intothe transport equation~37! and the expression for the eletrical current

JW~ t !5E ddk

~2p!d

kW

mbf ~kW ,t !, ~39!

linearize everything inEW , and so determine the proportionality betweenJW andEW .

It is useful to rewrite the equations in dimensionless qutities V5V/T, k5k/A2mbT, s5sb /(mbx f). Then it iseasy to see that the solution of the quantum Boltzmann etion at the critical coupling,mb50, is characterized by twoparameters

T5x f

2

G2~2mb!3T,

u5uz~3/2!

4p3/2

G

x f, ~40!

03511

ng

.r-d-

-

a-

where T is a reduced temperature, andu parametrizes thetemperature dependence of the effective ‘‘mass’’ of tbosons from Eq.~31!; G and x f are the parameters of thgauge propagator~17!. The linearized form of the Boltzmannequation~37! for the function

f 1~k![c~k/A2mbT! ~41!

is obtained as

2 f 08~ k!5E0

`

dk1@K1~ k,k1!c~ k!1K2~ k,k1!c~ k1!#

~42!

with f 08(x)5]/(]x2)@exp(x21uAT)21#21; the expressionsfor the functionsK1,2 are given in Appendix E. From thesolution of Eq.~42! one obtains the conductivity accordinto

s~ T,u!51

6p2ATE

0

`

dk k4 c~ k!. ~43!

The integral equation~42! was solved by straightforwardnumerical iteration on a logarithmic momentum grid. Wshow sample solutions for the functionk4 c( k) in Fig. 6. Thefinal results for the scaling function of the conductivity adisplayed in Fig. 7. For small temperatures, the logarithmdivergence ofsb(T) announced in Eq.~34! is clearly seen;for larger temperatures the conductivity is exponentially spressed due to the temperature-dependent boson mass.crossover region, the results could be fitted with a powerover a restricted temperature range of roughly one dechowever, no extended power-law regime emerges. In cparison with experiments, one has to keep in mind thatphysical resistivity is given by a sum of boson and fermiresistivities, and that the logarithmically decreasing lotemperature part of 1/sb(T) cannot be easily distinguishefrom a residual resistivity arising from impurities.

FIG. 6. Plot of the functionk4c( k) for a few values of the

reduced temperatureT and the interaction parameteru ~40!. c( k) isdefined in Eqs.~38! and ~41!, and has been obtained from the nmerical solution of the quantum Boltzmann equation~42!.

1-12

seAono

aha

herm

etheha

itiuad

n

io

thns

is-ons

w-the

ally

ithortllnd

ivitytheer.cha

eargap-eri-

toized

iskenot a

n isicnge a

lat-a

ilyrmiandst-ith

eticay

heat

sted

ially

verinx-

owmi-e

ua


D. SDW order

Our discussion so far has focused primarily on the croover between the FL to U~1! FL* phases, as this captures thprimary physics of the Fermi-volume changing transition.low T, we discussed in Sec. IV B that the longitudinal partthe gauge fluctuations may induce SDW order on the spiFermi surface of the FL* phase ~leading to the SDW*phase!. On the FL side of the transition, the gauge fluctutions are formally gapped by the Anderson-Higgs mecnism. They will, however, still mediate a repulsive~thoughfinite ranged! interaction between the quasiparticles at thot Fermi surface. Furthermore the shape of the hot Fesurface evolves smoothly from the spinon Fermi surfacthe FL* phase. Consequently, it is to be expected thatSDW order will continue into the FL region up to somdistance away from the transition. Thus it seems unlikely tthere will be a direct transition from SDW* to FL at zerotemperature. The actual situation then has some similarto the mean-field phase diagram in Fig. 4. However, flucttions will strongly modify the positions of the phase bounaries, and we expect that the U~1! FL* region actually occu-pies a larger portion of the phase diagram. Also there issharp transition between the FL and U~1! FL* regions~un-like the mean-field situation in Fig. 4!, and there is insteadexpected to be a large intermediate quantum-critical regas shown in Fig. 2.

VI. EXPERIMENTAL PROBES OF THE U „1… SDW* STATE

In this section, we discuss experimental signatures ofU~1! SDW* phase focusing particularly on the distinctio

FIG. 7. Scaling function for the boson conductivity,s

5sb /(mbx f), as function of the reduced temperatureT for differ-

ent values of the interaction parameteru ~40!. The results are ob-tained from the numerical solution of the quantum Boltzmann eq

tion ~42! together with Eq.~43!. Top panel: conductivitys(T) on a

log-log scale. Bottom panel: resistivity 1/s(T) on a linear scale.

03511

s-

tfn

--

iife

t

es-

-

o

n

e

with more conventional SDW metals.We begin by considering a U~1! SDW* phase in which a

portion of the spinon Fermi surface remains intact. As dcussed in Sec. IV, the coupling between the gapless spinand the gauge field leads to singularities in the lotemperature thermodynamics in this phase. In particularspecific heat behaves asC(T);T ln(1/T) at low tempera-ture. Thus this phase is readily distinguished experimentfrom a conventional SDW. Electrical transport in this U~1!SDW* phase will be through the conduction electrons wno participation from the spinons. Thus electrical transpwill be Fermi-liquid-like. In contrast thermal transport wireceive contributions from both the conduction electrons athe gapless spinons. Consequently the thermal conductwill be in excess of that expected on the basis ofWiedemann-Franz law with the free-electron Lorenz numb

The distinction with conventional SDW phases is mumore subtle for U~1! SDW* phases where the spinons havefull gap. In this case, there is a propagating gapless lindispersing photon which is sharp. The presence of theseless photon excitations potentially provides a direct expmental signature of this phase. It is extremely importantrealize that the emergent gauge structure of a fractionalphase is completely robust to all local perturbations, andnot to be confused with any modes associated with brosymmetries. Thus despite its gaplessness the photon is nGoldstone mode. In fact, the gaplessness of the photoprotectedeven if there are small terms in the microscopHamiltonian that break global spin rotation invariance. Beigapless with a linear dispersion, the photons will contributT3 specific heat at lowT which will add to similar contribu-tions from the magnons and the phonons of the crystaltice. In addition, the conduction electrons will contributelinear T term. The phonon contribution is presumably eassubtracted out by a comparison between the heavy Feliquid and magnetic phases. To disentangle the magnonphoton contributions, it may be useful to exploit the robuness of the photons to perturbations. Thus for materials wan easy-plane anisotropy, application of an in-plane magnfield will gap out the single magnon, but the photon will stgapless and will essentially be unaffected~at weak fields!.Thus careful measurements of field-dependent specificmay perhaps be useful in deciding whether the U~1! SDW*phase is realized.

Finally, quasielastic Raman scattering has been suggeas a probe of the U~1! gauge-field fluctuations46 in the con-text of the cuprates—the same prediction applies essentunchanged here to the fractionalized phases ind53.

Conceptually the cleanest signature of the U~1! SDW*phase would be detection of the gapped monopole. Howeat present we do not know how this may be directly doneexperiments. Designing such a ‘‘monopole detection’’ eperiment is an interesting open problem.

VII. DISCUSSION

The primary question which motivates this paper is hto reconcile a weak-moment magnetic metal with non-Ferliquid behavior close to the transition to the Fermi liquid. W

-

1-13

ileaetami

uath

thea

iourioeth

.te

or

onsi,ise

threalfewifis

dri

obtisticrmrgexrmtur.av

inmr.

n

ave

bye

pinialal

malil-s-e-veis

lsoid

e

ical

rsnsthethate

enaa

en-

A.

fulnstheR-6res

P.ity,

r-ace


have explored one concrete route toward such a reconction. The U~1! SDW* magnetic states discussed in this papmay be dubbed spin-charge separated spin-density-wmetals. They constitute a class distinct from both the convtional spin-density-wave metal and the local-moment mementioned in the Introduction. However, they share a nuber of similarities with both conventional metals. Just asthe conventional local-moment metal, in the U~1! SDW*state the local moments do not participate in the Fermi sface. Despite this the ordering moment may be very smIndeed this state may be viewed as a spin-density wavehas formed out of a parent nonmagnetic metallic state wismall Fermi surface. This parent state is a fractionalizFermi liquid in which the local moments have settled intospin liquid and essentially decoupled from the conductelectrons. The spinons of the spin liquid form a Fermi sface which undergoes the SDW transition—this transitdoes not affect the deconfinement property of the gauge fibecause the SDW order parameter is gauge neutral anddoes not effectively couple to the gauge-field excitations

We showed that in the region of evolution from this stato the conventional Fermi liquid, non-Fermi-liquid behaviobtains~at least at intermediate temperatures!. We also ar-gued that the underlying transition that leads to this nFermi-liquid physics is the Fermi-volume changing trantion from FL to FL* . Despite the jump in the Fermi volumethis transition is continuous and characterized by the vaning of the quasiparticle residueZ on an entire sheet of thFermi surface~the hot Fermi surface! on approaching thetransition from the FL side.

A specific heat that behaves asT ln(1/T) is commonlyobserved in a variety of heavy-fermion materials close totransition to magnetism. In the context of the ideas exploin this paper, such behavior of the specific heat is naturobtained inthree-dimensionalsystems. A small number oheavy-fermion materials exhibit such a singular specific heven in the presence of long-ranged magnetic order. Ashave emphasized, precisely such non-Fermi-liquid specheat obtains in one of the exotic magnetic metals discusin this paper@the U~1! SDW* phase with a partially gappespinon Fermi surface#. It would be interesting to check foviolations of the Wiedemann-Franz law at low temperaturesuch materials.

A general point emphasized in this paper is that theserved non-Fermi-liquid physics near the onset of magneactually has little to do with fluctuations of the magneorder parameter. Rather we propose that the non-Feliquid physics is associated with the destruction of the laFermi surface. The concrete realization of this pictureplored in this paper is that the destruction of the large Fesurface leads to a fractionalized Fermi liquid which evenally ~at low temperature! develops spin-density-wave ordeAs we discussed extensively, the resulting spin-density-wstate is an exotic magnetic metal.

It is also of interest to consider a different scenariowhich the small-Fermi-surface state is unstable at low teperature towardconfinementof spinons and magnetic ordeIt is particularly interesting to consider such a scenario ind52. The physics of the Fermi-volume changing fluctuatio

03511

ia-rven-l-

n

r-ll.atad

n-nld,us

--

h-

edly

atec

ed

n

-m

i-e-i-

e

-

s

is again described by a theory of condensation of a slboson field coupled to a Fermi surface of spinons by a U~1!gauge field. For a noncompact U~1! gauge field, such atheory has a number of interesting properties. As noticedAltshuler et al.,47 the spin susceptibility at extremal wavvectors of the spinon Fermi surface has~possibly divergent!singularities due to the gauge fluctuations. Indeed the sphysics of this model is critical and described by a nontrivfixed point. The dynamical susceptibility at these extremwave vectors and at a frequencyv is expected to satisfyv/Tscaling. For a general spinon Fermi surface these extrewave vectors will chart out one-dimensional lines in the Brlouin zone at which critical scattering will be seen in inelatic neutron scattering. A spin-density-wave instability can dvelop out of this critical state at a particular extremal wavector where the amplitude of the diverging susceptibilitythe largest. Arguments very similar to those in Sec. V B ashow that transport will be governed by non-Fermi-liqupower laws in this theory.

There is a strikingqualitativeresemblance between thesresults and the experiments on CeCu62xAux . At the criticalAu concentration neutron-scattering experiments see critscattering on lines in the Brillouin zone satisfyingv/Tscaling.5 Magnetic ordering occurs at particular wave vectoon this line. Furthermore, empirically the spin fluctuatioappear to be quasi-two-dimensional, suggesting thatideas sketched above may indeed be relevant. We notethey significantly differ from earlier proposals to explain thbehavior of CeCu62xAux .48 As mentioned in the text, thespecific heat in thed52 quantum-critical region will havethe formC/T;T21/3; interestingly, such a behavior has beobserved in YbRh2Si2 in the low-temperature regime nearquantum-critical point.49 On the theoretical front, there arenumber of conceptual issues50 related to the legitimacy ofignoring the compactness of the gauge field ind52. Devel-oping a more concrete theoretical description of these geral ideas is an interesting challenge for future work.

ACKNOWLEDGMENTS

We thank P. Coleman, M.P.A. Fisher, E. Fradkin,Georges, L. Ioffe, Y.-B. Kim, G. Kotliar, A. Millis, N.Prokof’ev, T. M. Rice, Q. Si, M. Sigrist, A. Tsvelik, andX.-G. Wen for useful discussions. T.S. is particularly grateto Patrick Lee for a number of enlightening conversatiothat clarified his thinking. This research was supported byMRSEC program of the U.S. NSF under Grant No. DM0213282 ~T.S.!, by U.S. NSF Grant No. DMR 009822~S.S.!, and by the DFG Center for Functional Nanostructuat the University of Karlsruhe~M.V.!. T.S. also acknowl-edges funding from the NEC Corporation, and the AlfredSloan Foundation and the hospitality of Harvard Universwhere part of this work was done.

APPENDIX A: OSHIKAWA’S ARGUMENT ANDTOPOLOGICAL ORDER

Oshikawa has presented51 an elegant nonperturbative agument demonstrating that the volume of the Fermi surf

1-14

emve

r,a

itadiu-be

a’r ae

la

a

x

ar

anlic

ornined

umrtte

-

r

m

a-gi-ed,

r ofns-

to a

spec-vo-r-

in

onsr,

in-


is determined by the total number of electrons in the systIn our previous work,11 and in the present paper, we haargued for the existence of a nonmagnetic FL* state with adifferent Fermi-surface volume. As we discussed earlie11

this apparent conflict is resolved when we allow for globtopological excitations in Oshikawa’s analysis; such exctions emerge naturally in the gauge theories we havecussed for the FL* state. In other words, Oshikawa’s argment implies that violation of Luttinger’s theorem mustaccompanied by topological order.

In this appendix, we briefly recall the steps in Oshikawargument, and show how it can be modified to allow fosmall Fermi surface in a FL* state. As far as possible, wfollow the notation of Oshikawa’s paper.51

For definiteness, consider a two-dimensional Kondotice with a unit cell of lengthsax,y . The ground state isassumed to be nonmagnetic, with equal numbers of up-down-spin electrons. Place it on a torus of lengthsLx,y , withLx /ax , Ly /ay coprime integers. Adiabatically insert a fluF52p (\5c5e51) into one of the holes of the torus~saythe one enclosing thex circumference!, acting only on theup-spin electrons. Then the initial and final Hamiltoniansrelated by a unitary transformation generated by

U5expS 2p i

Lx(

rnr↑D , ~A1!

wherenr↑ is the number operator of all electrons~includingthe local moments! with spin up on the siter. After perform-ing the unitary transformation to make the final Hamiltoniequivalent to the initial Hamiltonian, the final and initiastates are found to have a total crystal momentum whdiffers by

DPx52p

Lx

LxLy

v0

ra

2 S mod2p

axD , ~A2!

wherev05axay is a volume of a unit cell, the second facton the right-hand side~rhs! counts the number of unit cells ithe system, andra52ra↑ is the mean number of electronsevery unit cell. Clearly the crystal momentum is definmodulo 2p/ax , and hence the modulus in Eq.~A2!.

Now imagine computing the change in crystal momentby studying the response of the quasiparticles to the inseflux. As shown by Oshikawa, the quasiparticles associawith a Fermi surface of volumeV lead to a change in momentum which is

DPxq5

2p

Lx

V~2p!2/~LxLy!

S mod2p

axD , ~A3!

where the second factor on the rhs counts the numbequasiparticles within the Fermi surface. EquatingDPx andDPx

q , and the corresponding expressions forDPy andDPyq ,

Oshikawa obtained the conventional Luttinger theorewhich applies to the volumeV5VFL of the Fermi surface inthe FL state,

03511

.

l-s-

s

t-

nd

e

h

edd

of

,

2v0

~2p!2VFL5ra~mod 2!. ~A4!

In the FL* state, there are additional low-energy excittions of the local moments that yield an additional topolocal contribution to the change in crystal momentum. Indethe influence of an insertion of fluxF is closely analogous tothe transformation in the Lieb-Schultz-Mattis52 argument,and it was shown53,54that a spin-liquid state ind52 acquiresthe momentum change

DPxt 5

p

ax

Ly

ayS mod

2p

axD , ~A5!

where the second factor on the rhs now counts the numberows which have undergone the Lieb-Schultz-Mattis traformation. Now usingDPx5DPx

q1DPxt , we now obtain the

modified Luttinger theorem obeyed in the FL* phase:

2v0

~2p!2VFL* 5~ra21!~mod2!. ~A6!

It is clear that the above argument is easily extendedZ2 FL* state ind53. The case of U~1! FL* state ind53 issomewhat more delicate because there is now a gaplesstrum of gauge fluctuations which can contribute to the elution of the wave function under the flux insertion; nevetheless, the momentum change in Eq.~A5! corresponds to anallowed gauge flux, and we expect that Eq.~A5! continues toapply.

APPENDIX B: TOY MODEL WITH U „1…FRACTIONALIZATION AND A SPINON FERMI SURFACE

In this appendix, we will display a concrete modelthree spatial dimensions that is in a U~1! fractionalized phasein three dimensions, and has a Fermi surface of spincoupled to a gapless U~1! gauge field. As discussed earliethis spinon Fermi surface could eventually~at low energies!undergo various instabilities including in particular to a spdensity-wave state.

Consider the following model:

H5Htc1HD1Hb1Hu1HU ,

Htc52 (^rr 8&

t~c r†c r 81h.c.!,

HD5D (^rr 8&

eifrr 8~c r↑† c r 8↓

†2c r 8↑

† c r↓† !1H.c.,

Hb52w ([ rr 8r 9]

cos~f rr 82f rr 9!,

Hu5u (^rr 8&

nrr 82 ,

1-15

of

se

emfirtwe

ce

drg

i

hebein

ll-o

s

d

f in-

-be

tohes

l

ceili-

areuge

aw-r-

ari-her

the

gh

held.d in

iza-he

dg

i-


HU5U(r

~Nr21!2. ~B1!

Herec r destroys a spinful charge-1 electron at each sitecubic lattice in three spatial dimensions;eifrr 8 creates acharge-2, spin-0 ‘‘Cooper pair’’ that resides on the linknrr 8 is conjugate tof rr 8 and may be regarded as the Cooppair number associated with each link.Nr is the total chargeassociated with each site and is given by

Nr5 (r 8Pr

nrr 81c r†c r . ~B2!

The HamiltonianH may be regarded as describing a systof electrons coupled with strong phase fluctuations. Theterm in Hb represents Josephson coupling between‘‘nearest-neighbor’’ bonds.Hu penalizes fluctuations in thCooper pair number at each bond.HU penalizes fluctuationsin the total chargeNr that can be associated with each lattisite. The total charge of the full system clearly is

Ntot5(r

Nr . ~B3!

Depending on the various model parameters, severaltinct phases are possible. Here we focus on the limit of laU. DiagonalizingHU requires that the ground state~s! satisfyNr51 at all sitesr. There is a gap of orderU to states that donot satisfies this condition. Clearly the system is insulatingthis limit.

The conditionNr51 for all r still allows for a huge de-generacy of ground states which will be split once the otterms in the Hamiltonian are included. This splitting maydescribed by deriving an effective Hamiltonian that livesthe space of degenerate states specified byNr51. As dis-cussed in Ref. 14, this effective Hamiltonian may be usefuviewed as a~compact! U~1! gauge theory. This may be explicitly brought out in the present case by the changevariables

f rr 85e rarr 8 ,

nrr 85e rErr 8 ~B4!

c ra5 f ra for r PA,

c ra5 isaby f rb

† for r PB. ~B5!

Heree r511 on theA sublattice and21 on theB sublattice.In terms of these variables, the constraintNr51 reads

¹W •EW 1 f r†f r51 ~B6!

at each siter. ~We note that bothaW andEW may be regarded avector fields defined on the lattice.! At orderw2/U, u, t, D,the effective Hamiltonian takes the form

Heff5HK1Hu1H f ,

03511

a

.r

sto

is-e

n

r

y

f

HK52K(P

cos~¹W 3aW !,

Hu51u(r

EW 2,

H f52D (^rr 8&

~eiarr 8 f r†f r 81H.c.!. ~B7!

Here K52w2/U, and the sum(P runs over elementaryplaquettes.Heff together with the constraint may be vieweas a Hamiltonian for a compact U~1! gauge theory coupled toa gauge charge-1 fermionic matter fieldf. ~Note that to lead-ing order thet term does not contribute.! Heff still admitsseveral different phases depending on its parameters. Oterest to us is the limitK;w2/U@u. In this limit, mono-poles of the compact U~1! gauge field will be gapped. Consequently at low energies, we may take the gauge field tononcompact. The cos(¹W 3aW) term can then be expandedquadratic order to get the usual Maxwell dynamics for tgauge field. Thef particles form a Fermi surface which icoupled to this gapless U~1! gauge field. Note that in thelow-energy manifold withNr51 at all r, all excitations havezero physical electric charge. Thus thef particles are neutrafermionic spinons.

As with any Fermi surface, this spinon Fermi-surfastate could at low energies further undergo various instabties to other states~density waves, pairing, etc.! dependingon the residual interactions between the spinons. Therevarious sources of such interactions: First there is the gainteraction that is explicit inHeff in the leading order. Thenthe t term contributes toHeff at second order and leads toquartic spinon-spinon interaction as well. The specific loenergy instability of the spinon Fermi surface will be detemined by the details of the competition between these vous sources of interaction, and will not be discussed furthere for this model.

Apart from the deconfined phase discussed above,model possesses confined phases; for largeU those occur forsmalleru, and the deconfinement transition occurs throuthe condensation of monopoles in the gauge field.

APPENDIX C: MEAN-FIELD PHASE DIAGRAM IN ANEXTERNAL ZEEMAN MAGNETIC FIELD

In this appendix we briefly discuss the behavior of tmean-field theory of Sec. IV C in an externally applied fieA sample zero-temperature phase diagram is displayeFig. 8, which shows very rich behavior.

The phases at small fields are straightforward generaltions of the low-temperature zero-field phases of Fig. 4: TU~1! SDW* has weakly polarized conduction electrons,b050, nonzerox0 indicating spinon hopping, and a cantespinon magnetizationMW r with a staggered component alonx and a uniform component alongz. The SDW phase hassimilar characteristics, but nowb0Þ0 indicating a conven-tional weakly field-polarized magnet with confinement. Fnally, the FL phase hasb0Þ0, x0Þ0, weakly polarized

1-16

s-

de.

Osb

mak

Lgyrm

av

FLinn

acnangnon

eliengintior

theheeInuc-

theeroy

be

th


heavy quasiparticles, and the mean-field parameterMW r hasonly a uniformz component.

In the small-JK region, increasing external field progresively suppresses the effect ofJH . At intermediate field, aphase with ‘‘canted’’f moments arises, where nowx05b0

50 ~no spinon hopping!, and MW r is canted as describeabove. Larger fields fully polarize the local moments, i.MW r points uniformly alongz with maximum amplitude, andx05b050. This phase is also realized for largerJK andlarge fields—here the field quenches the Kondo effect.the Fermi-liquid side of the phase diagram, two more phaarise in the present mean-field theory which are labeledFL2 and FL3 in Fig. 8; both have nonzerob0 andx0. In FL2,the mean-field parameterMW r has both staggered and uniforcomponents, i.e., this phase describes canted, wescreened local moments. Turning to the FL3 phase, this high-field phase has the same symmetry characteristics as Fintermediate fields, but a different Fermi-surface topoloWhereas FL phase at intermediate fields has a single Fesurface sheet for one spin direction~the majority spins haveone full and one empty band whereas the minority spins hone partially filled and one empty band!, in FL3 the upperband of the majority spins becomes partially filled, too.and FL3 are separated by a strongly first-order transitionmean-field theory. There are numerous other phase trations associated with a change in the Fermi-surftopology—those do not display strong thermodynamic sigtures and are not shown. We note that for the field radisplayed in Fig. 8,uHW extu!t, the conduction electrons are igeneral weakly affected by the field; significant polarizatiof them occurs only at much higher fields.

Notably, smaller values of the decoupling parameterx ad-mit yet another field-induced transition in the small-JK re-gion: If the magnetism is very weak, i.e., the spinons havsmall gap compared to their bandwidth, then a small appfield can close the spinon gap without significantly affectitheir band structure. Such a transition would yield a kinkthe magnetization of the local-moment subsystem as funcof the applied field, implying a ‘‘metamagnetic’’ behavio

FIG. 8. Mean-field phase diagram ofHmf ~18! on the cubiclattice, now as function of Kondo couplingJK and external fieldHz

at T50. Parameter values are as in Fig. 4. For a description ofphases see text.

03511

,

nesy

ly

at.i-

e

si-e-e

ad

n

which is here generically associated with acontinuoustransition.

APPENDIX D: SPECIFIC-HEAT SINGULARITY

Here we present some details on the calculation ofsingular specific heat coming from gauge fluctuations. Tcalculations in the FL* phase and at the critical point arstandard. We will therefore only consider the FL phase.this phase close to the critical point, transverse gauge fltuations are described by the action

S5E d3k

~2p!3

1

b (vn

S uvnuk

1k21rsD uaW ~kW ,vn!u2.

~D1!

As explained in Sec. V, close to the transitionrs;b02. This

gives a free energy

F52

b (vn

E d3k

~2p!3lnS uvnu

k1k21rsD . ~D2!

To calculate the low-temperature specific heat, we needchange in free energy on going from zero to a small nonztemperature. After a Poisson resummation this is given b

dF~T![F~T!2F~0!

52EkW(

mÞ0E dv

2peibmvlnS uvu

k1k21rsD

52EkW ,vE

0

`

dl (mÞ0

keibmv

uvu1k~k21rs1l!

52EkW ,v,l

(mÞ0

kE0

`

dueibmv2u[ uvu1k(k21rs1l)] .

~D3!

The v,l integrals may now be performed to obtain

dF~T!54

pEkW(

m51

` E due2uk(k21rs)

u21~mb!2

5E0

L dk k2

2p3 Eu

@puT coth~puT!21#e2uk(k21rs)

u2.

In the last equation we have introduced an upper cutoffL forthe momentum integral. The remaining integrals can nowstraightforwardly evaluated for smallT, and we find

dF~T!5T2

12plnS L2

rsD . ~D4!

Thus the specific heat

C~T!5gT ~D5!

with g; ln(1/rs); ln(1/b0). Settingrs50, a similar calcu-lation also shows thatC(T);T ln(1/T) in the FL* phase.

e

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ans

a-nn

laq


For completeness, we mention the corresponding behior in two dimensions. In analogy to the above calculatiowe findC(T)}T2/3 in the quantum-critical and FL* regions.

APPENDIX E: DETAILS OF THE QUANTUMBOLTZMANN EQUATION

In the following we describe a few details of the derivtion of the linearized version of the quantum Boltzmaequation~42! in Sec. V C. Inserting the ansatz~38! into Eq.~37! leads to a scalar equation forf 1. The frequency integrais easily performed; the remaining momentum integral cbe split into radial and angular parts. This directly yields E~42!, with

K1~ k,k1!5@11 f 0~ k1!1n~ k22 k12!#

3k k1

2

4p2E21

1

dx KS x,k1

k,kATD , ~E1!

K2~ k,k1!5@ f 0~ k!2n~ k22 k12!#

3k k1

2

4p2E21

1

dxxk1

kKS x,

k1

k,kATD ~E2!

n-

.v.tu

03511

v-,

n.

with f 0(x)5@exp(x21uAT)21#21 and n(x)5(ex21)21.

The kernelK(x,k1 / k,kAT) is given by

K~x,a,l!5ImS 2 i12a2

Aa21122ax

1l~a21122ax!D 21S 12~ax21!2

a21122axD .

The integrals necessary for the evaluation ofK1,2 are of theform

E dyyn21/2

y31C

with n50, . . . ,3 and can beperformed analytically.The numerical solution of Eq.~42! is done by rewriting it

in the form

c~ k!52 f 08~ k!S E0

`

dk1FK1~ k,k1!1K2~ k,k1!c~ k1!

c~ k!G D 21

which allows for a stable numerical iteration.

ys.

sions

hys.de

-acteany

s.

tosedn,

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1-18

hebe. Ir-is

0

aa

ys

u

fiere

th

iz

n

er,

thein-nonshe

lec-

n-

, K.

r.


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03511

n

of

1

-

ld

is

es

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Weak magnetism and non-Fermi liquids near heavy-fermion critical points

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Transcript of Weak magnetism and non-Fermi liquids near heavy-fermion critical points