Ashvin Vishwanath- Dirac nodes and quantized thermal Hall effect in the mixed state of d-wave...

8/3/2019 Ashvin Vishwanath- Dirac nodes and quantized thermal Hall effect in the mixed state of d-wave superconductors

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Dirac nodes and quantized thermal Hall effect in the mixed state of d-wave superconductors

Ashvin VishwanathDepartment of Physics, Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544

Received 3 December 2001; revised manuscript received 7 May 2002; published 1 August 2002

We consider the vortex state of d-wave superconductors in the clean limit. Within the linearized approxi-

mation the quasiparticle bands obtained are found to posess Dirac cone dispersions band touchings at special

points in the Brillouin zone. They are protected by a symmetry of the linearized Hamiltonian that we callTDirac . Moreover, for vortex lattices that possess inversion symmetry, it is shown that there is always a Dirac

cone centered at zero energy within the linearized theory. On going beyond the linearized approximation and

including the effect of the smaller curvature terms that breakTDirac, the Dirac cone dispersions are found to

acquire small gaps ( 0.5 K T1 in YBCO that scale linearly with the applied magnetic field. When the

chemical potential for quasiparticles lies within the gap, quantization of the thermal-Hall conductivity is

expected at low temperatures, i.e., xy /Tn(2kB

2/3h) with the integer n taking on values n2 and 0. This

quantization could be seen in low-temperature thermal transport measurements of clean d-wave superconduct-

ors with good vortex lattices.

DOI: 10.1103/PhysRevB.66.064504 PACS numbers: 74.72.h, 74.60.Ec, 73.43.f

I. INTRODUCTION

Since the experimental verification of the dx2-y 2 nature of

superconductivity in cuprate materials,1 there has been muchactivity in studying the physics of quasiparticles in a d-wavesuperconductor. In contrast to the s-wave case, the d-wavesuperconducting gap vanishes at points on the Fermi surfaceleading to low-energy quasiparticles that behave like mass-less Dirac fermions. One question that is of great interest isthe behavior of these quasiparticles in the mixed state ofsuperconductorsa problem that is both theoretically rich aswell as of relevance to explaining several different experi-ments.

Experimentally, there have been several probes of quasi-

particle behavior in a magnetic field, including a measure-ment of the thermal Hall conductivity xy

2,3 the low tem-perature longitudinal thermal conductivity xx ,

4,5 and thespecific heat.6 A satisfactory explanation of the results ofthese experiments in terms of dx2-y 2 quasiparticles would bestrong support for the point of view that the superconductingphase of the cuprates is a conventional d-wave supercon-ductor. On the other hand, if such an explanation proveselusive, it may well point to some additional, and perhapsexotic, physics even in the superconducting state of the cu-prates.

This problem of Dirac quasiparticles in a mixed state hasbeen considered by several authors. Gorkov and Schriffer7

and Anderson8 proposed a Landau-level-like spectrum. In thelatter work, this was derived by mapping the problem onto aDirac particle in a uniform magnetic field, under the assump-tion that the superflow may be neglected. In Ref. 9, the au-thors studied a lattice model of the problem, first in the sameapproximation as Ref. 8, and then numerically with the su-perflow included. Topological aspects of relevance to thisproblem were pointed out.

A different approach was taken by Franz and Tesanovic,who considered the problem within the linearized approxi-mation and introduced a fictitious U1 gauge field to imple-ment the statistical interaction between the vortices and qua-

siparticles Franz-Tesanovic transformation.10 This led to a

numerical evaluation of the resulting quasiparticle bandstructure, which was extended by the detailed study of

Marinelli et al.,11 Vafek et al.,12 and Knapp et al.13

Here too we begin by considering the problem of d-wavequasiparticles in the vortex lattice state within the linearizedapproximation. In contrast to some earlier approaches, wefocus on identifying the symmetries of the problem and theirconsequences for the spectrum of the linearized Hamiltonian.Our results are easily statedthe Hamiltonian, as a conse-quence of linearization, posesses an additional symmetrywhich we call TDirac ) that preserves Dirac cones at certainspecial points in the Brilluoin zone. The energy dispersion, inthe vicinity of these points, is that of a massless Dirac par-

ticle. If the vortex lattice posesses inversion symmetry, thenthere is a Dirac cone centered at zero energy. While theseresults are in general agreement with the results obtained inearlier numerical work1013 we point out a subtle feature ofthe Franz-Tesanovic transformation in the linearized approxi-mation, which could yield spurious gaps in numerical simu-lations and which we believe to be the cause for discrepan-cies from the results derived here.

We then proceed to consider the effect of the curvatureterms, that were dropped when linearizing the Hamiltonian.These terms arise, for example, from the parabolic nature ofthe electron dispersion. In the parameter range of interestthey may be considered as small perturbations on the linear-ized Hamiltonian. However, as pointed out in Ref. 14, theyare crucial to generating a nonvanishing thermal Hall re-sponse. Since the curvature terms break the symmetry TDiracthat protects the Dirac nodes, they give rise to a small gap atthese nodes, and the energy dispersion in their vicinity isnow that of a massive Dirac particle. Thus for temperaturesthat are smaller than this gap scale, the curvature terms canhave a qualitative effect on the properties of the system.

In addition to information about the quasiparticle spec-trum, it is possible in some cases within our approach toderive consequences for low-temperature thermal transport.To this end, we first recall that massive Dirac particles in two

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dimensions exhibit a quantized Hall effect when the chemi-

cal potential lies in the gap.15,16 In the presence of a vortex

lattice with inversion symmetry, the chemical potential for

the superconductor quasiparticles will lie at the center of the

gap induced by the curvature terms, if we ignore Zeemansplitting. As a result, a quantized thermal Hall conductanceof xy /T1/2 in appropriate units, at low temperaturesis expected from each of the four nodes, and can give rise to

two scenarios xy /T2 or 0, while xx /T0 in bothcases. First, if the contribution of all four nodes is of thesame sign, xy /T2, we have a nontrivial quantized ther-mal Hall conductance that is expected to be attained for tem-peratures smaller than the gap. This situation is topologicallyidentical to i.e., has the same edge content as a puredx2-y 2idxy superconductor, that is also known to exhibit aquantized thermal Hall effect.17,18 A magnetic induction ofsuch a pairing symmetry in the cuprates was proposed inRefs. 19 and 20; here we will provide a concrete realizationof these general ideas, and layout the route to calculatingphysical parameters such as, for example, the size of theenergy gap. The second scenario is when the Hall conduc-

tances from the different nodes cancel leading to a xy /T0, that is topologically identical to a dx2-y 2is supercon-ductor, or any other thermal insulator. Which of these sce-narios is realized is a function of vortex lattice geometry andthe anisotropy of the Dirac dispersion in the homogenoussuperconductor. In order for the quantization to be visibleexperimentally the energy gap needs to be larger than theZeeman splitting, which plays the role of chemical potentialfor the quasiparticles. Since the energy gap is also found toscale linearly with magnetic field, and is roughly of the samemagnitude as the Zeeman energy, the question of which oneis larger in any particular material is a detailed quantitativeissue. Here we will demonstrate how fairly simple numerical

calculations within our theory can predict in a given situa-tion, which of the scenarios described is realized, as well asprovide a quantitative estimate of the energy gap and itsdependence on various physical parameters.

The rest of this paper is organized as follows. In Sec. II,we begin by laying out our assumptions and then derivingthe Bogoliubovde Gennes equation that describes d-wavequasiparticles in the mixed state. The quasiparticles areshown to couple to the superflow generated by the combinedeffect of the vortices and the magnetic field. In addition, theyacquire a Berry phase of1 on circling a hc/2e vortex. Wethen derive a linearized approximation that describes thelow-energy quasiparticle excitations in the magnetic-fieldrange of interest. These linearized equations describingd-wave quasiparticles in a vortex lattice are analyzed in Sec.III. For purposes of clarity, we first consider a vortex latticeof hc/e double vortices. The simplifying feature here isthat the Berry phase terms are absent and we only have tocontend with the effects of the superflow. A symmetry of thelinearized Hamiltonian that protects the Dirac nodes is iden-tified, and the role of inversion symmetry in maintaining aDirac node at zero energy is described. We then tackle thephysically more interesting case of a vortex lattice of hc/2evortices. Here a similar though more involved analysis ob-tains for us the same results as for a vortex lattice of hc/e

vortices. In Sec. IV, we go beyond the linearized approxima-tion by including the effect of curvature terms and discussimplications for low-temperature heat transport. A numericalcalculation of the gaps induced by the curvature terms for thehc/e square vortex lattice case is also presented. Finally, inSec. V we conclude with some brief comments on the effectof vortex disorder and a comparison of our theoretical expec-tations against available experimental data. Expressions for

the Chern number of quasiparticle bands are contained inthe Appendix. A short summary of this work appeared inRef. 21.

II. BDG EQUATIONS FOR d-WAVE QUASIPARTICLES

IN MIXED STATE

In this section we derive the equations governing quasi-particles in a dx2-y 2 superconductor, in the presence of a vor-tex lattice. We begin by detailing the assumptions and ap-proximations that we will make in what follows.

A. Assumptions and approximationsa Existence of quasiparticles: The systems that we will

mainly be interested in are cuprate high-temperature super-conductors, that are known to have dx2-y 2 gap symmetry. Weassume that this superconducting state is otherwise conven-tional, in particular that there are well defined quasiparticleexcitations in this phase, for which there is experimental sup-port from angle-resolved photoemission studies. For inho-mogenous situations, such as the vortex lattice state, we as-sume that the quasiparticles are governed by an appropriateBogoliubov-de Gennes BdG type equation.

b Neglecting vortex core contributions: The vortex coreis the region around the center of the vortex of size , the

coherence length, where the magnitude of the order param-eter is significantly supressed from its bulk value. At fieldsmuch smaller than Hc2, the vortex cores of extreme type-IIsuperconductors are significantly smaller than the separationbetween vortices. This is the situation, for example, in opti-mally doped YBa2Cu3O6.9 YBCO over the accessible fieldrange. The vortex cores have a size 15 A, while theintervortex separation is of order 500 A in a 1-T field. Sincethe vortex cores take up so little of the sample area 0.1% inthis example we neglect the modulation of the order param-eter magnitude, while retaining its phase variation.

c Perfect vortex lattice: We will make the assumptionthat the superconductor is clean and the vortices are arrangedin a perfect lattice. Of course, in any real situation, vortexdisorder is expected to be present. In some cases it may evenbe so large as to destroy the long-range positional order ofthe lattice, in which case, of course, the perfect lattice ap-proximation will not be a good starting pointfor example,in BSCCO in magnetic fields of around 1 T, neutron-scattering studies indicate that an ordered vortex lattice isabsent.22 However in YBCO, Bragg spots from the vortexlattice are seen23 for which the starting point of a perfectvortex lattice may be more justified. Despite the existence ofvortex disorder, in what follows we shall proceed with theassumption of a perfect vortex lattice as it provides us with a

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theoretically tractable starting point. In addition, there issome evidence from experiments5 and theory24 that the scat-tering from vortex disorder at low temperatures is rathersmall, and neglecting its effect may be permissible at a firstapproximation. Extending the theory to include the effects ofweak vortex disorder is left for future investigation, althoughwe briefly return to the topic of vortex disorder in Sec. V,while considering the stability of our results.

B. Bogoliubovde Gennes equations for d-wave

superconductors in the mixed state

Consider a two dimensional d-wave superconductor in thepure state, which may be described by the model Hamil-tonian

Hpurep

c pp cp

1

2 p cp

cp

H.c. 1

where cp is the creation operator for an electron of momen-tum p and spin projection and are elements of the unitantisymmetric matrix:

0 11 0

. 2Here we consider a parabolic dispersion for the electrons,

p 1

2mp2EF 3

and a dxy symmetry of the gap function equivalent to dx2-y 2rotated by 45) which could be taken to have, near the Fermi

momentum, the functional form

p 0

pF2

pxpy . 4

We now consider the effect of a magnetic field applied alongthe z axis, that gives rise to vortices. It is convenient at thisstage to define the d operators18 as

d1pcp ,

d2pcp ;

here the spin projection is taken in the direction of the mag-

netic field. The Hamiltonian for a layer of superconductor inthe mixed state can them be written as

H d2r d1 rd2 rHBdG d1 rd2 rHZeeman , 5where

HBdG ieA r e i(/2)i e i(/2)ei(/2)iei(/2) ieA r

6

where A is the vector potential A(r)B(r)z and (r) isthe phase of the order parameter; and

HZeemangB

2 d2rB r d1 rd1 rd2 rd2 r .

7

The Hamiltonian written in terms of the d particles contains

no anomalous terms which implies that the total number of dparticles is conserved. This is simply a reflection of the factthat the z component of the spin is conserved in this situa-

tion. Thus the density of d particles, D(r) d1(r)d1(r)

d2(r)d2(r) , is proportional to the spin density of the qua-

siparticles along the z axis, and so appears in the expressionfor the Zeeman coupling to the magnetic field. The uniformpart of the magnetic field density will therefore behave as achemical potential for the d particles. In what follows, weshall not retain the Zeeman term, but rather comment on itseffect at the very end.

In Eq. 6 the phase variation of the order parameter, in-duced by the vortices, has been written in the form

e i(/2)(i)e i(/2). This preserves gauge invariance and isconsistent with the formulas in Refs. 26 and 12. The varia-tion of the magnitude of the order parameter has beendropped, as discussed earlier. We shall take as given the

magnetic field distribution B(r) and the phase variation,which must satisfy the equation

2i

(2 ) rRiz, 8

where Ris denote the positions of the vortices and the sumruns over all vortices in the lattice.

Due to the conservation of the dparticles, and the fact that

they do not interact with each other at this level, we canwrite a wave equation which contains all the physics of thismany body system. This wave equation is the BogoliubovdeGennes equation,

HBdG u rv r

E u rv r

, 9where HBdG has been defined in Eq. 6 and can be written inthe compact form

HBdGieA rzz

e

i[(r)/2]

i

e

i[(r)/2]

H.c..

The s are the 22 Pauli matrices in the usual represen-tation. In this language, the Zeeman term takes the form

HZeeman(gB /2 )B(r)1.It is convenient to make a gauge transformation to elimi-

nate the phase variation from the order parameter Londongauge which may be affected by the unitary transformation:

Ue(i/2)(r)z 10

The transformed Hamiltonian takes the simple form

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HBdG UHBdGU1iPs rzz i x,

11

where Ps12 eA, a gauge invariant quantity, is the me-

chanical momentum carried by each member of Cooper pair

at point r. We will sometimes refer to this quantity as thesuperflow, though this terminology is not quite accurate.

Given this definition of Ps

, it is easily seen that

Ps 12 hi rRieB rz. 12In the presence of elementary hc/2e vortices, it must be

noted that unitary transformation 10 is not single valued,but changes sign on circling an odd number of such vortices,since it depends on the half angle (r)/2 that winds by anodd multiple of. Thus the transformed quasiparticle wavefunctions, (uv)T, defined by

u rv r

U u rv r

, 13are not single valued, but change sign on circling an oddnumber of hc/2e vortices. Methods for handling this statis-tical interaction between quasiparticles and vortices are de-scribed in Sec. II C.

Thus the effect of the magnetic field on the quasiparticlesis most transparent when written in this gauge. The quasipar-

ticles interact with the superflow ( Ps12 eA) generated

by the vortices and magnetic field, and also acquire a Berryphase factor of 1 on circling a hc/2e vortex.

C. Linearized approximation

In the limit of low temperatures, and magnetic fields

much smaller than Hc2, we can restrict our attention to thelow energy quasiparticle excitations near the gap nodes ofthe d-wave superconductor. We first recall the linearized ap-proximation for the pure case, which gives us the Dirac dis-persion of the nodal quasiparticles. The Hamiltonian for qua-siparticles with momentum p is Hp(p)z(p)x . Forlow-energy excitations in the vicinity of node 1 see Fig. 1

this can be expressed in terms of pp Fxp where p ismeasured from node 1, as

HvFpxzvpyx 12m p 2z 0pF2 pxpyx ,14

where vFpF /m and v0 /p F . The linearized approxi-mation consists of dropping the terms within brackets, inwhich case we obtain an anisotropic Dirac dispersion for thequasiparticles near the nodes,

ELp vFpx22py

2,

where vF /v is the anisotropy of the Dirac dispersion.Following Ref. 14 we estimate the temperature scale up towhich this approximation can be trusted. We require that thevalue of the quadratic terms at the typical momenta of the

thermally excited quasiparticles be smaller than the linearcontribution. For YBCO, this temperature turns out to beTma x200 K.

27

We now consider a similar low-energy approximation forthe vortex state of the dxy superconductor. First, we describea procedure to handle the non-single-valued nature of the

wave functions (v

u) of Eq. 138,10 which is called the

Franz-Tesanovic transformation, although we adopt aslightly different approach to its derivation here.

Franz-Tesanovic FT transformation: Recall that we wantto solve the eigenvalue equation

HBdG E, 15

with the condition that (x,y ) is not single valued, butacquires a negative sign on circling an odd number of hc/2evortices. To implement this condition, we write as a prod-uct of a fixed function that is multiple valued which preciselybuilds in the required sign changes (x,y ) , and a single-valued wave function FT(x,y ). Thus

x,y x,y FTx,y , 16

and the preceeding eigenvalue problem can be formulated asan equivalent problem for the single-valued wave functionFT . In order that this is also a Hermitian eigenvalue prob-lem, we choose (x ,y) from the class of functions,

qx ,y i

zz izzi

q i/4

, 17

where the q i are odd integers, and we have used the complex

coordinates zxiy and zxiy . The product runs overall hc/2e vortices, located at z ix i iy i . Note that thisfunction is pure phase, i.e., q(x,y ) 1.

28 The choice ofthe odd integers q i is arbitrary and represents a gauge degreeof freedom. Clearly, physical results cannot depend on thischoice, though in practice we may work with a particular set

FIG. 1. The gap structure for a dxy superconductor with nodes

along the coordinate axes.


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of q i that is convenient for calculation. Due to its singularnature, this transformation needs to be handled with somecare especially in the linearized theory. Further elaborationson this subtle point are contained in the appendices of Ref.29.

Inserting Eq. 16 into Eq. 15, we have

HFTFTEFT 18

HFTpaPsz zpax, 19

where a(x,y ) is a real vector field given by

ailogq r 20

2 i q iz

rri

rri2

, 21

which implies

a ri

q i(2 ) rri,

i.e., that solenoids of flux q i of the a gauge field have beenattached to the vortices. The sign change of the quasiparticleson circling a unit vortex is now accounted for by theAharonov-Bohm effect arising from this solenoid of flux.Thus a fictitious U1 gauge field has been invoked to handlethe 1 phase factors acquired by a quasiparticle on circlinga vortex. Clearly, this is a highly redundant descriptionin

principle the U1 gauge field a can account for any phasefactor, which is here being restricted to just 1. The mini-mal choice that would take care of just these two factors, isan Ising (Z2) gauge field, which however requires a realspace lattice for its formulation.25

Linearization for the vortex state: Following Ref. 14, weconsider low-energy excitations near the nodal points, andneglect the effect of inter-node scattering. Then we can ex-pand the wave function as

FT reikFx1 re

ikFx1 reikFy2 r

eikFy2 r, 22

where the functions i (i1, 1,2,2) are considered to be

slowly varying on the scale of kF1 . The problem then re-

duces to solving, at each node,

Hi

FTHi rEi r, 23

where the HiFT represent the linearized part of Hamiltonian

18,

H1FTH

1FT

vFpxax z1pyay xxPs r,

24

H2FTH

2FTvFpyay z

1pxax xyPs r25

with pi . The remaining part of the Hamiltonian, H,arises from the curvature of the electon dispersion and of thegap function and is given by

H1

2mpa2Ps

2 rz

1

2mpa, Ps r1

0

2p F2 p 1a 1 ,p2a 2x, 26

where , denotes antisymmetrization; A,BABBA.It may now be argued that the linear piece Hi

FT dominates

the curvature terms H for magnetic fields much smallerthan Hc2. Let us denote the typical separation between vor-

tices by d1/B. While the linear part of the Hamiltonianhas terms of order E1vF /d 50 K T

1/2 in YBCO or oforder E2v /d 3 K T

1/2 in YBCO, the largest terms in

H are of order E12/EF 0.5 KT

1 in YBCO, so long as weassume that the quasiparticle trajectories do not come tooclose to the vortex core. Thus, in the magnetic field range of

interest of a few T, we can at a first approximation neglectthe curvature terms H and simply solve the linearizedproblem. Subsequently we will include the effect of thesecurvature terms they are crucial to producing a finite ther-mal Hall signal, and verify that their effect is indeed a per-turbation on the linearized problem.

We are therefore led to consider the linearized problem,say at node 1:

H1FTE,

H1FTvFpxaxzpyay xP sx r1 27

We have already noted that, in addition to the interactionwith the amplitude variation of the gap which has been ne-glected, the quasiparticles interact with the vortices via the

superflow Ps and the Berry phase of 1 acquired on cir-cling a fundamental vortex. For the sake of clarity, it is help-ful to first consider a situation where the Berry phases areinactive, and the quasiparticles only see the superflow. Thisis formally accomplished by considering the case of a vortexlattice ofhc/e double vortices. Then there are no nontrivial

Berry phase factors, and hence the a fields which were in-ducted to keep track of these phase factors, may be dropped.Equivalently, it may be noted that for the case of hc/e vor-tices, transformation 10 is single valued. Below, we will

analyze the linearized problem in the presence of a hc/evortex lattice. Armed with this understanding, we then con-sider the physically relevant case of a lattice of hc/2e vorti-ces. Although our conclusions from the hc/e vortex latticecase go over unchanged, the reasoning there is a little moreinvolved.

D. Vortex lattice of hce vortices in the linearized

approximation

Consider the linearized Hamiltonian at node 1, for a con-ceptually simpler case of a vortex lattice of hc/e vortices,


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H1vFpxz1pyxPsx r1, 28

where the superflow Ps( Psx , Psy ) is given by the gaugeinvariant combination

Ps 12eA ,and satisfies

Ps hi

2 rRieB rz, 29

where the sum runs over all the vortices in the lattice, at

positions Ri . This differs from Eq. 12 in that the strengthof the vortices is now doubled. For a periodic lattice of vor-

tices, Ps , being a physical quantity, is also a periodic func-tion with the same period as the vortex lattice. Formally, thiscan be seen by noting that

unit celld2r Ps 0 30from flux quantization. Thus

B

Psd l0

on the boundary B of the unit cell, which is consistent with a

periodic superflow Ps .Consequently, the Hamiltonian H1 Eq. 28 is that of a

Dirac particle in the presence of a periodic scalar potential,

played by Psx (r

). Clearly, a band structure will result, andthe eigenstates can be labeled by the band index, and the

crystal momentum k that takes values within the Brillouinzone. What follows is a symmetry analysis of the spectrumof H1. One of the main issues we address is whether theDirac node at zero energy, that is present for the free DiracHamiltonian and corresponds to the node of a dxy supercon-ductor in the pure state survives in the presence of the pe-

riodic superflow term P sx (r). Our results are as follows.Dirac nodes band touchings survive in the spectrum of H1at those points in the Brilloin zone that are invariant under

the kk transformation. The symmetry that protects thesenodes, which we call TDirac , is obtained as a consequence ofthe linearization. Further, we find that there is a Dirac nodecentered at zero energy if the vortex lattice posesses inver-sion symmetry.

III. DIRAC NODES IN THE LINEARIZED PROBLEM

We begin by analyzing the conceptually simpler case of avortex lattice ofhc/e vortices, where the Berry phase factorsare absent, and we have only the coupling of the quasiparti-cles to the superflow to deal with. We then tackle the hc/2evortex lattice case along parallel lines.

A. Vortex lattice of hce vortices

We will consider the linearized Hamiltonian for node 1,

H1vFpxz1pyxP sx r1,

which is Eq. 28 above, where Ps(r) for an arbitrary latticeof vortices, is a periodic function with the same period as thelattice and satisfies Eq. 29. This gives rise to a band struc-ture for the quasiparticles, and the eigenstates are labeled by

a band index and a crystal momentum (k), which takes val-ues within the Brillouin zone. We consider for generality anoblique vortex lattice, which posess no additional symme-tries. The Brillouin zone for such a vortex lattice is depicted

in Fig. 2. A symmetry of the linearized Hamiltonian: Consider an

eigenstate k of the linearized Hamiltonian H1 with eigen-

value E and crystal momentum k:

H1kEk .

Then the transformed wave function kk* is also an

eigenstate with energy Ebut crystal momentumk, where is the dimension two antisymmetric matrix:

FIG. 2. The Brillouin zone for an oblique lattice of vortices. The

points of the Brillouin zone that are taken to themselves under

time-reversal symmetry are shown as , A, B, and M points. At

these points degenerate doublets and hence Dirac cones are ex-

pected within the linearized theory for all lattices of hc/e vortices.

The same holds for lattices of hc/2e vortices.


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0 11 0

This can be verified by noting that the linearized Hamil-tonian is taken to itself under the transformation

H1*H1 , 31

where we have used the identities

21,

i*i .

Since this transformation is formally equivalent to the time-reversal operation for Dirac particles, we will call it TDirac ,although, as explained below, it is distinct from the physicaltime-reversal transformation for this problem. This symme-

try ensures that states with crystal momentum k and k

have the same energy. For those points in the Brillouin-zonethat are taken to themselves under time reversal modulo a

reciprocal-lattice vector: kk mod G there is a degen-

erate pair of states k and k* . These states are orthogonal,

from the antisymmetry of . Hence we find a degeneratedoublet at these special points in the Brillouin zone, which

are shown in Fig. 2 as the (k0), A, B, and Mpoints. Thespectrum at these special points of the Brillouin zone is com-posed entirely of degenerate pairs, and as we will see shortly,on moving a little bit away from these points in crystal mo-mentum, the states split and give rise to the energy disper-sion of a massless Dirac particle.

The symmetry operation TDirac that operates on the qua-siparticle excitations at a single node is distinct from the

physical time reversal operation that would transform statesat one node into states at the opposite node. Rather, TDirac isobtained as a symmetry as a consequence of linearizing theelectron dispersion, and it is easily seen that the curvatureterms, such as for example (1/2m)p 2z, violate this symme-try. Thus, although TDirac is not a symmetry of the entireproblem, since the curvature terms that violate it are sosmall, it is still a good approximate symmetry. For the lin-earized theory of course, it is an exact symmetry.

Dirac cones from degenerate doublets: As we move awayfrom the special points in the Brillouin zone at which degen-erate doublets are found, the crystal momentum splits thesestates and gives rise to a Dirac cone dispersion. This is mosteasily appreciated by analogy with the pure case. There the

free Dirac Hamiltonian H freevFpxzvpyx has a de-

generate pair of states at the center of the Dirac cone at p

0) given by the two constant spinors that are eigenstateswith zero energy. Moving away in momentum, which is ex-actly analogous to turning up the Zeeman field on a spinone-half particle, the states are split into E pairs, with thesplitting being linearly proportional to the momentum. Thisgive rise to the massless Dirac dispersion that we expect forthis Hamiltonian. In a similar way, here we find that Diraccones arise centered at the special points at which the degen-erate doublets are present.

Consider a pair of degenerate states, (r) and *(r), atone of these special points in the Brillouin zone. The effect

of moving away from this point by crystal momentum k

can be accounted for by adding the piece

HkvFkxzvkyx 32

to the Hamiltonian and leaving the boundary condition for

the wave function unchanged. If the deviation in crystal mo-mentum is small compared to the reciprocal-lattice vectors,this additional piece will barely mix the different pairs ofenergy eigenstates, and so can be treated in degenerate per-turbation theory within the two dimensional subspace of thedegenerate doublet. The perturbation, projected into the sub-

space of(r) and *(r), takes the form

Hk Proj rHk

rTHk

rHk*

rHk

,33

where the integral r runs over the unit cell and we have

used the notation (r)*T(r). Clearly, an explicit knowl-edge of the wave functions is needed to diagonalize thisHamiltonian, but a few general observations can be maderight away. First, the eigenvalues obtained here are the en-ergy splitting (Ek) of the previously degenerate pair ofstates, which will appear in a plus-minus pair since the pro-

jected Hamiltonian has vanishing trace. Also, if we denote

the angle the vector k makes with the kx axis by k , theenergy splitting may be written as:

Ek

vFAk

k

, 34which is a massless Dirac particle with an anisotropic disper-sion. Calculating the anisotropy function A(k) requires anexplicit knowledge of the wavefunctions.

If the vortex lattice posesses reflection symmetry aboutthe y axis (x axis, then the dispersion of the Dirac nodes ofthe linearized Hamiltonian at node 1 2 takes a particularlysimple form. The dispersion can then be written as

EkvF 2kx

2v

2ky

2, 35

where again an explicit knowledge of the wave functions is

required to compute the renormalized velocities vF and v .

The effect of the renormalization can be substantial andtherefore the energy scale at which the Dirac node may beexpected to make its presence felt can be quite different fromE2, as one might naively expect, as pointed out in Ref. 11.

Inversion symmetry and the Dirac node at zero energy: Ifthe vortex lattice posesses inversion symmetry, that is, if it is

invariant under the transformation rr assuming the ori-gin is the center of inversion, then it is easy to see from Eq.

29 that the superflow Ps satisfies

PsrPs r. 36


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This leads to a particle hole symmetry of the linearizedHamiltonian. If (x,y ) is an eigenstate of the HamiltonianH1 Eq. 28 with energy E, then (x,y ) is also aneigenstate but with energy E.30 Inversion symmetry en-sures that there is a degenerate doublet of the linearized

Hamiltonian H1 at the (k0) point at zero energy. Theargument is as followslet us focus on the spectrum at the point; in which case we need to solve for the eigenstates ofH1 on the unit cell with periodic boundary conditionsi.e.,

on a torus. First, consider the case without the Doppler term,that is a free Dirac particle on a torus,

H freevFpxzvpyx

which can easily be solved. Clearly, this has a pair of statesat zero energy, given by the product of the constant solutiontimes any spinor. The rest of the states also occur in degen-erate pairs, and for every pair of states at energy E0 thereis a pair of states at energy E; a consequence of the freeDirac Hamiltonian respecting the TDirac and inversion sym-metries. This spectrum is sketched in Fig. 3a. Thus, in thefree case, the spectrum consists of an odd number of de-generate pairs, due to the existence of the pair at zero energythis can be made more rigorous by introducing an ultravioletcutoff, and hence a finite number of states. Now, turning onthe Doppler term P sx for inversion-symmetric vortex lattices,preserves both the TDirac as well as particle-hole symmetry.Therefore, we still have states coming as degenerate dou-blets, in a particle-hole symmetric spectrum. Since the totalnumber of pairs of states cannot change from the free case,we are forced to have a degenerate doublet at zero energy, sothat the total number of pairs of degenerate states remainsodd, as it was for the free case.31 This argument is illustratedin Fig. 3.

Note that this argument is also valid perturbatively. If weimagine the value of the Doppler term, continuously turningup from zero, the zero-energy doublet that is present for thefree Dirac Hamiltonian can neither split it is protected byTDirac) nor move away from zero energy thanks to particle-hole symmetry and hence continues to exist at zero energyeven for the Hamiltonian H1 that includes the superflow ofan inversion-symmetric vortex lattice. This can also be veri-

fied to all orders in perturbation theory, the details of whichmay be found in Ref. 29.

This doublet of states at zero energy for inversion-symmetric lattices will give rise to a Dirac cone centered atzero energy, by our previous arguments. Thus, in this situa-tion we have been able to access the nature of the low-energyphysics solely via the use of symmetry arguments.

B. Vortex lattice of hc2e vortices

We now turn to the physically relevant case of a vortexlattice ofhc/2e vortices within the linearized approximation.For convenience, here we collect the relevant formulas de-rived in Sec. III; the linearized Hamiltonian at node 1 is

H1FTvFpxax zpyay xP sx r1,

where the a fields implement the Berry phase factor for cir-cling a vortex which we have taken to be

ailogq r,

qz,zi

zz izzi

q i/4

,

where the q i are arbitrary odd integers. More generally, the a

fields just need to satisfy

a rhi

q i(2 ) rri, 37

which attaches solenoids ofq i fictitious flux to the vortices.Clearly, physical results should not depend on the choice ofq i . However, since we have a vortex lattice, it will be con-

venient to pick a set of q i that gives rise to a periodic a field.Imagine choosing a set of q i , so that the unit cell for theproblem now contains n vortices. The periodicity of the vec-

tor potential a requires that the total flux corresponding tothis vector potential vanishes over the unit cell, i.e.,

1

n

q0, 38

where the sum is over the vortices in a unit cell. Since q areodd integers, this requires that the number of vortices in theunit cell, n, be an even number. Therefore, a vortex latticethat has one physical vortex per unit cell will require at leasta doubling of the unit cell when considered in this way. Onceagain, the eigenfunctions can be labeled by crystal momenta

k that takes values in the appropriate Brillouin zone which is

FIG. 3. Dirac node at zero energy for inversion-symmetric lat-

tices. a Energy spectrum of a free Dirac particle on a torus (

point. There are an odd number of degenerate pairs of energy

states, due to particle-hole symmetry and the pair at zero energy. b

In the presence of the linearized Doppler term ( Psx ) for inversion

symmetric lattices of hc/e vortices, the particle-hole symmetry of

the spectrum is preserved, and the degenerate pairs of states cannot

split due to symmetry under TDirac . To keep the count of energy

levels the same as for the free case, there has to be a pair of statesat E0. Similar considerations apply for the case of inversion-

symmetric lattices of physical hc/2e vortices.


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now determined, not just by the periodicity of the vortexlattice, but also by the choice of q is.

Invariance under combinedTDirac and gauge transforma-

tion: It is easily seen that the Hamiltonian H1FT is not invari-

ant under TDirac ; i.e., under the transformation *, the

sign of the vector potential a is inverted. However, since the

flux associated with the a field is exactly times an odd

integer, it is possible to arrange for a gauge transformationthat would reverse its sign, which when used in combinationwith TDirac , would leave the Hamiltonian invariant. Thus, if

k(r) is an eigenfunction of H1FT with energy E and crystal

momentum k, then

k rq2

rk* r 39

is also an eigenfunction of the Hamiltonian with energy E.

The factor of q2 performs a gauge transformation that in-

vert the sign of the gauge fields a. It is crucial that 2(r) is

a single-valued function, so this gauge transformation is onlyallowed because we have the special case of flux sole-

noids. Also, it is easily shown that 2(r) is a periodic func-

tion, so the wave function k(r) carries a crystal momen-

tum k. Thus, once again, states with crystal momentum k

and k have the same energy. At those points in the Bril-louin zone that are taken to themselves under time reversal,

modulo a reciprocal-lattice vector kk mod G ) the en-ergy levels appear as degenerate doublets. This follows from

the fact that the two wave functions k(r) and q2

k*(r)]

are always orthogonal from the antisymmetry of . Thesedegenerate doublets will lead to Dirac cones in their vicinity

by the same argument as before. Hence, for any vortex latticewe expect the spectrum of H1

FT to posess Dirac cones bandtouchings.

As we have noted, we are only allowed to make the gaugetransformation above because we are dealing with (2 m

1) solenoid fluxes of the fictitious gauge field a, that

leads to q2 (r) being a single-valued function. For a gen-

eral value of the fictitious flux, the time-reversal symmetryof the Dirac equation is broken in an essential way and can-not be fixed by such a gauge transformation. This also im-plies that, in general, the doubly degenerate states that weobtain are not accessible within perturbation theory startingwith a free Dirac equation. In doing perturbation theory weare implicitly assuming that we can gradually crank up fromzero the value of the fictitious fluxhowever, since the dou-blet only appears at the flux values, it is missed in pertur-bation theory unless some additional symmetry, that pre-serves the degenerate doublets for all values of the fictitiousflux, is present. This issue as well as some subtle aspects ofthe gauge transformation that we have just used were con-sidered in more detail in Ref. 29.

Inversion symmetry and the Dirac cone at zero energy:Consider a vortex lattice that posesses inversion symmetryabout the origin, say, i.e., is invariant under the transforma-

tion rr. We argue here that in this case there exists adegenerate doublet, and hence a Dirac cone centered at zeroenergy.

For such a lattice it is always possible to make a gaugechoice of the set q i for which, under inversion, q iq i . A couple of examples are shown in Fig. 4. In casewe begin with a set of q i that do not satisfy this condition,one can always make a gauge transformation that does notalter any of the physics but brings the q is into this form,

which happens to be convenient for the following analysis.In this gauge we have that the a fields are even under inver-sion, i.e.,

ax,y ax,y , 40

which is easily seen from Eq. 37 for the curl of a and thefact that the position of a q vortex is taken by a q vortexunder inversion. Of course we still retain the fact that thesuperflow is odd under inversion:

psx ,y psx ,y 41

These facts imply a particle-hole symmetry for the Hamil-

tonian H1FT

. If(x ,y) is an eigenfunction of H1FT

with en-ergy E, then *(x ,y) is also an eigenfunction but withenergy E.11

Inversion symmetry ensures that the degenerate doublet

of Hamiltonian H1FT at the k0 () point is at zero energy.

The argument is as followsfor the point we need to

solve for the eigenstates of Hamiltonian H1FT on a torus. First

consider the case without the Doppler term and the a field,that is, a free Dirac particle on a torus:

H freevFpxz1pyx

This has a pair of states at zero energy, given by the constantsolution and any spinor. The rest of the states also occur inpairs due to TDirac ) and for every pair at energy E0 thereis a pair of states at energy E from the particle-hole sym-metry. Thus, in the free case, as sketched in Fig. 3 a, thereis an odd number of degenerate doublets due to the pair atzero energy this can be made more rigorous by introducingan ultraviolet cutoff, and hence a finite number of states.

Now, turning on the Doppler term and the gauge field a, theenergy levels will again appear as degenerate pairs, fromsymmetry under the combined effect of TDirac and a gaugetransformation, as discussed earlier. For the case of inversionsymmetric vortex lattices, particle-hole symmetry is pre-

FIG. 4. Different FT gauge choices ( q i) for a vortex lattice of

two physical vortices per unit cell. a Smallest unit cell possible;

under inversion the q and

q vortices are interchanged. b Dou-bling of the unit cell due to the gauge choice. Once again the ficti-

tious gauge field flux has been chosen, for convenience, to map

onto its negative under inversion about the origin *.


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served as well. Therefore we have the degenerate doubletsappearing in a particle-hole symmetric fashion. Since the to-tal number of pairs of states cannot change from the freecase, where we had an odd number of pairs, we are forced tohave a degenerate doublet at zero energy, as illustrated inFig. 3b. The doublet of states at zero energy will give riseto a Dirac cone centered at E0 by our previous arguments.Note that this is a nonperturbative argument for the existence

of a zero-energy pair of states in inversion symmetric latticesofhc/2e vortices. In general it is not possible to access thesestates within perturbation theory since they only appear

when the flux of the gauge field a takes on values that areodd multiples of. Thus, for the case of an inversion sym-metric lattice of hc/2e vortices as well, the linearized theorypredicts the existence of a Dirac node at zero energy.

IV. BEYOND THE LINEARIZED APPROXIMATION:

MASSIVE DIRAC QUASIPARTICLES

AND QUANTIZED THERMAL HALL EFFECT

We now consider the effect of the subdominant curvature

terms (H) on the spectrum obtained from the linearizedequations. In view of the smallness of these terms comparedto the linearized Hamiltonian, their primary effect will be tolift degeneracies that are present in the spectrum of the lin-earized problem. Therefore, we study the effect of theseterms near the Dirac cones band touchings of the linearizedproblem. In order to simplify the discussion we shall con-sider the the situation of a vortex lattice of hc/e doublevortices. For the physically relevant case of the hc/2 e vortexlattice, the discussions runs on very similar lines, althoughH in that case involves more terms arising from the Franz-

Tesanovic gauge field a Eq. 26. We do not present thedetails of that case here.

A. Massive Dirac quasiparticles

The problem at hand then is to consider the effect of

H1

2mp 2Ps

2 rz

1

2mp, P s r1

0

pF2

pxpyx,

42

the curvature terms, on the spectrum of the linearized Hamil-tonian at, say, node 1:

H1vFpxz1pyxPsx r1.

We have seen that the band structure of the linearizedproblem posesses Dirac cones at special points in the Bril-louin zone where there is a pair of degenerate states. Thisdegeneracy is a result of the invariance of the linearizedHamiltonian under the symmetry operation TDirac . The cur-vature terms however do not respect this symmetry underTDirac , H changes sign and so are expected to split thedegenerate doublets, giving rise to a gap in the dispersionsee Fig. 5. This may be analyzed within degenerate pertur-bation theory, the details of which are below.

Consider a degenerate doublet (,*) of the linearizedHamiltonian at one of the special points in the Brillouinzone. We may treat the effect of H within degenerate per-turbation theory, since the other energy levels at this crystalmomentum are separated by energies of order E1 or E2, thatare much larger than the strength of the perturbation H. Onprojecting H into this two dimensional subspace, it can bewritten as

H Pr o j eBm nH, 43where are the Pauli matrices that act within this two di-mensional subspace, and there is no term proportional to 1,since H changes sign under TDirac . This defines for us the

vector nH . The dependence on the magnetic field B is ex-plicitly exhibited in the prefactor. Further, if we neglect thelast term in H Eq. 42, that arises from the curvature ofthe gap and is smaller than the other terms by a factor of

/EF , then nH defined above is only a function of theanisotoropy , and the vortex lattice geometryi.e., the typeof lattice and its orientation relative to the nodes.

We now derive the effect of the perturbation H Pr o j onthe dispersion around the special Brillouin-zone points. Con-

sider making a small excursion in crystal momentum k

(k2/d the typical reciprocal lattice vector from thispoint. As in Eq. 32, we obtain an additional term in thelinearized Hamiltonian HkvF kxz

1kyx whichwhen projected into the two dimensional space of the degen-erate doublet takes the form

Hk Pr o jvFkxnkx1kynk

y, 44

where, once again there is no term proportional to 1, since

Hk changes sign under TDirac . nkx and n

kyare, in general, a

FIG. 5. On including the effect of curvature terms, the Dirac

cone obtained at zero energy for inversion symmetric lattices a

now acquire a small mass gap 2 mD b. The chemical potential for

quasiparticles, that is set by the Zeeman energy EZ , is shown here

as lying within the gap.


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pair of linearly independent vectors, that are defined by theabove equations. It may easily be seen that the dispersionresulting from the sum of these projected Hamiltonians H Pr o jHk Pr o j , is that of a massive Dirac particlewith mass

mD

eB

m nH

nk

xnk

y

n

kxn

ky, 45

i.e. the component ofnH perpendicular to the plane defined

by nkx and n

ky. This is most easily seen for the case when nkx

and nky

are orthogonal.32 Then, if nkx

(nky

) is the unit vector

in the direction of nkx (n

ky) we can write

HHk Pr o jvF kxkx0 nkx

v kyky0nky

mD nkx nky

. 46

The center of the dispersion is now at (kx0 ,ky

0)

(eB/mvF)(n

H nkx,n

H nky) and the renormalized

velocities are vFvFnkx and vvn

ky and the mass

term mD(eB/m)nH(nkx nky

). In an appropriately

chosen basis the above projected Hamiltonian will take theform:

HHk Pr o jvF kxkx01v kyky

0 2mD3 .47

Clearly, the resulting dispersion is

E kvF kxkx02v kyky

0 2mD2 , 48

i.e., that of a massive Dirac particle with mass mD , centeredat the crystal momentum (kx

0 ,ky0) as shown in Fig. 5. Of

course, to calculate these quantities requires an explicitknowledge of the wave function . Thus, the Dirac nodesobtained within the linearized approximation acquire smallgaps upon the inclusion of the curvature terms. An explicitnumerical calculation of these gaps is presented later in thissection. Below, we invoke some well-known properties ofmassive Dirac particles in two dimensions, to derive someconsequences for quasiparticle transport in the vortex latticestate.

B. Quantized thermal transport

The low temperature transport properties of quasiparticlesdepends strongly on the nature of the spectrum at the chemi-cal potential. For vortex lattices that possess inversion sym-metry, the particle-hole symmetry of the spectrum at eachnode will allow us to make some precise statements regard-ing the low-temperature transport propertiestherefore inwhat follows we specialize to the case of inversion symmet-ric lattices.

For inversion symmetric lattices, within the linearized ap-

proximation, there exists a Dirac node at the (k0) pointcentered at zero energy. On including the effect of the cur-

vature terms, H, we have seen that a gap is induced and thespectrum near zero energy is that of a massive Dirac particle,that retains particle-hole symmetry, and whose dispersion iscentered near the point. All the negative-energy states arethen occupied in the d particle representation of quasiparti-cles that has been adopted since the quasiparticle chemicalpotential, in the absence of Zeeman splitting, is at zero en-ergy.

This situation is topologically identical to a free Diracparticle in two dimensions, with a mass term. In other words,the spectra of the two systems can be continuously deformedinto one another without closing the gap at the chemicalpotential. The massive Dirac equation in two dimensions has

the dispersion E(p)p 2m2 and the negative-energybranch is completely filled in the ground state. There, it iswell known that the zero temperature Hall conductance of

the Dirac particles is quantized xy12 (eD

2/h)sgnm where

eD is the charge associated with the Dirac particle.15,16 Su-

perconductor quasiparticles of course do not carry a welldefined electrical chargehowever, for the case of interestthe component of the quasiparticle spin along the applied

magnetic field is conserved, and plays the role of eD ; as wehave seen the density of the conserved d particles corre-sponds to the spin density in the direction of the field. Hence,in the present case of quasiparticles we expect a quantizedspin Hall conductance,17 18 with eD replaced by the quantumof spin /2 carried by the quasiparticles. Thus, if we define

0S(/2)2/h ,

xyS

1

20

SsgnmD, 49

xxS0 50

for a single node.It is easily seen that for inversion-symmetric lattices, op-

posite nodes contribute equally to the spin Hall conductivity.

The Hamiltonian at node 1, H1H, under the inversion

operation 1(r)1(r), is found to be identical to H1H. Therefore they have the same gap mD(1) mD(1) , and since the Hall conductance is unchanged bythe inversion rotation by an angle of), have identical spin

Hall conductances xyS (1)xy

S (1). Similarly we can relate

the gaps and spin Hall conductances between nodes 2 and

2.33

If no further symmetry of the vortex lattice is assumed,

the size of the gap and the sign of the contribution to xyS

from the nodes 2 and 2 are not simply related to their values

at the nodes 1 and 1. Thus, as regards the low-temperatureHall transport, two scenarios present themselves. The first inwhich the contribution from both pairs of nodes have the

same sign, in which case xy ,To tS

4xyS (1 )

20S sgn mD(1) and the second, when contributions from

the two pairs of nodes are opposite and cancel, xy ,To tS

2xyS (1)2xy

S (2 )0, and the spin Hall conductance isquantized to zero. In both cases the quantization begins to setin at temperatures lower than the smallest gap. These two


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cases are topologically equivalent to homogenous supercon-

ductors with pairing wave functions that break time reversal

symmetry, dx2-y 2idxy in the first case and dx2-y2is , or any

other thermal insulator, in the second case.

When the vortex lattice possesses an additional symmetry,

that of reflection about an axis that bisects the nodal direc-

tions dashed lines in Fig. 1, we can relate the physics at the

nodes (1 1

) with that at the nodes (2 2

). We then havemD(1 ) mD(2 ) and the spin Hall conductances from allthe nodes add. Thus, the first scenario, with xy ,To t

S

20S sgn mD(1 ) will be realized. For lattices that weakly

break this symmetry, the exact equality of the gaps mD(1 )and mD(2 ) no longer holds, but the spin Hall conductanceis still expected to remain equal to 20

Ssgn mD(1) . A finite

deformation of the lattice is at least required to make a tran-

sition to the topologically distinct state with xy ,To tS

0.

Effect of Zeeman splitting on quantization: So far we have

neglected the Zeeman splitting of the quasiparticles by the

magnetic field, which, as we have seen earlier, plays the role

of the chemical potential for the d particles. The magnitude

of this term, EZeeman 12 gBB0.7 K T1, is of the sameorder as the expected mass gap in the vortex state of YBCO

mDE12/EF0.5 K T

1 and has the same field dependence.

Therefore, the Zeeman term plays an important role in deter-

mining whether or not the exact quantization of the spin Hall

conductance discussed above will be realized in these mate-

rials. If the quasiparticle gap mDminmD(1),mD(2)exceeds the Zeeman term mDEZeeman , then the quasi-particle chemical potential lies in the gap, and quantization

of the spin Hall conductance is expected, although the tem-

perature below which this will set in is now given by the

difference mDEZeeman . If, however, the Zeeman split-

ting exceeds the gap, then the chemical potential for the qua-siparticles lies in a region where extended states are present,

and the exact quantization will be lost. When disorder is

taken into account, these states may be localized and then

exact quantization will be recovered. The question ofwhether the quasiparticle gap exceeds the Zeeman energydepends on several details of the problem such as the valueof the anisotropy and the vortex lattice geometry, as wellas on the value of the band mass and g factor of the electronfor the material in consideration.

Quasiparticle heat transport: While so far we have beendiscussing the spin conductance of the quasiparticles, a farmore accessible transport quantity in experiments is the ther-mal conductance. In the superconductor, heat is transportedby the quasiparticles as well as the phonons. The electronicpart of the thermal conductivity is usually isolated in one ofthe following ways. For the longitudinal part of the thermalconductivity (xx ), the phononic contribution at low tem-peratures varies as T,35 and can typically be separated out. Asfor the thermal Hall conductance, that is expected to arisepurely from the quasiparticles, since phonons are not ex-pected to be skew scattered by the magnetic field.2

At low temperatures, a Widermann-Franz relation be-tween the thermal and spin conductances of the quasiparti-cles is expected to hold:

limT0

T 23 kB

2

/22 S. 51

Thus, for the cases discussed earlier, we are led to expect, atlow temperatures, a quantized thermal Hall conductivity,

in the sense xy /Tn (2/3)(kB

2/h) , where n2,0 for

the two possible cases, and (xx /T)0.

Violation of Simon-Lee scaling. In Ref. 14, Simon andLee showed that the leading contribution to the thermal Hallconductivity takes the scaling form

xy T2Fxy T/H

1/2vFv . 52

Clearly, the quantized Hall conductivity derived abovedoes not satisfy this scaling relation. The reason for this iseasily seen. In deriving Eq. 52 for the vortex lattice follow-ing the arguments in Ref. 14, we need to assume that thespectrum of states in the linearized theory at a given crystalmomentum, are well separated in energy, and so are onlyweakly mixed by the perturbing curvature terms (H).While this assumption is true over most of the Brillouin

zone, it breaks down at the isolated points where doublydegenerate states are present. It is precisely the lifting of thisdegeneracy by H that leads to the mass term for the Diracquasiparticles, and the quantized thermal Hall conductivitythat violates the scaling form above. For temperatures ex-

ceeding the induced gap mDE12/EF , Simon-Lee scaling is

recovered.

C. Numerical evaluation of gaps

In order to make the preceeding discussions more con-crete, here we present a numerical evaluation of the gapsinduced by the curvature terms. These results confirm ourassumption that their effect on the spectrum of the linearized

Hamiltonian is indeed small. We consider the simple situa-tion of a square lattice of hc/e double vortices, orientedalong the nodal directions. The mass term induced by Hfor the Dirac cone at zero energy is calculated at node 1 fortwo values of the anisotropy parameter (1,2). Reflectionsymmetry of the lattice about the 45 line allows us to relatethe mass term at the other nodes to mD(1). The linearizedHamiltonian to consider is

H1vFpxz1pyxxPs r1 , 53

where the explicit form of Ps(r) that we use is

P

s r

h

d2 G0zG

G 2 eiGr

, 54

where G(2/d)( m ,n) are the reciprocal vectors for asquare vortex lattice with lattice parameter d, which fromflux quantization for a hc/e vortex lattice satisfies d

h/eB. This form for Ps may be derived from Eq. 29 ifwe assume that a Ps is divergence free, i.e., Ps(r)0,which is valid if the superfluid density can be taken as uni-form; and b the magnetic field is constant, which is justifiedif the penetration depth is large compared to the magneticlength d/1.


064504-12


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By our previous arguments we are assured of a pair ofdegenerate states at E0 for this linearized Hamiltonian,i.e., there exists a pair of linearly independent wave func-

tions that satisfies H1(r)0. The pair of degenerate statesare obtained numerically, and the effect of H is evaluatedin degenerate perturbation theory within this two dimen-sional subspace, to yield the Dirac mass mD Eq. 45. Thus,the curvature terms

H1

2mp 2z

1

2mPs

2 rz

1

mpPs1 0

p F2

pxpyx55

generate a Dirac mass at this node;

mDm1m2m3m4,

where each term m i results from the corresponding piece inH. The term m4, which is smaller by a factor of /EF , canbe neglected. The remaining terms can be written in the fol-lowing form that explicitly displays their dependence on themagnetic field B and band mass ( m).

m1,2,31,2,3eB

2m,

To t123 ,

mDTo teB

2m,

where is are dimensionless quantities that depend only onthe lattice geometry and the value of the anisotropy .Therefore, it is convenient to present the numerical results interms of them. Physical quantities of interest are easily re-

lated to To t . Its magnitude sets the size of the gap while itssign determines the sense of the quantized thermal Hall ef-fect. Thus, the gap is given by 2 mD2To t(eB/2 m) and is the same at all nodes due to the reflection symmetryabout the 45 line of the square lattice being considered. Thethermal Hall response from all these nodes add to give thequantized thermal Hall conductance:

xy /T22kB

2

3hsgn To t eBm .

We formulate the numerical problem in reciprocal momen-tum space, retaining (2L1) 2 reciprocal lattice vectors.The null space of H

1(H

10) at the point is obtained

and the Dirac mass terms calculated using Eqs. 4345.These are shown in Fig. 6 for two values of (1,2),where To t as well as the separate contribution of the differ-ent curvature terms, the is are plotted as a function of L. Wesee that satisfactory convergence is obtained by L16. Themagnitude of the Dirac mass, in both cases turns out to beroughly mD1.2eB/2m which is 0.8 K T

1, if m istaken as the electron mass. Thus, for fields of a few T, thegap is much smaller than the energy scale E1 50 K T

1/2 inYBCO, which justifies our treatment of the curvature termsas perturbations on the linearized Hamiltonian. Incidentally,

the mass term in this particular example, exceeds the Zee-

man energy 0.7 K T1 with g2), and so the chemical

potential for the quasiparticles will lie within the gap.

The overall sign of the quantized thermal Hall conductiv-ity is clearly seen to be a function of ; it reverses sign ongoing from 1 to 2 (To t takes opposite signs forthese two values of). Its dependence on the sign of eB/mis expected, since on inverting the sign of any of thesee,B,m, the quasiparticles are expected to deflect in the op-posite direction.

Thus, given a real material whose vortex lattice structureand microscopic parameters are known, one can follow therecipe above to a calculate if a quantized Hall conductanceis expected for the clean lattice case i.e., does mD exceed

FIG. 6. Numerical evaluation of the Dirac mass term for a

square vortex lattice of hc/e vortices for two values of the anisot-

ropy: 1.0 and 2.0. The magnitude of the dimensionless quantity

To t controls the gap size, while its sign determines the sign of the

quantized thermal Hall effect. Here To t , as well as the separate

contributions i arising from the different curvature terms, are plot-

ted as a function ofL, that is a measure of the number of reciprocal

lattice vectors ( 2L1) 2, retained in the numerical calculation.


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EZeeman ), and if so b the quantization value (2,0 in ap-propriate units, and the temperature scale set by mD EZeeman) below which quantization is expected.

We have already noted the similarity of the state withquantized thermal Hall conductivity of 2 to the puredx2-y 2idxy systemboth are gapped and have identical lowtemperature thermal conductivities. Other authors19,20 con-sidered the possibility of a magnetic field inducing an idxycomponent in a dx2-y 2 superconductor. We believe that ourwork for the vortex lattice provides a concrete realization ofthe general ideas of Refs. 19 and 20. Further, we have de-scribed a procedure that allows us to determine when such astate is to be expected, and the size of the energy gap in-duced.

V. DISCUSSION

In summary, we have considered the problem of d-wavequasiparticles in a mixed state within the perfect lattice ap-proximation. Within the linearized theory we find a masslessDirac dispersion for the low-energy quasiparticles, so long as

the vortex lattice possesses inversion symmetry. On goingbeyond the linearized approximation and including thesmaller curvature terms, a small gap is found to open and thelow-energy excitations now behave like massive Dirac par-ticles. The size of the gap is proportional to the applied mag-

netic field, and is roughly of order E12/EF (0.5 K T

1 inYBCO. If the sign of the Dirac mass term appropriatelydefined is identical at the four nodal points, then we obtaina topologically nontrivial statei.e., one that has gaplesschiral quasiparticle modes at the edge, which in this case arethe same as in a pure dx2-y 2idxy superconductor. When thechemical potential for quasiparticles lies within thegap a quantized thermal Hall conductivity of xy /T

2(

2

kB2

/3h) is expected at low temperatures. Thechemical potential for these quasiparticles is controlled bythe Zeeman energy EZ , and if this is smaller than the gapscale mD the quantized thermal Hall effect will be observedat low temperatures ( T mDEZ). We also described acomputationally simple procedure for evaluating the numeri-cal value of the gap and the quantized themal Hall conduc-tance, which depend on the microscopic material parametersas well as the value of the anisotropy and the geometry of thevortex lattice.

Although in general the quantized thermal Hall conduc-tivity can take on any integer value see the Appendix, onthe basis of the energetics for this particular situation wehave argued that the thermal Hall coefficient is quantized toone of the discrete set of values (xy /T2, 0 in appropri-ate units. This is a consequence of a the independent nodeapproximation, and b that at each of the four nodes, the

only low-energy feature compared to E12/EF) is a single

Dirac cone centered at zero energy within the linearizedproblem for inversion symmetric lattices. For special situa-tions where the linearized theory may posess additional lowenergy features, for example as reported in Refs. 10, 11, and13 for the square lattice of vortices at large anisotropy ( ),other integer values of the quantized thermal Hall conduc-tance may be realized.

Throughout, we have treated the four nodes as being in-

dependent, and neglected the effect of internode scattering.

This is expected to be important when there is a high degree

of commensuration between the reciprocal vectors associated

with the vortex lattice, and the momenta separating the fournodes at the Fermi surface. In that case, as demonstrated inRef. 12 the nodal points acquire a gap due to the formationof a quasiparticle density wave. However, away from such

special commensuration, the independent node approxima-tion used here is expected to hold.

Our discussion in this paper has been restricted to theclean limit but of course disorder is always present in thephysical system. The topological character of the states wehave been considering renders them insensitive to the effectsof weak disorder. Further, if the quasiparticle chemical po-tential lies outside the gap, while a quantized thermal Halleffect would not be expected in the pure case, disorder couldlocalizes the quasiparticle states at the chemical potential sta-bilizing the quantization. However, for larger disorderstrengths, a completely different approach that does not relyon the Bloch nature of the quasiparticle states will be re-

quired. With these caveats in mind we turn to survey some ofthe relevant experiments in YBCO.

Low temperature thermal Hall measurements would pro-vide the most direct test of the occurrence of such topologi-cally ordered states in superconductors. Currently, such mea-surements on YBa2Cu3O7 go down to temperatures of about12 K in a field of 14 T,3 which is presumably still too high toobserve the quantization, should it exist. Indeed, although themeasured xy at a given temperature is found to saturate forstronger fields, the value at this plateau scales as T2, ratherthan linear in T as would be expected if xy /T was quan-tized. However it is an intriguing fact that the plateau valueofxy at the lowest temperature measured 12.5 K is very

close to what would be expected from a quantized thermalHall conductance ofxy /T2 in appropriate units. Whilethis experiment is not conclusive with regard to the low-temperature state of the quasiparticles, we can look tospecific-heat measurements for a signature of the chemicalpotential lying in a gap. Such low-temperature specific-heatmeasurements down to 1 K in magnetic fields of 14 T inYBa2Cu3O7 were reported in Ref. 6. However, they show noevidence of a gap, in fact a finite density of the quasiparticlestate that scales as the square root of the field was found. Inprinciple, a quantized thermal Hall condutance could stillarise in such a system if the states at the chemical potentialare localizedin all cases the quantization of the thermalHall conductance will be accompanied by the vanishing ofthe longitudinal thermal conductance xx /T0 at tempera-tures below the mobility gap scale. But low-temperatureT0.1 K longitudinal thermal conductivity measurementsin fields up to 8 T in YBa2Cu3O6.9 Ref. 5 reveals xx /Tsaturating to a nonzero value that rules out a quantized ther-mal Hall effect in this material down to these low tempera-tures. Thus, for this case of YBCO, either the Zeeman split-ting causes the chemical potential to lie outside the gapregion, or else the vortex lattice in this case is so disorderedthat the perfect lattice assumption we start with requires se-rious modification. Nevertheless, given the large number of


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potentially different experimental systems with a d-wave gapthat are available, it is not unreasonable to expect that thequantized thermal Hall effect, realized in the manner de-scribed, will be observed in the future.

ACKNOWLEDGMENTS

The author thanks F. D. M. Haldane for several stimulat-

ing conversations, and L. Balents, B. Halperin, D. Huse, M.Krogh, L. Marinelli, A. Melikyan, N. P. Ong, T. Senthil, Z.Tesanovic, and O. Vafek for useful discussions. Support fromNSF Grant No. DMR-9809483 and the C.E. Procter fund, aswell as hospitality at the ITP, Santa Barbara, where part ofthis work was done, are acknowledged. After the completionof this work we learnt of Ref. 34, where a numerical com-putation of Chern numbers of quasiparticle bands is reported,which, in the region of overlap, agrees with our results.

APPENDIX: CHERN NUMBERS FOR QUASIPARTICLE

BANDS

Consider calculating the spin Hall conductivity of the full

quasiparticle Hamiltonian HBdG the linearized Hamiltonianhas vanishing spin Hall conductance due to invariance underTDirac). We use the Kubo formula that evaluates the responseof the quasiparticle current jxHBdG /px to a perturbationcorresponding to a uniform Zeeman gradient e Hey 1along the orthogonal direction.

At zero temperature the spin Hall conductivity for an iso-lated filled band takes a particularly simple form, just as inthe case of the electrical Hall effect with charged particles ina periodic potential.35 It is proportional to a topological in-variant, the Chern number,9 that can be asigned to the qua-siparticle band labeled by n via the formulas

xyspin n 1

h 2 2

Cn , A1

Cn1

2

B .zonekA kd

2k. A2

The vector potential is defined in terms of the wave functionsof the quasiparticles in that band by

A ki d2rV

u nk* rku nk rvnk* rkvnk r,

A3

where the integral is over the unit cell, and we have writtenout the components of the quasiparticle wavefunctions ofband n as nk(r) u nk(r)vnk(r)

T that we choose to benormalized according to

1unit cell

d2r

V unk r

2 vnk r

2, A4

where V is the area of the unit cell. Clearly, a phase redefi-nition of the wave function at each point of the Brillouinzone

nk

e i( k)nk

corresponds to a gauge transformation

A(k)A(k)k(k). This does not affect the value of theChern numberwhich is thus defined despite the phase am-biguity of the wave functions. Following standard argumentsdeveloped for electrons,36 it is easily established that for thecase of superconductor quasiparticles as well the Cn are in-tegers and hence the spin Hall conductivity, in appropriateunits, will be quantized to integers for this case of quasipar-ticles of a spin-singlet superconductors, so long as thechemical potential for quasiparticles lies in a gap.17,18 For thecase of homogenous superconductors that break time-reversal and -parity symmetries, the appropriate limit of ex-pressions A2 and A3 reduces to the topological invariantof Refs. 17 and 37.

The quasiparticles therefore, could be said to blend to-gether Bloch wave and Landau-level properties. While theyform bands that are labeled by Bloch crystal momenta, thesebands, like Landau levels, carry a Chern number that is re-lated to the spin Hall conductivity.

1 See D. J. van Harlingen, Rev. Mod. Phys. 67, 515 1995.2 K. Krishana, J. M. Harris, and N. P. Ong, Phys. Rev. Lett. 75,

3529 1995; B. Zeini et al., ibid. 82, 2175 1999; K. Krishana

et al., ibid. 82, 5108 1999.3 Y. Zhang et al., Phys. Rev. Lett. 86, 890 2001.4 M. Chiao et al., Phys. Rev. B 62, 3554 2000.5 May Chiao et al., Phys. Rev. Lett. 82, 2943 1999.6 Y. Wang et al., Phys. Rev. B 63, 094508 2001.7 L. P. Gorkov and J. R. Schrieffer, Phys. Rev. Lett. 80, 3360

1998.8 P. W. Anderson, cond-mat/9812063 unpublished.9 Y. Morita and Y. Hatsugai, cond-mat/0007067 unpublished.

10 M. Franz and Z. Tesanovic, Phys. Rev. Lett. 84, 554 2000.11 L. Marinelli, B. I. Halperin, and S. H. Simon, Phys. Rev. B 62,

3488 2000.

12 O. Vafek, A. Melikyan, M. Franz, and Z. Tesanovic, Phys. Rev. B

63, 134509 2001.13 D. Knapp, C. Kallin, and A. J. Berlinsky, cond-mat/0011053 un-

published.14 S. H. Simon and P. A. Lee, Phys. Rev. Lett. 78, 1548 1997.15 F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 1988.16 A. W. W. Ludwig, M. P. A. Fisher, R. Fisher, R. Shankar, and G.

Grinstein, Phys. Rev. B 50, 7526 1994.17 G. E. Volovik and V. M. Yakovenko, J. Phys.: Condens. Matter 1,

5263 1989; G. E. Volovik, Pisma Zh. Eksp. Teor. Fiz. 66, 492

1997 JETP Lett. 66, 522 1997.18 T. Senthil, J. B. Marston, and Matthew P. A. Fisher, Phys. Rev. B

60, 4245 1999.19 R. B. Laughlin, Phys. Rev. Lett. 80, 5188 1998.20 T. V. Ramakrishnan, J. Phys. Chem. Solids 59, 1750 1998.


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21 Ashvin Vishwanath, Phys. Rev. Lett. 87, 217004 2001.22 R. Cubitt et al., Nature London 365, 407 1993.23 B. Keimer et al., Phys. Rev. Lett. 73, 3459 1994; S. T. Johnson

et al., ibid. 82, 2792 1999.24 I. Vekhter and A. Houghton, Phys. Rev. Lett. 83, 4626 1999.25 T. Senthil and M. P. A. Fisher, Phys. Rev. 62, 7850 2000.26 Jinwu Ye, Phys. Rev. Lett. 86, 316 2001.27 At temperature T, the largest typical momentum for 1 is

py ma xT/vF . The quadratic term is also of the same orderif the temperature reaches Tma x1/ 2mpy ma x

24EF /

2. For

YBa2Cu3O6.9 , 14, EF3, and 300 K, and there is a mass

anisotropy that we have not explicitly included which boosts

Tma x by a factor of 3 to give Tma x200 K eventually.28 Note, that each of these factors can also be written as (zz i /zz i)

(1/2)qi which is familiar from Chern-Simons transforma-

tions for the QHE problem.29 A. Vishwanath, cond-mat/0104213 unpublished.30 In fact, particle-hole symmetry may be obtained at every point on

the Brillouin zone Ref. 11, by combining the previous inver-

sion transformation with TDirac . Thus the transformation

(x ,y)*(x ,y) leaves the crystal momentum unaf-fected, while changing the sign of the energy.

31 Strictly speaking, we just need to have an odd number of doublets

at zero energy, which in the generic case will be just one dou-

blet.32 This is guaranteed at node 1,1 (2,2) if the vortex lattice is sym-

metric under the reflection xx (yy).

33 In fact, this equality of mD for the opposite nodes may be provedmore generally, and holds even in the absence of inversion sym-

metry. It derives from the intrinsic particle-hole symmetry

present in the Boguliubovde Gennes equation, that relates the

opposite nodes.34 O. Vafek, A. Melikyan, and Z. Tesanovic, Phys. Rev. B 64,

224508 2001.35 D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs,

Phys. Rev. Lett. 49, 405 1982.36 M. Kohmoto, Ann. Phys. N.Y. 160, 343 1985.37 N. Read and D. Green, Phys. Rev. B 61, 10267 2000.


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