Relativistic framework for microscopic theories of ... · Relativistic framework for microscopic theories of superconductivity. I. The Dirac equation for superconductors K. Capelle

Relativistic framework for microscopic theories of superconductivity.I. The Dirac equation for superconductors

K. CapelleandE. K. U. GrossInstitut fur Theoretische Physik, Universitat Wurzburg, Am Hubland, D-97074 Wurzburg, Germany�

Received14 October1997�We�

presenta unified treatmentof relativistic effectsin superconductors.The relativistically correct � Dirac-type� �

single-particleHamiltoniandescribingthe quasiparticlespectrumof superconductorsis deducedfromsymmetry� considerationsandtherequirementof thecorrectnonrelativisticlimit. We providea completelist ofall� orderparametersconsistentwith the requirementof Lorentzcovariance.This list containsthe relativisticgeneralizations� of theBCSandthe triplet orderparameters,amongothers.Furthermore,we presenta symme-try�

classificationof the order parametersaccordingto their behaviorunder the Lorentz group, generalizingprevious treatmentsthatwerebasedon theGalilei group.Theconsiderationsin this paperarebasedonly on theconcepts of pairing andLorentzcovariance.They can thereforebe appliedto all situationsin which pairingtakes�

place. This includes BCS-type superconductors,as well as the heavy-fermioncompounds,high-temperature�

superconductors,pairing of neutronsandprotonsin neutronstars,andsuperfluidhelium 3.�S0163-1829�

99� �

11305-5�

I. INTRODUCTION

This is the first in a seriesof two papersdevotedto aninvestigationof theeffectsof relativity in superconductors.Ithas�

beennotedbeforeby many authorsthat relativistic ef-fects�

can have a profound influenceon superconductivity.Spin-orbit�

coupling, a relativistic effect of secondorder in� /�c� ,� is known to influence the symmetry of the order

parameter,� 1–4 the�

spin-susceptibilityand the Knight shift,5,6�

magnetic impurities in superconductors,7�

Josephson�

currents,� 1,8–10 the�

value of the uppercritical field,11,12 H�

c� 2 ,�and the magnetoopticalresponseof superconductors,13–15

among other quantities.Relativistic correctionsto the Coo-per� pair masshave beenevaluatedtheoreticallyand mea-sured! experimentally.16–19 The self-consistentscreeningofthe�

currentswhich givesrise to theMeissnereffect is duetothe�

current-currentinteraction,20,21"

which# is of relativisticorigin$ andof secondorderin % /

�c� . Plasmafrequencyanoma-

lies which were observedin somehigh-temperaturesuper-conductors� havebeensuggestedto be dueto current-currentinteractions&

aswell.22 In'

theanyonictheoryof superconduc-tivity�

it was recentlyarguedthat one needsto start from arelativisticLagrangianin orderto obtaina completedescrip-tion�

of theMeissnereffect.23 In'

studyingtheelectrodynamicsof$ vorticesin high-temperaturesuperconductorsit wasfoundnecessaryto employ a relativistically covariantwave equa-tion�

in orderto explainthe experimentaldata.24"

The(

relativ-istically&

covarianttheorypresentedin the presentpaperpro-vides) a unified frameworkwithin which sucheffectscanbeinterpretedandanalyzed.

Moreover,*

wheneverthereareelementswith atomicnum-ber+

Z , 40 in the lattice,25"

the�

bandstructurehasto becalcu-latedusingrelativisticmethods.2

"6–28 Many interestingsuper-

conductors,� e.g.,theheavy-fermioncompoundsandthehigh-temperature�

superconductors,do indeedcontainvery heavyelements,- suchas mercury(Z

. /80)0

, uranium(Z. 1

92)2

, bis-muth (Z 3 83)

0, thallium (Z 4 81)

0, platinum(Z 5 78)

6, etc. In

Sec.�

V of the presentpaperan equationis derived whichallows oneto performsuchrelativistic band-structurecalcu-lationsfor superconductors.

In spiteof thefact thatrelativity thusobviouslyis relevantfor�

a largenumberof interestingeffectsin superconductors,a unified andcovariantrelativistic approachto superconduc-tivity�

hasnot beenworkedout, until very recently.In a pre-vious) paper29

"we# presenteda first steptowardssucha rela-

tivistic�

theory of superconductivity.That theory led to arelativistic7 generalization of the Bogolubov–de Gennesequations- of superconductivity.By performing a weaklyrelativistic7 expansionup to secondorderin 8 /

�c� ,� where9 is

&a

typical�

velocity of theparticlesinvolved,we foundtheusualrelativistic corrections : spin-orbit! coupling, mass-velocitycorrection,� and Darwin term; in

&a form appropriatefor su-

perconductors.� Furthermore,a number of new relativisticcorrections� of the same< order$ in = /

�c� emerged,- which exist in

superconductors! only. Thesenew termscould be identifiedas thesuperconductingcounterpartsof thespin-orbitandtheDarwin term, with the pair potentialtaking the placeof thelatticepotential.Theappearanceof suchtermscanbe tracedback+

to the complexinterplaybetweenrelativistic symmetrybreaking+

and superconductingcoherence.The theory hasmeanwhilebeenshownto lead to potentiallyobservableef-fectson, e.g.,theenergyspectrumof a superconductor29

"and

on$ the magneto-opticalresponseof superconductors.13–15

The(

presentpaperprovidesa more generalapproachtorelativistic effectsin superconductors.It is organizedasfol-lows: after this introduction we show, in Sec. II, how theDirac>

equationis generalizedto describesuperconductorsbyintroducing the order parametersas 4 ? 4 matricesinto theDirac>

Hamiltonian.In'

Secs.III and IV we presenta detailedanalysisof theresultingHamiltonian.We discussLorentzcovarianceof theformalism and investigatethe behaviorof the order param-eters- underLorentztransformations.This leadsto a classifi-cation� of all possibleorder parameterswith respectto theirbehavior+

undertheoperationsof theLorentzgroup.Next we

PHYSICAL REVIEW B 1 MARCH 1999-IIVOLUME@

59, NUMBER 10

PRB 590163-1829/99/59A 10B /7140C D

15E /$15.00C

7140 ©1999The AmericanPhysicalSociety

show! that theseorderparameterscanbe interpretedin termsof$ the symmetriesof the underlyingCooperpair states.ThisgivesF rise to a relativistic generalizationof the conceptsofsinglet! and triplet superconductivityand allows one tospecify! what kind of physicsis describedby the variousor-derG

parameters.In'

Sec.V we discussthe diagonalizationof the Hamil-tonian.�

This leads to a set of differential equationswhichgeneralizeF the Bogolubov–de Gennesequationsof the non-relativistic theory.

WhileH

the presentpaperthus dealswith the Dirac equa-tion�

for superconductors,the secondpaperof this series30I

will# treat the Pauli equationfor superconductors.In that pa-per� we takeup the topic of weakly relativistic correctionstothe�

conventional theory of superconductivityand discusssome! observableconsequencesof the new terms.

Unfortunately,J

the respectiveterminologiesof relativityand superconductivityare ratherdifferent and thereis littleoverlap$ in the literatureon theseapparentlydistinct fields ofphysics.� To aid thenonspecialist,we havethereforeincludedtwo�

sectionsin which we briefly review somepertinentas-pects� of themicroscopictheoryof inhomogeneoussupercon-ductorsG K

Sec.�

II A L and of relativisticcovarianceM Sec.�

III A N .It shouldbe stressedfrom the outsetthat we do not at-

tempt�

to formulatea fully relativistic interactingO

field theoryof$ superconductivity:We quantizeonly the electrondegreesof$ freedomand treat the externalfields as classicalfields.Furthermore,the effectsof relativity areconsideredonly onthe�

single-particle level, i.e., on the level of theBogolubovP

–de Gennesequations.A relativistic treatmentofthe�

interactionis beyondthe scopeof the presentpaper,al-though�

we offer someremarksconcerningthis topic in Sec.V CQ

.

II.R

GENERAL FRAMEWORK FOR A RELATIVISTICTHEORY OF SUPERCONDUCTIVITY

A. A basis for the nonrelativistic order parameter

In'

the nonrelativisticcaseit is well known that the orderparameter� S OP

T Ustructure! is determinedby very generalsym-

metry considerations,as follows:3I1–34 Cooper

Vpairing takes

place� betweensingle-particlestateswhich are,for any givenstate! W nX Y[Z ,� constructedby applicationof the time-reversaland parity operators.If nX stands! for a completesetof quan-tum�

numbersneededto label a normal-stateeigenfunction,we# thushavethe states

\nX ]_^ ,� ` Ta ˆ nX b[c ,� d Pe ˆ nX f[g ,� h Pe ˆ T

a ˆ nX i[j k 1lavailable for pairing, where T

a ˆ and Pe ˆ are the time-reversal

and parity operators,respectively.Thepossiblepairingstatesof$ the superconductorcan thenbe characterizedin termsofthese�

statesas3I

1–34

mSn oqpsr

nX t ,� TnX u_vxwsy PTnX z ,� PnX {[| ,� } 2~�T �s�q�s� nX � ,� PnX �_� ,� � 3� �

�T �s�x�s� PTnX � ,� TnX �[� ,� � 4�

�T0� �x�s�

nX � ,� TnX �[�x�s� PTnX � ,� PnX �_ . ¡ 5¢ £

The physicalsignificanceof this constructionis easily seenby+

consideringa homogeneouselectrongas,wherespin andmomentumaregoodquantumnumbers.If ¤ nX ¥[¦q§s¨ k ©«ª ,� then¬Ta ˆ nX [®q¯s°²± k

³ ´¶µ,� · Pe ˆ nX ¸[¹qºs»½¼ k

³ ¾¶¿,� and À Pe ˆ T

a ˆ nX Á[ÂxÃsÄ k³ Å¶Æ . Tak-ing&

into accountthateachof thetwo-particlestatesis a Slaterdeterminant,G

the configuration-spacerepresentationof Ç Sn Ètakes�

the formÉrÊ ,� rÊ ËÍÌ Sn ÎxÏsÐÒÑ

kÓ Ô rÊ Õ×ÖÙØ k

Ó Ú rÊ ÛÝÜßÞáàÙâkÓ ã rÊ ä×å k

Ó æ rÊ çéèëêíìïîñðqòôóöõø÷úùöûüôýöþ ÿ

,� �6� �

where# the � kÓ (� r)�

arenormal-statesingle-particlewave func-tions�

andthe �� are theusualspin functions.Obviously,thespatial! part of Sn � is anevenfunctionunderexchangeof thetwo�

particles,while thespinpart is odd.This statedescribesthe�

conventionalsinglet Cooperpair.35,36I

Representing�

thePauli

spinorsas �� (�1,0)T and �� (

�0,1)T,� this canbewrit-

ten�

as �rÊ ,� rÊ �� Sn �� "! k

Ó # rÊ $&%(' kÓ ) rÊ *,+.-0/(1 k

Ó 2 rÊ 3&4 kÓ 5 rÊ 687�9 m: ˆ 1 ; 76 <

with#

m: ˆ 1

0 1�= 1 0

. > 80 ?The(

otherthreestatesareevenfunctionsof thespinvariablesand odd functionsof the spatialcoordinates.They describetriplet�

Cooperpairs, as found in superfluidhelium 3 @ Ref.37� A

and, possibly, organic superconductors38I

and heavy-fermion�

compounds.39I

The(

spin partsof the triplet statesarerepresentedby

BTa CED�F

m: ˆ 2" 1 0

0 0� ,� G 92 H

IT JE K � L m: ˆ 3

I 0 0�0 1� ,� M 10N

OTP

0� QSR

m: ˆ 4

0 1�1 0

. T 11UThesematricesform thebasisfor manyapproachesto super-conductivity.� In particular,the Balian-Werthamerparametri-zationV for the order parameterof triplet superconductorsissimply! a linear combinationof the matrices W T XEY ,� Z T []\ ,�and ^ TP 0

� _.40,41 The

(fact that the singlet OP is composedof

time-reversed�

single-particlestatesis thebasisfor the theoryof$ impurities in BCS superconductors.42,43

`Furthermore,the

above definedstates a 2b c – d 5e f were# takenas a startingpointfor�

investigationsof theorder-parametersymmetryin uncon-ventional) superconductors,e.g.,in Refs.31–34.

Multiplying thesematrices from the left and from theright7 with spinorswhoseentriesare functionsof rÊ ,� yields areal-space7 representationof theOPwhich is commonlyusedin theBogolubov–de Gennesequations43

ànd in thedensity-

functional�

theory of superconductivity.21,41,44"

–49 Since�

thiswill# beof considerableimportancebelow,we demonstrateitexplicitly- for the singlet case. The conventionalBogolubov–de GennesHamiltonianfor a singlet supercon-ductorG

is43`

PRB 59 7141RELATIVISTICg

FRAMEWORK FOR . . .g

. I. ...

Hnon-relh di 3Ir jlknm k† o rp pq 2

2m: rtsvu rw.xzy {|&} r~� d

i 3Ir d� 3

Ir� ��* � rÊ ,� rÊ �,�S� ˆ �� rÊ ,� rÊ �,�.� H.c.

� �,� �

12�where# the �� are secondquantizedfield operators,� (

�rÊ ) i�

sthe�

lattice potential, �� (� rÊ ,� rÊ � )� is the pair potential,and theexpectation- value of � ˆ �� (� r,� r ¡ )� ¢�£

(�r)� ¤�¥

(�r ¦ )� is the § nonlo-

cal� ¨ singlet! OP.43`

In Sec.V C the role of this Hamiltonianindensity-functionalG

theory for superconductorsis further dis-cussed.�

The termsin Eq. © 12ª referring to superconductivitycanbe+

rewrittenas

di 3Ir d� 3

Ir� «¬�®�¯* ° rÊ ,� rÊ ±,²S³ ˆ ´�µ�¶ rÊ ,� rÊ ·,¸

¹ 1

2di 3Ir d3

Ir º¼ »�½�¾* ¿ r,� r À,Á

Â Ã�ÄÅ�Æ T 0 1�Ç 1 0

È�ÉÊ�Ë Ì 13Íand similarly for its Hermitianconjugate.We havemadeusehereof the fact that the pair potential for a singlet stateiseven- under interchangeof (r)

�and r Î . This is guaranteed

because+

thespinpartof thesingletstateis anoddfunctionofthe�

spins,and the full pair potential,having the symmetriesof$ a fermionicpair wavefunction,mustbeoddunderparticleexchange.-

It is seenthat thematrix Ï 80 Ð appears naturallyin Eq. Ñ 13Ò .In'

exactlythe sameway the matricesÓ 92 Ô – Õ 11Ö appear in thegeneralizationF of the Bogolubov–de Gennesequationstotriplet�

superconductors,as demonstratedexplicitly in Ref.41. The correspondingpair potentialsare odd functionsofthe�

spatialcoordinates.In'

the mostgeneralcase,all theseOP arepresentsimul-taneously.�

Then all the order parametersenter the Hamil-tonian,�

multiplied by the appropriatepair fields and the su-perconducting� term in the Hamiltonianbecomes

1

2di 3Ir di 3Ir × Ø�Ù�Ú rÛÜ�Ý�Þ

rßT à

já â j

á ã r,� r ä,å m: ˆ já æ�ç�è r é8êë�ì�í

r î8ï ,� ð14ñ

where# m: ˆ já are the four matricesdefined in Eqs. ò 80 ó – ô 11õ .

Since�

m: ˆ 2 and m: ˆ 3I have only one entry, the factor 1/2 is

present� only for jö ÷

1,4. øWeH

can replace it by 12ù (1� úEû j

á2"üEý

já3I )� to havea closedexpressionfor all cases.þ

It'

is by no meansnecessaryto usethis particularset ofmatrices.ÿ Any linear combinationof them � respectively,7 ofthe�

states� 2� – � 5e �� can� beusedjust aswell. Writing thegen-eral- OP as ��

r� ��rÊ �

T

m: ˆ � rÊ ,� rÊ �� r ��rÊ �� ! 15"

with# an arbitrary 2 # 2b

matrix m: ˆ (�rÊ ,� rÊ $ )� , we can expand

m: ˆ (�r,� r % )� in any completebasisin spin spaceinsteadof the

m: ˆ já ,� suchas,for example,the setof the threePauli matrices

and the 2 & 2 unit matrix.

B. A basis for the relativistic order parameter

WeH

arenow in a positionto formulatetherelativisticgen-eralization- of the Bogolubov–de GennesHamiltonian ' 12(or,$ equivalently, the superconductinggeneralizationof theDirac Hamiltonian. Proceedingin the spirit of a single-particle� theory it is obvious, that the first ) i.e., the Schro-dingerG *

part� of Eq. + 12, is&

to be replacedby the correspond-ing&

Dirac Hamiltonian. In analogyto Eq. - 15. the�

secondpart� canalwaysbeexpressedin termsof a generalandasyetundetermined/ 4 0 4

1matrix M

2 ˆ (�rÊ ,� rÊ 3 )� , i.e.,

H 4 di 3Ir 5¯ 6 r798 c� :ˆ pq ; mc: 2 < q= >ˆ ? A @BA�CED rF

G 1

2b di 3Ir d� 3

Ir� H�IKJ T L rÊ M MN ˆ O rÊ ,� rÊ P�Q�RES rÊ T�U9V H.c.

� W,� X

16Ywhere# Z (

�r)�

is a four-componentfield operator[ Dirac-spinoroperator$ \ ,� ]¯ (

�rÊ )� ^ †(

�rÊ )� _ˆ 0

ànd A

a bis&

the electromagneticfour-potential� 50,51

�

Aa ced 1

qf gih rÊ j9kml ,� n Ao

. p 17qqf is the charge of the particles involved. In writing theabove, we madeuse of the summationconvention,i.e., asummation! overgreekr indices

&which appearonceasanupper

and onceas a lower index is implied. We now expandthematrixÿ M

N ˆ (�rÊ ,� rÊ s )� in a basis set of 16 linearly independent

4 t 4 matricesM já with# expansioncoefficientsu j

á* (�r,� r v ):�

w T x ry M z r,� r {�|�}E~ r ��9�� T � r� �já � j

á* � r,� r �� M já �E� r �� .�

18�Just�

asin the nonrelativisticcase,in principle,any completeset! of 4 � 4

1matricescanbechosenasa basisfor this expan-

sion.! Below we employoneparticularsuchset,namelythatof$ the �ˆ matricesÿ definedby Eqs. � 19� – � 34

� �. This setwill be

shown! to ensurerelativistic covarianceof the formalism � cf.�Sec.�

III � and to permitan interpretationof thephysicsof thepaired� statesanalogousto Eqs. � 2b � – � 5e ��

cf.� Sec. IV � . Forclarity� we displaybelowonly thosematrix entrieswhich aredifferentG

from zero:

ˆ1¡ 1

1¢ 1

,� £ 19¤

7142 PRB 59K. CAPELLE AND E. K. U. GROSS

¥ˆV0`

1¦ 1 §1

1

,� ¨ 20b ©

ªˆV1

1 «1¬ 1

1

,� 21®

¯ˆV2"

i°

i°

± i°

² i° ,� ³ 22

µˆV3I

¶ 1· 1

1

1

,� ¸ 23¹

ºˆ 5�

1» 1

1¼ 1

½ i° ¾ˆ ¿ˆ

V0` À

ˆV1 Áˆ

V2 Âˆ

V3I

,� Ã 24b Ä

ÅÂ0`

1Æ 1Ç 1

1

ÈEÉˆ Êˆ 5� Ë

ˆV0`

,� Ì 25b Í

ÎÂÏ1

1 Ð1 Ñ

1

1

ÒEÓˆ Ôˆ 5� Õ

ˆV1 ,� Ö 26×

ØÂÏ2

i°

i°

Ù i°

Ú i°

ÛEÜˆ Ýˆ 5� Þ

ˆV2 ,� ß 27à

áÂ3I

â 1ã 1

1

1

äæåˆ çˆ 5� è

ˆV3I

,� é 28b ê

ëˆT01`

ì 1

1í 1

1

îæïˆ ðˆV0` ñ

ˆV1 ,� ò 29

b ó

ôˆT02` õ

i°

i°

i°

i° öæ÷ˆ øˆ

V0` ù

ˆV2"

,� ú 30� û

üˆT03`

1

1

1

1

ýEþˆ ÿˆV0` �

ˆV3I

,� � 31� �

�ˆT12

i°

i°

i°

i° ��ˆ �ˆ

V1 �ˆ

V2 ,� 32

�

�ˆT13 �

1

1

1

1

��ˆ �ˆV1 �ˆ

V3I

,� � 33� �

�ˆT23"

i°

� i°

i°

� i°��ˆ �ˆ

V2" �

ˆV3I

. � 34� �

These16 matricesare linearly independentand hencecon-stitute! a completeset in which every 4 � 4

1matrix can be

expanded.- 52�

The(

labeling of the �ˆ matricesÿ refers to thetransformation�

behaviorof thecorrespondingOPunderLor-entz- transformations,aswill beexplainedin Sec.III B andinTableI.

Clearly,V

the nonrelativistic limit of Eq. � 16� is&

theBogolubovP

–de GennesHamiltonian for singlet and triplet�

OP! "

Refs.43 and41#%$ the�

spin degreesof freedomarenatu-rally containedin the relativistic framework& ,� while thenon-superconducting! limit ' all (

já ) 0

�) is the conventionalDirac

Hamiltonian.WeH

cannow also seeanotherreason,why it is useful tointroduce&

the matrix notation for the order parameter:Thecorresponding� termsin the Hamiltonianareall of the form

s< pinor * 41 + 41

matrix: , s< pinor f-

ield,� . 35� /

just0

asarethe termsof the Dirac Hamiltonian.In particular,the�

pairing fields 1 já and the electromagneticfield A

2 3enter-

the�

Hamiltonianon the samefooting. The four-currentden-sity! is defined,as usual,by j

ö 465c� 7¯ 8ˆ 9;: ,� while the order

PRB 59 7143RELATIVISTICg

FRAMEWORK FOR . . .g

. I. ...

parameters� are analogouslygiven by < ˆ já =?> TM

@ ˆjá A . The

four-currentdensityand the order parametersare thus alsotreated�

in a parallelway. This is particularlyappealingfromthe�

point of view of the density-functional theory ofsuperconductivity,! 21,41,44–49 where# the Bogolubov–deGennesB

equationsbecomethe Kohn-Shamequationsfor su-perconductors� andthe orderparametersenterthe formalismas ‘‘anomalousdensities,’’in additionto the normaldensitynC (�r)�

and,if necessary,thecurrentdensityjD. We call Eq. E 16F

the�

generalizedDirac–Bogolubov–de GennesHamiltonianbecause+

it containstheDirac–Bogolubov–de GennesHamil-tonian�

previouslysuggested29 as a specialcase.

III. SYMMETRY ANALYSIS WITH RESPECTTO THE LORENTZ GROUP

A. Bilinear covariants of the conventional Dirac equation

EveryG

relativistic theory needsto be covariant,i.e., theequations- musthavethe sameform in everyinertial system.We,H

therefore,have to verify that our equationsare forminvariant under Lorentz transformations, i.e., invariantmoduloÿ a Lorentz transformationof the arguments.This isclearly� thecasefor theconventionalDirac theory,50,51,53

�i.e.,&

without# the termscontainingthe H já . To determinewhether

the�

additional terms in Eq. I 16J are covariant,we needto

TABLE I. This tableis a completelist of all order-parametermatriceswhich areconsistentwith the requirementof Lorentzcovariance.The first column denotesthe transformationbehavior of the OP. The secondcolumn containsthe familiar bilinear covariantsof theconventionalDirac equation.The third columncontainsthe correspondinganomalousbilinear covariants.Every individual componentofeachentry in this tableis a 4 K 4 matrix.ThelabelsSE, SO, TE, andTO

L, assignedto eachcomponentof theanomalousbilinearcovariants

refer to the spin character(SM N

singlet,� T O triplet)�

andthe behaviorunderspaceinversion(E P even,Q OR S

odd)T of the correspondingpairs.Theselabelsareexplainedin detail in Secs.IV B andIV C.

Transformationbehavior Conventionalbilinear covariants Anomalousbilinear covariantsunderLorentztransformations of the Dirac equation of Eq. U 43

V WX¯ (Yr)Z

... [ (Yr)Z \ T(r) . . . ] (

Yr^ _ )Z

Scalar ` aˆ (YSEM

)Z

Four vector bˆ cedfˆ 0ghˆ 1

iˆ 2jkˆ 3l mˆ

Vneopˆ

V0gqˆ

V1

rˆV2jsˆ

V3l

SEMSOMSOMSOM

Pseudoscalar tˆ 5u v

i wˆ 0g x

ˆ 1 yˆ 2j z

ˆ 3l {

ˆ 5u |

i }ˆ ~ˆ V0g �

ˆV1 �ˆ

V2 �ˆ

V3l

(SO)

Axial four vector �ˆ 5u �ˆ �e�

�ˆ 5u �ˆ 0g

�ˆ 5u �ˆ 1�ˆ 5u �ˆ 2j

�ˆ 5u �ˆ 3l

�Â� �e��ˆ �ˆ 5

u �ˆ

V�TO

TE

TE

TE

Antisymmetrictensor � i� � ˆ �� ˆ

T ;¡�¢0 £ˆ 0

g ¤ˆ 1 ¥ˆ 0

g ¦ˆ 2j §

ˆ 0g ¨ˆ 3l

©«ªˆ 0g ¬ˆ 1 0 ˆ 1 ®ˆ 2

j ¯ˆ 1 °ˆ 3

l±«²ˆ 0g ³ˆ 2 ´«µˆ 1 ¶ˆ 2 0 ·ˆ 2 ¸ˆ 3

l¹»ºˆ 0g ¼ˆ 3l ½«¾

ˆ 1 ¿ˆ 3l À«Á

ˆ 2 Âˆ 3l

0Ã Äˆ

0 Åˆ V0g Æ

ˆV1 Çˆ

V0g È

ˆV2 Éˆ

V0g Ê

ˆV3l

Ë�ÌˆV0g Í

ˆV1 0

Î ÏˆV1 Ðˆ

V2j Ñ

ˆV1 Òˆ

V3l

Ó�ÔˆV0g Õ

ˆV2 Ö�×ˆ

V1 Øˆ

V2 0

Î ÙˆV2 Úˆ

V3l

ÛÝÜˆV0g Þ

ˆV3l ß�à

ˆV1 áˆ

V3l â�ã

ˆV2j ä

ˆV3l

0

0 TO TO TO

TO 0Î

TE TE

TO TE 0 TE

O TE TE 0Î


explicitlyå study their transformationbehaviorunderLorentztransformations.æ

Inç

principle,a discussionof Lorentzinvarianceshouldbebasedè

on the Lagrangiandensity and not the Hamiltonianbecauseè

theformeris invariantitself, while thelatter is not.50é

The Hamiltonian formulation, though, is much more com-monê and convenientin solid-statephysics. Since the La-grangianë density ì is related to the Hamiltonian Hí?î di 3ïr� ð by

èñóòõô

i ö˙ i

÷;øù˙

i úû

,ü ý 36þ ÿ

all� termsin�

which� do not dependon �˙ i � such� asthe termsdescribing�

superconductivityin Eq. � 16� appear� in � and� �with� oppositesign, but the sametransformationbehavior.We

can thus carry on with the Hamiltonianformulation, ifwe� keep in mind that the actual invariant quantity is theLagrangian�

density.Theprescriptionhow to performa Lorentztransformation

for Dirac spinorscan be found in many textbooks.50,51,53é

If�is the original spinor,thenthe spinorin a moving inertial

system� is relatedto it via

��x� �� S

�ˆ �� x� � ,ü 37þ !

where� x� " (#ct$ ,ü rÊ )% andx� &(' (

#ct$ ) ,ü rÊ * )% arethespace-timecoordi-

natesin the original and the new inertial system,respec-tively.æ

For a transformationto an inertial systemmovingwith� uniform velocity + k

Ó along� the k,

axis� (k, -

x� ,ü y. ,ü z/ )% , thetransformationæ

matrix is given by

S�ˆ 0 cosh1 2

23 465 sinh� 7

23 0

8 9ˆ

kÓ

: ˆkÓ 0

8 ,ü ; 38þ <

where�

tanhæ =

23 >

?kÓ

c$ @ 39þ A

and� B ˆkÓ are� the usual2 C 2 Pauli matrices.This form applies

toæ

finite, homogeneousD notE involving translationsF ,ü ortho-chronous1 G not involving time reflectionsH and� pure I not in-volvingJ spatialrotationsandreflectionsK Lorentztransforma-tionsæ L

i.e.,M

‘‘Lorentz boosts’’N .One!

cannow form all possiblecombinationsof theDirac-Oˆ matricesandinvestigatethetransformationbehaviorof thequantitiesP 50,51,53

éQ¯ RTSˆ UWV(XZY�[ † \ˆ 0

] ^ˆ _W`badc S� egf † hˆ 0

] iˆ j S� kml n

40o p

underq Lorentztransformations,where rˆ stands� for anyof thelinearlys

independentDirac matrices.Sincethereare 16 lin-earlyå independent4 t 4 matrices,onecanfind 16 suchquan-titiesæ

which can be classifiedaccordingto their transforma-tionæ

behavior. Explicitly, one finds that one can form ascalar,� a pseudoscalar,a u polarv w four vector, an axial fourvectorJ and an antisymmetrictensor of rank two. Each oftheseæ

transformsunder Lorentz transformationsand spaceinversionM

accordingto an irreducible representationof theLorentzgroup54

é xsee� Sec.III C for moredetailsy . Thesefive

entitieså are normally called the bilinearz

covariants of{ the

Dirac equation.They are listed in the secondcolumn ofTable|

I. Their constructionand a detailedanalysisof theirpropertiesv is foundin manystandardtextbookson relativisticquantumP field theory.50,51,53

éTogether|

they have16 compo-nents.Sincewe hadonly 16 matricesto startwith, we haveexhaustedå all possiblecombinations.For the Dirac equationthereæ

areexactly thesefive andno otherbilinear covariants.Their|

importancestemsfrom thefact thattheHamiltonian}or{ the Lagrangian~ needsto be expressedin termsof these

bilinearè

covariantsin orderto be manifestlycovariantitself.Furthermore,�

sincethereis just a small numberof differentbilinearè

covariants,onecanclassifyall interactionswith re-spect� to thebilinearcovariantswhich oneneedsfor a properdescription.�

It turnsout, that almosteveryinteractioncanbedescribed�

in termsof just one bilinear covariant.The mostprominentv exceptionis the weak interactionof high-energyphysics.v A descriptionof this interactionrequiresthe pres-enceå of a � polarv � four

�vectorand� an� axial four vector in the

Lagrangian.�

Sinceundera spatialreflectionthe formerpicksupq a relative sign with respectto the latter, the weak inter-action� violates parity invariance.51,53

éWe

have discussedtheseæ

factsat lengthbecausethey will turn out to be highlyrelevant� for the superconductingcaseaswell.

B. Symmetry classification of all possible order parameters

We

first considerthe transformationbehaviorof the OPformed with �ˆ . Given Eq. � 38

þ �,ü it is a matter of simple

matrixê multiplication to verify that

�W� T �ˆ �W�b�d� S�ˆ �g� T �ˆ � S�ˆ �g��Z� T �ˆ � ,ü � 41o �

i.e.,M � T �ˆ � is

Mform invariantunderLorentztransformations.�ˆ can1 be expressedin termsof the �ˆ matricesas �ˆ 1 ˆ 3

ï.

The bilinear covariantwhich is formed with these ¡ˆ matri-ces,1 namely ¢¯ £ˆ 1 ¤ˆ 3

ï ¥,ü is, on the other hand,not¦ invariant

Munderq Lorentz transformations.As displayedin the secondcolumn1 of TableI, it transformsasthecomponent§ ˆ 13 of{ theantisymmetric� tensor ˆ ©«ª .

A¬

similar situationis found for all OP: their transforma-tionæ

behaviorcannotsimply beobtainedfrom their represen-tationæ

in terms of the ˆ matricesin Eqs. ® 19 – ° 34þ ±

and� acomparison1 with the correspondingrepresentationof theconventional1 bilinear covariantsin termsof the samematri-ces.1 Thereasonfor this is, that theOPcontain² T instead

Mof³¯ ´Zµ † ¶ˆ 0

]. Therefore, it is the transposedtransformation

matrixê S�ˆ T which� entersandnot, asusual,theHermitiancon-

jugate0

S�ˆ †. Since S

�ˆ is,M

in general,a complex matrix, thisdifference�

leads,togetherwith the additional ·ˆ 0],ü to a differ-

entå transformationbehavior,ascomparedto theconventionalcase.1 This phenomenonis known alsofrom the nonrelativis-ticæ

triplet OP, whereit leadsto usingthe Balian-Werthamermatricesinsteadof the Pauli matricesin order to obtain aquantityP which transformsas a three vector under spatialrotations.� 37,40,55

ïWe

can,of course,investigatethe transformationbehav-ior underspacereflectionsin the sameway. Herethe trans-formation�

matrix is simply ¸ˆ 0]

and� it is readily verified that

PRB 59 7145RELATIVISTIC¹

FRAMEWORK FOR . . .¹

. I. ...

º�» T ¼ˆ ½W¾(¿dÀÂÁˆ 0] ÃmÄ

T Åˆ ÆÂÇˆ 0] ÈmÉ�ÊZÈ

T Ëˆ Ì Í 42Î Ï

is invariant under this operationas well. We concludethatÐ T Ñˆ Ò isM

aLorentzscalar,whichdoesnot changesignunderspatial� reflections Ó the

ælatter property excludesa pseudo-

scalar,� which would changesignÔ . We do not needto con-sider� spatial rotationsand translationsbecausethey do notinfluenceM

the classificationof the bilinear covariants.50,51,53é

Theseoperationsdo not, in general,leave the underlyinglattice of the superconductorinvariant.The symmetrygroupof{ the lattice canbe usedto further classify the OP.1,33,56–58

The|

considerationsof thepresentsectioncanbeviewedasanextensionå of theseworks from usingtheGalilei groupto theLorentzgroup.

In the sameway as for the Õˆ -OP we can investigatethetransformationæ

behaviorof all the other15 matrices.To de-termineæ

their transformationbehaviorwe perform Lorentztransformationsæ

alongall threespatialaxes.This allowsustodistinguish�

scalars,vectors,and tensors.To further distin-guishë betweenproperscalarsandpseudoscalarsandbetweenÖpolarv × four

�vectorsandaxial Ø pseudov Ù four

�vectors,we also

needE to investigatethebehaviorunderspatialreflections.Thecovariant1 quantitiesone finds in this way will be termedanomalousÚ bilinear

ècovariants.59

éWe

summarizethe resultsof{ this investigationin Table I, wherewe display the con-ventionalJ and the anomalousbilinear covariants,classifiedaccording� to their transformationbehavior.

The 16 componentsof the five distinct covariantquanti-tiesæ

of the Dirac equationcan all be expressedin a well-knownÛ

way50,53é

inM

terms of just five matrices, namelyÜˆ 0],ü Ýˆ 1,ü Þˆ 2,ü ßˆ 3

ï,ü which form the à polarv á four vector, and the

unitq matrix â ,ü which is the scalar.In just the sameway, the16 componentsof thefive distinctanomalouscovariantscanall� be expressed in terms of five matrices, namelyãˆ

V0]

,ü äˆ V1 ,ü åˆ V

2æ

,ü çˆ V3ï

,ü which form the è polarv é four vector,andtheêˆ matrix, which is the scalar.There is thus a far reachingcorrespondence1 betweentheconventionalbilinearcovariantsof{ theDirac equationfor normalelectronsandtheanomalousbilinearè

covariants ë order{ parametersì of{ the Dirac–Bogolubov–de Gennesequationfor superconductors.

Inç

view of this correspondenceit is not surprising thatthereæ

existsa transformationwhich, whenappliedto oneoftheæ

16 componentsof theconventionalcovariants,yields thecorresponding1 anomalouscovariant.If everyentryin thesec-ond{ columnof TableI is multiplied with íˆ 1 îˆ 3

ï,ü thereresultsï

upq to factors ð 1) the correspondingentry in the third col-umn.q In thecaseof thepseudoscalarñˆ 5

é,ü for instance,we findòˆ 1 óˆ 3

ï ôˆ 5é õ÷ö

ˆ 5é,ü while the tensor component øˆ 0

] ùˆ 2æ

becomesè

úˆ 1 ûˆ 3ï ü

ˆ 0] ý

ˆ 2 þ ÿ �ˆ �ˆV0] �

ˆV2æ

. This rule transformsa conventionalbilinearè

covariantin thecorrespondinganomalouscovariant,which� hasthe sametransformationbehaviorunderLorentzboostsè

andspatialreflections.It summarizesin a concisewaytheæ

lengthy calculationsnecessaryto determinethe matrixrepresentations� of the 16 �ˆ -OP by performing the Lorentztransformationsæ

explicitly.This rule,extractedfrom TableI, canalsobearrivedat by

means of a generalizationof the nonelativistic Balian-Werthamer

prescriptionfor the constructionof matricesfortheæ

singletandtriplet OP’s: Observingthat ordinarily a sca-

lar is expressedin termsof theunit matrix, while a vectorisexpressedå in termsof the threePaulimatrices,onefindsma-tricesæ

yielding scalarandvectorOPsimply by multiplicationwith� thenonrelativistictime-reversalmatrix i

° � ˆy� . Theresult-

ing matricesi° � ˆ

y� and� i° � ˆ

y� ˆ x ,ü i° � ˆ y� � ˆ y� ,ü i° ˆ y� � ˆ z� ,ü respectively,arethoseæ

employedin theBalian-Werthamerparametrizationfortheæ

nonrelativisticOP.In thesamespirit all matricesdescrib-ingM

relativistic OP can be obtainedfrom the conventionalDirac matricesby multiplication with the relativistic time-reversal� matrix �ˆ 1 �ˆ 3

ï. Although this procedureis highly

plausible,v its generalvalidity can only be verified by per-forming the Lorentztransformationsasdescribedabove.

Just�

asin thenonsuperconductingcase,the importanceoftheæ

anomalousbilinear covariantsstemsfrom the fact that aLagrangian�

expressedin termsof themis manifestlycovari-ant� andfrom the possibility to classifysystemswith respecttoæ

theappropriatecovariants.If for anygivensuperconductormoreê then one of the five covariantsappearin the Hamil-tonian,æ

a very generalsymmetry,suchas parity, would bebroken.è

Inç

thespirit of a mean-field� Hartree-type� �

approximation,�where� onehasa linear relationbetweenthe particle-particleinteractionandtheeffectivesingle-particlepotential,we canclassify1 the interactionleadingto superconductivitywith re-spect� to the symmetryof the correspondingeffective pairpotential.v If, for instance,for a certainsuperconductoronlytheæ

scalarOP is presentin the Hamiltonian,the interactionleadingto superconductivitymustbesuch,thatthepair fieldsmultiplyingê theotherbilinearOP’svanishidentically.As wewill� show below, this examplecorrespondsto the conven-tionalæ

BCS-OP.In this way we canclassify the interactionsas� ‘‘scalar interaction,’’ ‘‘vector interaction,’’ etc.

We

cannow formulatethe following theorem:No�

matterwhat� the OP symmetry and the detailed nature of the pairinginteraction°

leading to superconductivity, there are only fiveOP�

with a total of 16 components, consistent with the prin-ciple$ of general covariance. Each of these is a bilinear formof� the field operators � T andÚ � transforming� according toanÚ irreducible representation of the Lorentz group. Thistheorem,æ

derivedhereby explicit constructionof a completeset� of matrices,is rederivedin Sec.III C by purely group-theoreticalæ

considerations.The resultinganomalousbilinearcovariants1 are listed in Table I. Using theseresultswe cannowE write the Hamiltonian � 16� with� Eq. � 18 as�

H ! di 3ïr "¯ # r$&% c$ 'ˆ pq ( mc) 2

æ *q+ ,ˆ - A .0/2143 r5

6 1

23 di 3ïr d� 3ïr� 798;: T < rÊ =&>@?ˆ A * B rÊ ,ü rÊ CEDGFIHˆ 5

é JP* K rÊ ,ü rÊ LEM

NIOˆVPRQ V S* T rÊ ,ü rÊ UWVGXIYˆ

AZ@[ A \* ] rÊ ,ü rÊ Ê_Ìaˆ

Tbdcfe T, gih* j rÊ ,ü rÊ kEl2m2n4o rÊ pEqGr H.c.� s

. t 43Î u

Nowv

every contribution to the superconductingpart of theHamiltonian�

is expressedin termsof covariantquantities:thescalar� is formedwith wˆ and� x ,ü thepseudoscalarwith yˆ 5

éand�z

P ,ü the four vectorwith {ˆ V| and� }V ~ ,ü theaxial vectorwith�ˆ

AÏ � and� �

A � and� the tensor,finally, with �ˆ T�i� and� �T, �i� .


Notev

the formal correspondencebetweenthe four-vectorproductv �ˆ � A

� �,ü describingthe coupling of the four current,�¯ �ˆ �@� ,ü to the four potential,A

� �,ü andthe product �ˆ V�@� V � ,ü

describing�

the coupling of the four vector OP, � T �ˆV�@� , tü o

theæ

four potential,� V � .Equation� �

43Î �

isM

one of the central equationsof thepresentv work. It constitutesthe desiredrelativistic Hamil-tonianæ

for superconductorsandcontainsall possibleanoma-louss

bilinear covariants.

C. Irreducible representations of the Lorentz group

Themethodusedaboveto determinetherelativisticOPisbasedè

on explicit expressionsfor eachOPin termsof a 4� 4Î

matrix.ê The main conclusionof the previous section can,however,alsobe found without any recourseto explicit ma-trices,æ

simply by exploiting the propertiesof the irreduciblerepresentations� IR’s

ç ¡of{ the Lorentzgroup.To this endwe

first¢

recapitulatea numberof propertiesof suchIR’s whichwill� be usedbelow.6

£0–62

The|

IR’s of the Lorentzgroup,denoted¤ ,ü arecharacter-ized by two labels,p¥ and� q+ ,ü which cantakepositive integerand� half-integervaluesonly. ¦ Infinite dimensionalrepresen-tationsæ

characterizedby continuouslabelsexist as well, butare� not relevantfor the presentconsiderations.§ The

|dimen-

sion� of the IR ¨ pq© is (2p¥ ª 1)(2q+ « 1). A four vector, forinstance,M

transformsaccordingto the ¬ four-dimensional�

rep-�resentation�

®1/2 1/2 ¯±° 1/2 0 ²´³ 0

]1/2 ,ü µ 44

Î ¶while� a four-componentDirac spinortransformsaccordingtotheæ

direct sumrepresentation· 1/2 0 ¸´¹ 0]

1/2 .Productrepresentationsof such IR’s are, in general,re-

ducible,�

and can be written as a direct sum of irreduciblerepresentations.� This is greatly facilitatedby noting that theIR’sç

of the Lorentzgroupsatisfy

ºpq© »´¼ nm½ ¾ ¿À

p© Á n½ Â Ã kÓ Ä

p© Å n½ÆqÇ È mÉ Ê Ë l

Ì ÍqÇ Î mÉ

ÏklÓ . Ð 45

Î Ñ

Obviously,Ò

this is just the familiar rule for the coupling oftwoæ

angularmomenta Ó i.e.,M

for forming the productof twoIR’sç

of the rotationgroupÔ ,ü applied,however,to eachof thetwoæ

labelsof theIR’s individually. EquationÕ 44Ö is a specialcase1 of this rule.Anotherexampleis a generaltensorof ranktwo.æ

Being definedas the direct productof two vectors,ittransformsæ

accordingto

×1/2 1/2 Ø´Ù 1/2 1/2 Ú±Û 00

] Ü´Ý11Þ´ß 10à´á 01

] . â 46Î ã

Here� ä

00] denotes�

theidenticalrepresentation.Symmetricten-sors� transformaccordingto the ten-dimensionaldirect sumrepresentationå 00

] æèç11 é notethata symmetrictensorof rank

two,æ

definedin a four-dimensionalvectorspace,hasten in-dependent�

entriesê . Antisymmetrictensorstransformaccord-ingM

to the six-dimensionalIR’s ë 10ì´í 01] î an� antisymmetric

tensoræ

in a four-dimensionalvector spacehassix indepen-dent�

entriesï .The task of enumeratingall possiblerelativistic OP is

nowE reducedto finding all distinct IR’s accordingto whichtheæ

anomalousbilinearformsof Dirac spinorstransform.Ex-

plicitly,v the bilinear form ð T ñî ò transforms,

æfor any óˆ i ,ü

according� to the direct productrepresentationof two spinorrepresentations,i.e.,

ô;õ1/2 0 ö´÷ 0

]1/2øRùèú;û 1/2 0 ü´ý 0

]1/2þ . ÿ 47

Î �The resultingproductrepresentationis immediatelyseentobeè

reducible.Using Eq. � 45Î �

one{ finds��1/2 0 �� 0

]1/2� �� 1/2 0 � � 0

]1/2�� 00

] ��00] ��

1/2 1/2��1/2 1/2 �� 10�� 01

] .!48"

We

find that therearetwo Lorentzscalars# twiceæ

the identi-cal1 representation$ ,ü two four vectors % twice

æ &1/21/2)

%andone

antisymmetric� tensorof rank two containedin the productrepresentation.� An analysis of the proper Lorentz groupalone� does not suffice to distinguish betweenscalarsandpseudoscalars,v or vectorsandaxial vectors,respectively.Wehave'

thereforeobtainedtwo scalarsand two vectors.Paritythenæ

servesto further classify thesequantitiesaccordingtotheiræ

behaviorunderspaceinversion.It shouldbe notedthatexactlyå the sameanalysisgoesthroughfor the conventionalbilinearè

covariants(¯ ) ˆ * ,ü where+ ˆ is anyof the16 conven-tionalæ

Dirac matrices.As thesequantitiesareformedwith theadjoint� spinor they transformaccordingto

,�-1/2 0* .�/ 1/2 0* 021�3 1/2 0 4�5 0

]1/26 ,ü 7 49

Î 8where� 9

pq©* is the complexconjugaterepresentationof : pq© .For theLorentzgroupanIR is not generallyequivalentto itsconjugate1 IR. It follows immediately that anomalousandconventional1 bilinear covariantsdescribedistinct physicalobjects.{ The transformationbehaviorof theseobjects,how-ever,å dependsonly on the labelsp¥ and� q+ and� is thereforethesame� for the conventionalas for the anomalousbilinear co-variants.J The resultingclassificationof the conventionalbi-linears

covariantsinto scalars,vectors,andan antisymmetrictensoræ

of rank two is, of course,not new,but found in manystandard� textbooksby way of explicit manipulationsof ;ˆmatrices.ê 50,51,53

éAs far astheanomalousbilinearcovariantsareconcerned,

Eq.� <

48Î =

constitutes1 a rederivationof the abovetheorembypurelyv group-theoreticalconsiderations,without having towrite� down explicit matricesor Dirac-typeequations.

IV.>

SYMMETRY ANALYSIS WITH RESPECTTO?

DISCRETE SYMMETRIES

A. Discrete symmetries of the Dirac equation

We

now generalizethe considerationsof Sec.II A to therelativistic� case,i.e., we constructa symmetryadaptedbasisset� of two-particle statesinto which any OP can be ex-panded.v In a relativistic situation one has three� symmetry�operations,{ insteadof two, namelytime-reversalT,ü parity P

and� chargeconjugationC.50,51,53é

Every�

solution, @ nA B ,ü of theDiracC

equationyields four linearly independentstates,ac-cording1 to

DnA E ,ü F TnA G ,ü H PnA I ,ü J PTnA K . L 50

M NThe correspondingcharge-conjugatestatesare

PRB 59 7147RELATIVISTICO

FRAMEWORK FOR . . .O

. I. ...

PCnA Q ,ü R CT

S ˆ nA T ,ü U CPV ˆ nA W ,ü X CP

V ˆ TS ˆ nA Y . Z 51

M [As will turn out below, a completeset of OP’s are onlyobtained{ if both types of states,Eqs. \ 50

M ]and� ^ 51

M _,ü are

included.M 63

£The|

BCSprescriptionfor pairingrequiresthatthesingle-particle� statesare pairedto yield a two-particlestatewith� zerocenter-of-massmomentum.ThePauliprinciple re-quiresP that thesetwo-particlestatesbe antisymmetricunderparticlev exchange.We now constructtheappropriatesymme-tryæ

adaptedbasis functions, which are the counterpartstoEqs. ` 2a – b 5M c in the nonrelativisticcase.To this end,we firstconsider1 homogeneoussystems,in which themomentumis agoodë quantum number, and introduce four-componentspinors� of the form d

re f ki, g�hji

kk l re m�n i ,ü o 52

M pwhere�

q1 r

1

080808

s2 t

081

0808

u3ï v

08081

08

w4 x

0808081

. y 53M z

We

will refer to thespacedefinedby the label i°

as� i°

space.� ItisM

the formal generalizationof the nonrelativisticspin space.The three symmetry operators act on { k

k |i according�

toæ 50,51é

T} ˆ ~��

kk �

1��j� kk* � 2 ,ü P

V ˆ ��kk �

1��j�� kk �

1 ,ü C �� kk �

1 ��j�kk* � 4 ,ü etc.Using this notation,the relativistic counterpart

of{ the state � S� � is, for homogeneoussystems,given by

�1 �� ki

,,ü Tki, ��

PTki,

,ü Pki, �� ¡

C PTki,

,ü C Pki, ¢

£�¤ Cki,

,ü CT} ˆ ki, ¥

. ¦ 54M §

It is readily verified that the configuration-spacerepresenta-tionæ

of this stateis¨rr ©«ª 1 ¬��®�¯ k

k ° r±�²�³ kk ´ r µ·¶¹¸jº¼» k

k ½ r¾�¿ kk À r Á·ÂÄÃÅ�ÆÈÇ

1 ÉËÊ 2 Ì Í 2 ÎËÏ 1 Ð�Ñ 3ï ÒËÓ

4 Ô Õ 4 ÖË× 3ï Ø . Ù

55M Ú

Ûrre ÜÞÝ 1 ß is

Mseento beevenin thespatialcoordinatesandodd

in the i°

space� coordinates.In this senseit describesthe rela-tivisticæ

counterpartof a singlet state.Performingthe tensorproductsv of the à i yieldsá the matrix

1â 1

1ã 1

,ü ä 56M å

which� is just æˆ ,ü as definedin Eq. ç 19è .In inhomogeneoussystemsk

,is not a goodquantumnum-

berè

anymore.In completeanalogyto thenonrelativisticcaseone{ thereforedefinesthe generalbasisstateas

é1 ê�ë�ì niA ,ü TniA í�î�ï PTniA ,ü PniA ð�ñ�ò C PTniA ,ü C PniA ó

ô�õ CniA ,ü CTniA ö ,ü ÷ 57M ø

where� nA stands� for a completesetof quantumnumbersnec-essaryå to label a solution of the Dirac equation.Evidently,Eq.� ù

57M ú

contains1 Eqs. û 54M ü

and� ý 23 þ as� specialcases.In thesame� way in which ÿ 1 � is mappedonto the matrix �ˆ ,ü thematrices �ˆ V

0]

toæ �ˆ

T23 correspond,1 in this order, � upq to global

factors � 1,� i�)%

to the basisstates

�23 � �

niA ,ü T} ˆ niA �� PV ˆ T} ˆ niA ,ü PV ˆ niA �� CP

V ˆ T} ˆ niA ,ü CP

V ˆ niA ��

CniA ,ü CT} ˆ niA � ,ü � 58

M ��3þ ��

niA ,ü CPV ˆ T} ˆ niA �� CT

} ˆ niA ,ü PV ˆ niA !�" # T} ˆ niA ,ü CPV ˆ niA $

% &CniA ,ü PV ˆ T

} ˆ niA ' ,ü ( 59M )

*4 +�, - niA ,ü C PTniA .�/ 0 CTniA ,ü PniA 1�2 3 TniA ,ü C PniA 4

5 6 CniA ,ü PTniA 7 ,ü 8 609 :

;5M <>=@?

niA ,ü CniA A�B C C PniA ,ü PniA D�E F TniA ,ü CTniA GH I C PTniA ,ü PTniA J ,ü K 61

9 LM69 N>O@P

niA ,ü CniA Q�R S C PniA ,ü PniA T�U V TniA ,ü CTniA WX Y

CPV ˆ T} ˆ niA ,ü PV ˆ T

} ˆ niA Z ,ü [ 629 \

]7^ _>`@a

niA ,ü CniA b�c d CPV ˆ niA ,ü PV ˆ niA e�f g T} ˆ niA ,ü CT

} ˆ niA hi j

CPV ˆ T} ˆ niA ,ü PV ˆ T

} ˆ niA k ,ü l 639 m

n8o p�q r

niA ,ü PV ˆ niA s�t u PV ˆ T} ˆ niA ,ü T} ˆ niA v�w x CP

V ˆ T} ˆ niA ,ü CT

} ˆ niA yz { CniA ,ü C PniA | ,ü } 64

9 ~

�9� ��

niA ,� PniA �� PTniA ,� TniA �� C PTniA ,� CTniA ��

CniA ,� C PniA � ,� � 659 �

�10�� niA ,� TniA �� PTniA ,� PniA �� C PTniA ,� C PniA �

� �CniA ,� CTniA � ,� � 66

9

¡11¢�£ ¤ niA ,� C PTniA ¥�¦ § CTniA ,� PniA ¨�© ª TniA ,� C PniA «

¬ CniA ,� PV ˆ T

} ˆ niA ® ,� ¯ 679 °

±12²�³ ´ niA ,� CP

V ˆ T} ˆ niA µ�¶ · CT

} ˆ niA ,� PV ˆ niA ¸�¹ º T} ˆ niA ,� CPV ˆ niA »

¼ ½ CniA ,� PV ˆ T} ˆ niA ¾ ,� ¿ 68

9 À

Á13Â>Ã Ä niA ,� CniA Å�Æ Ç CP

V ˆ niA ,� PV ˆ niA È�É Ê T} ˆ niA ,� CT} ˆ niA Ë

Ì Í C PTniA ,� PTniA Î ,� Ï 699 Ð

Ñ14Ò�Ó Ô niA ,� TniA Õ�Ö × PTniA ,� PniA Ø�Ù Ú C PTniA ,� C PniA Û

Ü Ý CniA ,� CTniA Þ ,� ß 70^ à


á15â>ã ä niA ,� PV ˆ niA å>æ ç PV ˆ T

} ˆ niA ,� T} ˆ niA è>é ê CPV ˆ T} ˆ niA ,� CT

} ˆ niA ëì í CniA ,� C PniA î ,� ï 71

^ ð

ñ16ò>ó ô niA ,� PniA õ>ö ÷ PTniA ,� TniA ø>ù ú C PTniA ,� CTniA û

ü ýCniA ,� C PniA þ ,� ÿ 72

� �respectively.The states � 1 � – � 16� are� independentof thechoice� for the label i

�in

the sensethat i�

2,3,4�

lead,up toglobal� phasefactors,to the same16 matricesas i

� 1. These

matrices� aredeterminedonly up to within an overall unitarytransformation.�

The same applies, evidently, to the basisstates� � 57

M �– � 72� �

. The presentchoice is simply a matter ofconvenience.�

This�

connectionbetweenthe discretesymmetriesof theDirac�

equationandthe behaviorof the relativistic OP underLorentz transformations64

£can� now be used to discussthe

physical� meaningof the varioustermsin Eq. � 43� .B.�

Order parameters involving charge conjugation

The�

charge-conjugationoperationC relates� positive andnegative-energy� solutions of the Dirac equation to eachother.� The natureof the correspondingpairs can be under-stood� by expressingtheorderparametersin termsof creationand� annihilationoperatorsof single-particlestatesratherthanin terms of field operators.The field operators� have anexpansion� of the form:53

!#"%$

p© &s' ( 1

4

f)

s' ,p© b*ˆ

p© + s' , ,� - 73� .

where/ the b*ˆ

p© (0 s' )1 annihilate� positive energy electronsfor s23 1,2 and negativeenergyelectronsfor s2 4 3,4.5

The coeffi-cient� f

)s' ,p© can� be expressedin termsof solutionsof the cor-

respondingDirac equation.53

Theorderparametersthuscon-tain�

operators for two positive-energy electrons 6 e.g.,�b*ˆ

p© (1)0

b*ˆ

p© (2)0

)7, two negative-energyelectrons8 e.g.,� b

*ˆp© (3)0

b*ˆ

p© (3)0

) o7

r a

positive� and a negative-energyelectron 9 e.g.,� b*ˆ

p© (1)0

b*ˆ

p© (3)0

).7 65:

Itfollows;

that thereareno electron-positronpairscontainedinour� theory,asthesewould requiretermslike b

*ˆp© (1)0

(<b*ˆ

p© (3)0

)7 †.

The appearanceof pair statesformed with one C opera-�tion,�

suchas = ni> ,� Cni> ? ,� in the basissetthusreflectsthe pos-sibility� of pairs consisting of a positive and a negative-energy� electron.Suchpairsrequirethe pairing interactiontobridge@

thegapof 2mcA 2B. As this energyis far biggerthanthe

typical�

energiesof pairing mechanismsdiscussedin connec-tion�

with ordinarysuperconductors,suchpairsareunlikely tobe@

realizedin solid-statesituations.In particular,in the non-relativistic limit the negative-energystatesare completelydecoupledC

from the positive-energystates,so that the pairsformedwith one C operation� do not contributeat all in thislimit.

OnÒ

theotherhand,it is necessary> to�

includethesepairsinthe�

formal developmentof the theory,becausewithout themone� doesnot recoverthe completesetof 16 4 D 4

Ematrices.

FurthermoreF

it shouldbe notedthat suchpairs are not for-bidden@

by either relativity or symmetry.Whetherthey are

actually� realized in naturedepends,therefore,only on theexistence� of a suitableinteraction.

InG

the nonrelativisticlimit we only have the stateswithzeroH or two C operations,� i.e., pairsbetweentwo positiveortwo�

negative-energystatesat our disposalto form Cooperpairs.� Consideringthesestatesit is importantto notethat thetheory�

is exactlysymmetricwith respectto the operationofcharge� conjugation.To everysingle-particlestatein Eq. I 50

M Jcorresponds� one in Eq. K 51

M Lwhich/ differs from it just by

application� of C. Furthermore,aswe did not specifythesignof� the externalpotential, A M ,� we have in fact no way todistinguishC

superconductivityinvolving two electrons,mov-ing in a lattice of ordinary atoms,from that involving twopositrons� in a latticecomposedof anti atoms.It is thereforeadefiniteC N

and� ratherplausibleO prediction� of the theorythat inan� ‘‘antiworld’’ superconductivitytakesplacebetweentwopositrons� insteadof two electrons.Sinceboth electronsandpositrons� aresolutionsof a Dirac equationtherearerelativ-istic

correctionsto both typesof pairs.AlreadyP

to secondorderin Q /RcS ,� at the weakly relativistic

level,T

thesecorrectionslead to important effects, such asspin-orbit� coupling, the Darwin term, etc. From the aboveconsiderations� it appearshighly plausiblethat therearesuchcorrections� to the order parametersas well, which indeedturns�

out to be the caseU see� the secondpaperin this seriesV .In Table I eachmatrix is assignedtwo labelsto indicate

the�

natureof theunderlyingbasisstates.Thesecond2 of� theselabelsT

refersto the behaviorof the pairsunderspaceinver-sion.� The matricescorrespondingto the statesformed withzeroH or two C operations� areall evenunderspaceinversion.These�

matricesare labeledEW

. Thosecorrespondingto pairsformed with one applicationof C turn

�out to be odd under

inversion

andare labeledO�

.

C.X

Order parameters involving triplet pairing

InG

the nonrelativisticcase,wherespin is a goodquantumnumber,� all pair statescan be classifiedas either singlet ortriplet�

states.This classificationis reflectedin theform of thepair� statesY 2Z – [ 5M \ . In the relativistic case,singletandtripletstates� losetheir meaningbecausespin is not a goodquantumnumber� anymore.However,aslong asa centerof inversionis present,a classificationaccordingto the behaviorunderparticle� exchange is still possible.34,1

]OrderÒ

parameterswhich/ areevenin i

�space� andodd in r space� arethe gener-

alization� of triplet OP. ThoseOP which are odd in i�

space�and� evenin r space� generalizesingletOP.

The symmetry in i�

space� can be immediately read offfrom;

the matrix representationsof the various OP in Eqs.^19_ – ` 34

5 a. All matricessatisfying bˆ T cedˆ are� evenin i

�space�

and� lead to triplet OP’s in the nonrelativistic limit. Thesematrices� correspondto the states7–16. In TableI thesema-trices�

carry a T} f

for;

triplet pairg as� a first label. Matricessatisfying� hˆ T ikjelˆ ,� on the other hand,give rise to singletOPÒ

in the nonrelativisticlimit. Thesematricescorrespondtothe�

basisstates1–6. In TableI they carry an Sm n

for a singletpair� o as� a first label.

The�

matriceslabeledTE}

in

TableI p corresponding� to thestates� 8,9,10,14,15,16)describetriplet pairs betweentwopositive� or two negative-energystates.Thesearethe relativ-

PRB 59 7149RELATIVISTICq

FRAMEWORK FOR . . .q

. I. ...

istic

generalizationsof theBalian-Werthamermatricesi� r ˆ

ys t ,�to�

which theyrigorouslyreduceu upv to phasefactors w 1, x i�

which/ area consequenceof our definitionof the yˆ matrices� zin the nonrelativisticlimit.

D.{

Order parameters of the generalized BCS-type

It is perfectlypossibleto formulatea relativistic theoryofsuperconductivity� on the basisof the completeHamiltonian|43E }

. Onewould carryon with 16 differentorderparametersand� the associatedpair fields, which transformas the com-ponents� of the variousbilinear covariants.

Obviously,Ò

this Hamiltonian leadsto very generalequa-tions�

of motion whosesolution will be rather involved. Inmost, if not all, realisticsituationsthe interactionleadingtosuperconductivity� will pick one of the various couplingchannels� ~ i.e.,

bilinearcovariants� and� favor one � or� a few� of�

the�

greatvariety of order parameters.In the following sec-tions�

we will thusnot continuewith the full equation� 43� .We�

assumeinsteadthat the nonelativistic limit of ourequations� must describea singlet superconductorwith pair-ing betweentwo positive-energystates.OP’s with such anonrelativisticlimit will in the following be termed‘‘orderparameters� of the generalizedBCS type.’’

ForF

ananalysisof triplet superconductorsandmoreexoticpairs� oneneedsto go backto Eq. � 43� and� selecttherelevantterms�

from Eqs. � 19� – � 345 �

,� just aswe shall do below for theBCS�

case.By inspectionof TableI we find that only two out of the

16 possibleOPareof thegeneralizedBCStype.Oneof them

is

theLorentzscalar,formedusingthematrix �ˆ . Theotheristhe�

zeroth componentof the � polar� � four vector, which is

formed;

with the matrix �ˆ V0�

.If we took both of thesetwo OP’s into account,the re-

sulting� expressionfor the Lagrangianwould not be covari-

ant,� because�ˆ V0�

is only a single componentof a bilinear

covariant.� This on its own doesnot disqualify the �ˆ V0�-OP as

a� valid OP, but it has the immediateconsequencethat toform a covariantquantitywe needto combineit with theOP�ˆ

V1 ,� �ˆ V

2 ,� and �ˆ V3]

which/ were alreadyexcludedabove,be-cause� they correspondto pairs betweena positive and anegative-energy� electron.Thereforewe will focusmainly onthe�

scalarOP in the remainderof this paper.For laterreferencewe now explicitly write out theHamil-

tonian�

containingthe generalizedBCS-typeOP

� ˆ � r,� r �� T � r ¢¡ˆ £¥¤ r ¦¨§ ,� © 74� ª

namely�

H« ˆ ¬ d

3]r® ¯¯ ° r± ²�³ cS ´ˆ pµ ¶ mcA 2

B ·q¸ ¹ˆ º A

» ¼¾½À¿¥Ár± Â

Ã75� Ä

This is the BCS versionof Å 43Æ or,� equivalently,the relativ-istic versionof Ç 12È and� É 13Ê . The otherOP of the general-ized BCS-type, Ëˆ V

0�

,� leadsto the term

Ì76� Í

If Î 76� Ï

is included in Eq. Ð 75� Ñ

,� the resulting Hamiltoniancontains� all termswhich in thenonrelativisticlimit reducetoBCS-type�

OP’s, but the Lagrangianit correspondsto is notcovariant.�

The formulationof the relativistic theoryof superconduc-tivity�

given in Ref. 29 is a specialcaseof thepresentformu-lation,T

in which only the center-of-massdegreesof freedomof� theCooperpair areconsideredandonly thescalarOP,Eq.Ò74� Ó

,� formedwith Ôˆ ,� is included.Thenonrelativisticlimit ofthe�

approachof the presentsectionhasbeenshownin Ref.29 to lead to the standardBogolubov–de Gennestheory ofinhomogeneoussuperconductors43

Õwith/ a local OP.

The�

essenceof this reductioncanalreadybeseenfrom thestructure� of the Öˆ -OP. Using the anticommutationrelationsof� the four componentsof the × ,� Ø i ,� and the symmetryofthe�

pair field Ù (<r± ,� r± Ú )7 under particle exchange,we can re-

write/1

2d 3]r d3]r ÛÝ Ü T Þ rßáà â¥ã r ä� åçæ * è r,� r é� ê ë 77

� ìas�

d 3]r d3]r íçîðï 1 ñ rò�ó 2 ô r õ¨ ö�÷ùø 3

] ú rûýü 4 þ r ÿ�� * � r,� r �� . 78�

Here�

we have essentiallyperformedthe steps in reverse,which/ led from Eq. � 12 to

�Eq. � 13� in the nonrelativistic

case.� We now see the physical significanceof the matrixentries� in �ˆ . The first product, � 1(

<r)7 �

2(<r � )7 , is just the rela-

tivistic�

counterpartto the familiar �� (< r± )7 �� (< r± � )7 , to which itrigorously� reducesin the nonrelativisticlimit.


The�

secondproduct, � 3] (< r± )7 � 4

Õ (< r± � )7 , is the analogoustermfor;

the lower componentsof the Dirac spinor. Since in arelativistic theory upper and lower componentsmust betreated�

on thesamefooting, theappearanceof thesetermsishighly�

plausible.In the nonrelativisticlimit the lower com-ponents� areby a factor of � /

RcS smaller� than the uppercom-

ponents.� Therefore, the second product, being of order(< �

/RcS )7 2B,� doesnot contributein this limit.

From theseconsiderationsit follows that it is not neces-sary� to appealto the discretesymmetriesof the Dirac equa-tion�

in order to identify the relativistic generalizationof theBCS OP, as we havedonein the beginningof this subsec-tion.�

It is sufficient to look at the upper left corner of thevarious !ˆ matrices,� which mustbe of the form

0 1"# 1 0

,� $ 79� %

in order to reduceto the BCS OP &�' (< r)7 (�)

(<r * )7 in the non-

relativistic limit. Obviously,both ways to identify the rela-tivistic�

generalizationof theBCSOPleadto thesameresult.

V. THE RELATIVISTICBOGOLUBOV–DE GENNES EQUATIONS

A. Relativistic Bogolubov-Valatin transformation

Equation + 75� ,

definesC

the Dirac–Bogolubov–de GennesHamiltonian�

for the generalizedBCS-OP -ˆ . This Hamil-tonian�

can be diagonalizedby a unitary canonicaltransfor-mation� from the field operators. (

<r± )7 to new operatorsa/ k

0 .1Usually2

thesenew operatorsare labeled 3 k0 in the literature

on� superconductivity.We usea/ k0 in orderto avoidconfusion

with/ the Dirac matrices 4ˆ .) In the nonrelativisticcase,thetransformation�

which diagonalizesEq. 5 126 is

given by theBogolubov-Valatintransformation43,66

Õ–68

7�8:9r± ;=<?>

k0 u@ k

0 A r± B a/ k0 C�DFE

k0 G r± H * a/ k

0 I† ,� J 80K L

M�N:OrP=Q?R

k0 u@ k

0 S rT a/ k0 U�VFW

k0 X rY * a/ k

0 Z† ,� [ 81K \

where/ the coefficientsu@ k0 (< r)

7and ] k

0 (< r)7

aredeterminedfromthe�

requirementthat the transformedHamiltonianbe diago-nal.� Obviously,thespinof thequasiparticlesentersthetrans-formation only in a fixed combination.To treat magneticimpurities,spin-orbitcoupling,triplet pairing,etc.,this trans-formation;

needs to be replaced by the more generalform; 1,37,41,43,45,69

^`_:arb=ced f

k0 g u@ hji k

0 k rl a/ m k0 nFoqpsr

k0* t ru a/ v k

0† w ,� x 82K y

where/ thespindegreesof freedomareinvolved in the trans-formationaswell. In the relativistic case,the spinlike quan-tum�

numbers z and� { have�

to be replacedby componentlabelsT

of the Dirac spinors.The relativistic generalizationofEq. | 82

K }is thus

~i � r± �=�?�

jk� � u@ i jk � r± � a/ jk

� �F�i jk* � r± � a/ jk

�† � . � 83K �

Thereare severalconditionsthe transformation� 83K �

hastosatisfy.� 49,70

ÕFirst of all it needsto be unitary � i.e., preserve

the�

normalizationof the quasiparticlewave functions� and�canonical� � i.e.,

preservetheanticommutationrelationsof the

field operators� .70�

Unitarity2

requiresthat

d 3]r® �

i

��i jk � r± �� i j � k0 �* � r± �= u@ i jk ¡ r± ¢ u@ i j £ k0 ¤* ¥ r± ¦�§©¨«ª

kk0 ¬®

j j� ¯°84K ±

and�d 3]r ²

i

³µí jk ¶ r· u@ i j ¸ k0 ¹»º r¼=½ u@ i jk ¾ r¿�À i j Á k0 Â®Ã rÄ�Å©Æ 0,

"Ç85K È

while/ the conditions

Ék j0 Ê u@ i jk Ë r± Ì�Í i Î jk�* Ï r± ÐÒÑ=ÓFÔ

i jk* Õ r± Ö u@ i × jk� Ø r± ÙÒÚ�Û©Ü 0" Ý

86K Þ

and�ßk j0 à u@ i jk* á r± â u@ i ã jk� ä r± åÒæ=çFè

i jk é r± ê�ë i ì jk�* í r± îÒï�ð©ñ«òii óõô÷ö r± ø r± ùÒú û

87K ü

ensure� that the transformationis canonical.As in thenonrel-ativistic� case,it turns out that the samerelationsare alsoobtained� by demandingthat the solutions of the resultingsingle-particle� equationsbe completeand orthonormal.45,49

ÕExplicitlyý

we find thatcompletenessof thesolutionsfollows,if

the transformationis canonical,while orthonormalityfol-lows, if it is unitary.

B.�

Dirac–Bogolubov�

–deþ

Gennes equations

FurtherF

conditionson the coefficientsu@ i jk(<r± )7 and ÿ i jk(

<r± )7

follow;

from demandingthat the Hamiltonian � 75� �

be@

diago-nal in the new creationandannihilationoperatorsa/ jk

� :

H ��jk� E

�jk� a/ jk

�† a/ jk� � E

�0� ,� � 88

K �

where/ E0� is

the ground-stateenergyand the a/ jk� create� and

annihilate� elementaryexcitations Bogolons with/ energyE�

jk� . In the sameway as in the nonrelativisticcase,43 one�

finds from Eqs. � 83K �

and� 88K �

that�

the u@ i jk(<r)7

and � i jk(<r)7

which/ diagonalizeEq. � 75� �

satisfy� a setof coupledintegrod-ifferential

equationsof the Bogolubov–de Gennes type.Theseequationsare most convenientlywritten in a matrixnotationas

h�ˆ �

�� * � h�ˆ *

u@ jk� � r��jk� � r� � E jk

� u@ jk� � r� jk� ! r" . # 89

K $

Here�

h�

is

the kernelof the Dirac Hamiltonian

h�ˆ %�&ˆ 0

� 'cS (ˆ pµ ) mcA 2 * 1 +�,ˆ 0

� -/.q¸ 0ˆ 1 A

» 243. 5 90

6 7

PRB 59 7151RELATIVISTICq

FRAMEWORK FOR . . .q

. I. ...

The�

term mcA 2(1< 8:9ˆ 0

�)7

arisesfrom subtractingmcA 2 from;

theenergy� eigenvaluesof Eq. ; 75

� <before@

diagonalization,i.e.,measuring� the energiesrelative to the rest energy. = is

an

integral

operatorthat containsthe pair potentialaskernel

>@?d 3]r A . . . BDC r,� r EGFIHˆ . J 91

6 KForF

the caseof a local pair potentialit reducesto the multi-plicative� operatorL (

<R)7 Mˆ ,� whereR is thecenter-of-massco-

ordinate� of theCooperpairs.Eachentry in thematrix in Eq.N89O P

is thusa 4 Q 4 matrix. Accordingly,the four-componentspinors� u@ jk

� (<r± )7 and R jk

� (<r± )7 aregiven by

u@ jk� S r± TVU

u@ 1 jk�

u@ 2 jk�

u@ 3]

jk�

u@ 4Õ

jk�

Wjk� X r± YVZ

[1 jk�

\2 jk�

]3]

jk�

^4Õ

jk�

. _ 926 `

Equation a 89O b

with/ Eqs. c 906 d

– e 926 f

constitutes� the relativisticgeneralization� of theBogolubov–de Gennesequation.It willin

the following be referredto as the Dirac–Bogolubov–deGennesg

equation.We can immediatelyverify that a numberof� importantspecialcasesis containedcorrectlyin Eq. h 89

O i:j

i k The nonrelativistic> limit is obtainedif we neglect thelowerT

two componentsof the spinors u@ jkl (<r± )7 and m jk

l (<r± ),7

which/ are small in the weakly relativistic limit and zero inthe�

nonrelativisticcase.The8 n 8O

equationo 89O p

then�

reducest�o a 4q 4 equation,which is identical to the nonrelativistic

4r 4 spin-Bogolubov–de Gennesequation.43,45,69Õ s

ii t In thenonsuperconducting> limit,� u@v 0,

"we obtainthe conventional

Dirac Hamiltoniansfor electronsandholes. w iii x In the localy

limity

the�

integraloperatorz becomes@

a multiplicativeopera-tor� {

(<r± )7 and we obtain the local version of the Dirac–

Bogolubov�

–de Gennesequation,derivedin Ref. 29.Equationsof a similar algebraicform as Eq. | 89

O }were/

previously� proposedin thecontextof nuclearphysicsandofHartree-Fock-Bogolubov�

theoryby KucharekandRing71~

and�by@

Zimdahl.72~

The�

presentderivationwithin the frameworkof� superconductivityand density-functionaltheory, the de-tailed�

symmetryanalysisof the orderparameter,andthe in-vestigation of the weakly relativistic limit and its conse-quences,� presentedin the presentandthe following paper,30

]however,�

arenot containedin theseolder works.

C.�

Density-functional aspects

The�

nonrelativisticHamiltonian � 12� is

the startingpointfor manymicroscopicinvestigationsof superconductivity,ofwhich/ we canjust discussa few.43,69,73

Õ–79

InG

thenonrelativisticcasethepotentials� (<r± )7 and � (

<r± ,� r± � )7

appearing� in Eq. � 12� are� eitherusedasparametersin orderto�

simulate, e.g., superconducting multilayers andheterostructures� 74,75

~or� determinedmicroscopicallyin a self-

consistent� fashion. The first approachcan be used in therelativistic caseaswell. If, for example,oneof the two ma-terials�

at the interfacecontainsheavy atoms,a relativisticdescriptionC

is calledfor.The�

self-consistentnumericalcalculationsareoften donein a mean-field framework43,76

Õ–79,69 or,� more recently,

usingv the apparatus of the density-functional theory

�DFT� . 21,41,44

B–49 In theDFT for superconductorsthe interac-

tion�

leadingto superconductivityand the Coulombinterac-tion�

are formally eliminated in favor of suitably chooseneffective� pair, � (

<r,� r � )7 , andlattice, � (

<r)7, potentials.The po-

tentials� �

(<r± )7 and � (

<r± ,� r± � )7 aredeterminedself-consistentlyas

functionalsof the densityand the orderparameter,by solv-ing theKohn-ShamBogolubov–de Gennesequations.Theseeffectice� potentialscanthusbe viewedasa convenientwayto�

dealwith the interactionsat hand.21,41,44–49

That suchrelativistic calculationscan becomenecessaryfor realistic superconductorscontaining heavy elementsisexemplified� by the resultsof Singh and co-workers,28 who/performed� � conventional� � relativistic� band-structurecalcula-tions�

for Ba(Sn,Sb)O3] and� concludedthat the absenceof

superconductivity� in thesematerialsis dueto relativistic ef-fects on the band structure. The Dirac–Bogolubov–deGennesg

equationsderivedaboveprovide the opportunitytoimprove

suchcalculationsby treatingtheeffectsof relativityand� superconductivityon the samefooting.

A proper relativistic DFT for superconductorshas notbeen@

formulatedasyet. Themainreasonfor this is theprob-lemT

of variational stability of the relativistic electron gaswhich,/ from a puristspoint of view, is only partly solved,even� for normal � i.e., nonsuperonducting� systems.� 25

BIt is not

the�

intentionof the presentpaperto tackle this questionforthe�

superconductingcase.It is, however,a definite conse-quence� of thepresentpaperthattheKohn-Shamequationsofany� conceivablerelativistic DFT for superconductorsmustAhave�

the algebraic form of Eq. (89� . In lieu of a microscopicprescription� how to determinetheeffectivepotentialsA � and��

in

this ‘‘Kohn –Sham–Dirac–Bogolubov–de Gennes’’equation,� we suggestthat theybe treatedeitherasadjustableparameters� to model realisticmaterials,as,e.g., in Refs.74and� 75, or by parametrizingthemin termsof the underlyingorbitals� of the systemunderstudy as,e.g., in Refs.47 and48.E

VI. SUMMARY AND OUTLOOK

The�

main resultsof this work aresummarizedin Table Iand� Eq. � 43� . In Table I we classifiedall possibleorderpa-rametersconsistentwith therequirementof relativisticcova-riance,� accordingto their transformationbehaviorunderLor-entz� transformations. This table therefore generalizesprevious� symmetry classificationsof the order parameterfrom the Galilei groupto the Lorentzgroup.The tablecon-tains�

the relativistic generalizationsof the standardBCS or-derC

parameter,as well as that of the triplet � Balian-�

Werthamer� �

order� parameters.It also predictsthat thereareseveral� other typesof order parameterswhich havenot yetbeen@

consideredin the literatureon superconductivity.Equationý �

43E �

contains� all theseorder parametersin amanifestlycovariantfashion.It expressesthetheoremthatallorder� parametershaveto transformlike bilinearcovariantsofthe�

Dirac equationin order to ensureLorentz invariance.Many of theseresultsare derived from more than one

viewpoint. In particular, the classificationof all relativisticOP’s�

asscalar,four vector,etc.,expressedin theabovetheo-rem� and Table I, follows from either one of the followingmethods: � i � A decompositionof bilinear forms of Diracspinors� accordingto irreduciblerepresentationsof the Lor-


entz� group.This methodis completelygeneralandalgebra-ically extremelysimple.However,it doesnot yield explicitexpressions� for the OP. ii ¡ The

�constructionof explicit ma-

trices�

having the indicatedtransformationproperties.Suchmatricesin turn canbe found in at leastthreeways: ¢ iia£ Bymaking educatedguessesas to the form of the matrix, fol-lowedT

by verification through explicit Lorentz transforma-tions.�

This methodis laboriousbut explicit. ¤ iib ¥By�

gener-alizing� the Balian-Werthamer construction ¦ i.e.,multiplication of the Pauli matriceswith the nonrelativistictime-reversal�

matrix§ to�

therelativisticdomain ¨ i.e.,

multipli-cation� of Dirac matriceswith the relativistic time-reversalmatrix© . ª iic « By forming linear combinationsof pair statesconstructed� usingthediscretesymmetriesof theDirac equa-tion,�

insteadof the Lorentzgroup.The�

latterconstructionallowsfor a physicalinterpretationof� the16 orderparameters,leadingto thedistinctionbetweenbetween@

triplet andsingletpairsand to identifying pairs in-volving positive and negative-energysolutionsof the Diracequation.�

We�

identifiedoneof the 16 orderparametersasthe rela-tivistic�

versionof theBCSorderparameter.TheHamiltoniancontaining� this orderparameterwasdiagonalized.Theresult-ing single-particleequationscan be regardedas the Diracequivalent� of the Bogolubov–de Gennesequations.

All¬

our considerationsin this paperarebasedon thecon-cepts� of pairing andLorentz invariance.They can thereforebe@

appliedwheneverpairing takesplace,not merely in the

case� of proper superconductors.Other situationsto whichour� results apply are superfluid helium 3,37,41

]nuclear

matter,� 80 and� the pairing of neutronsandprotonsin neutronstars.� 81,82

In order to predict observableconsequencesof the rela-tivistic�

termsit is advisableto proceedto theweakly relativ-istic

limit. This will bethesubjectof thesecondpaperin thisseries,� in which a numberof reductiontechniquesareappliedto�

the Dirac–Bogolubov–de Gennesequations.Thesemeth-ods� allow one to recoverthe familiar nonrelativisticequa-tions�

in zerothorderandto deriverelativistic correctionsinhigherorderof /

RcS . Explicit formsfor thesecorrectionswill

be@

derivedandit will bepointedout in which situationstheyare� relevantfor realisticsuperconductors.

ACKNOWLEDGMENTS

Many®

helpful discussionswith M. Luders,C

S. Kurth, T.Kreibich, M. Marques,J. Annett, and B. L. Gyorffy aregratefully� acknowleged.This work hasbenefitedfrom col-laborationsT

within, and has been partially funded by, theHCM�

network ‘‘ Ab-initio»

(from electronic structure) calcu-lationy

of complex processes in materials (Contract No.ERBCHRXCT930369)’’ and the program ‘‘ Relativistic ef-fects¯

in heavy-element chemistry and physics’’ of the Euro-pean� ScienceFoundation.Partial financial support by theDeutsche Forschungsgemeinschaftis gratefully acknowl-eged.�

1M. Sigrist andK. Ueda,Rev.Mod. Phys.63,° 239 ± 1991² .2J.³

Appel andP. Hertel,Phys.Rev.B 35´

,° 155 µ 1987¶ .·3¸G.¹

E. Volovik andL. P. Gorkov, JETPLett. 39´

, 674 º 1984» .·4¼J.³

Annett,Adv. Phys.39´

, 8° 3 ½ 1990¾ .5¿J.³

Appel, Phys.Rev. 139,° A1536 À 1965Á .·6ÂP.Ã

W. Anderson,Phys.Rev.Lett. 3´,° 325 Ä 1959Å .·

7ÆP.Ã

FuldeandK. Maki, Phys.Rev. 141,° 275 Ç 1966È .8ÉV.Ê

B. GeshkenbeinandA. I. Larkin, JETPLett. 43,° 395 Ë 1986Ì .9ÍA. J. Millis, D. Rainer, and J. A. Sauls,Phys. Rev. 38,° 4504Î

1988Ï .10G.¹

S. Lee,PhysicaC 292, 171 Ð 1998Ñ .11N.Ò

R. Werthameret al., Phys.Rev. 147, 296 Ó 1966Ô .12K. Maki, Phys.Rev. 148,° 362 Õ 1966Ö .·13K.×

Capelle,E. K. U. Gross,andB. L. Gyorffy,Ø Phys.Rev. Lett.78Ù

,° 3753 Ú 1997Û .·14K. Capelle,E. K. U. Gross,andB. L. Gyorffy, Phys.Rev.B 58,

473 Ü 1998Ý .·15K.×

Capelle,E. K. U. Gross,andB. L. GyorffyØ Þ unpublishedß .16B.à

Cabrera,H. Gutfreund,and W. A. Little, Phys.Rev. B 25á

,°6644â ã

1982ä .·17B. CabreraandM. E. Peskin,Phys.Rev.B 39

´,° 6425 å 1989æ .·

18S.ç

B. Felchetè al.,° Phys.Rev.B 31´

,° 7006 é 1985ê .·19B. Cabreraet al., Phys.Rev.B 42, 7885 ë 1990ì .20í

T. Holstein,R. E. Norton, and P. Pincus,Phys.Rev. B 8, 2649î1973ï .

21W.ð

Kohn, E. K. U. Gross,and L. N. Oliveira, J. Phys. ñ Franceò ó

50ô

,° 2601 õ 1989ö .·22í

M. Weger ÷ privatecommunicationø .23í

M.ù

Eliashvili andG. Tsitsishvili, Phys.Rev.B 57, 2713 ú 1998û .

24í

T. Ichiguchi,Phys.Rev.B 57ô

, 638 ü 1998ý .·25E. Engeletè al., i° n Density-Functional

þTheory,° Vol. 337of NATO

ÿAdvanced�

Study Institute, Series B: Physics,° editedby E. K. U.GrossandR. M. Dreizler � Plenum,New York, 1995� .�

26í

D. D. Koelling and A. H. MacDonald,in Relativistic Effects inAtoms,�

Molecules and Solids, Vol.�

87 of NATOÿ

ASI Series B:Physics, editedby G. L. Malli � Plenum,New York, 1983� .�

27í

B. L. Gyorffy et al., in The Effects of Relativity in Atoms, Mol-ecules and the Solid State,° editedby S.Wilson, I. P.Grant,B. L.Gyorffy � Plenum,New York, 1991� .�

28D. J. Singh, D. A. Papaconstantopoulos,J. P. Julien, and F.Cyrot-Lackman,Phys.Rev.B 44,° 9519 1991 .�

29í

K. CapelleandE. K. U. Gross,Phys.Lett. A 218,° 261 � 1995� .30¸

K. CapelleandE. K. U. Gross,following paper,Phys.Rev.B 59ô

,°7155 1999� .�

31¸

K. UedaandT. M. Rice,Phys.Rev.B 31, 7114 � 1985� .32¸

M. R. Norman,Phys.Rev.Lett. 72,° 2077 � 1994� .33¸

P. Fulde,J. Keller, andG. Zwicknagl, in Solid�

State Phyics: Ad-vances in Research and Applications, edited� by H. Ehrenreichand D. Turnbull � AcademicPress,SanDiego, 1988� , Vol. 41,p. 1.

34¸

P. W. Anderson,Phys.Rev.B 30´

, 4000 � 1984� .�35¸

J. Bardeen,L. N. Cooper,and J. R. Schrieffer,Phys.Rev. 108,°1175 � 1957� .�

36¸

J. R. Schrieffer,Theory�

of Superconductivity � Addison-Wesley,Reading,MA, 1988� .

37¸

D. Vollhardt andP. Woelfle, The Superfluid Phases of Helium 3�Taylor&Francis,London,1990� .

38¸

T. IshiguroandK. Yamagij, Organic Superconductors,° Springer

PRB 59 7153RELATIVISTIC

FRAMEWORK FOR . . .

. I. ...

Series!

in Solid StateSciences,Vol. 88 " Springer-Verlag,Berlin,1990# .

39¸

H.$

Tou etè al.,° Phys.Rev.Lett. 77,° 1374 % 1996& .�40R.

Balian andN. R. Werthamer,Phys.Rev. 131'

, 1553 ( 1963) .�41¼

K. Capelleand E. K. U. Gross,Int. J. QuantumChem.61,° 325*1997+ .

42P.,

W. Anderson,J. Phys.Chem.Solids11, 25 - 1959. .�43P.,

G. de Gennes,Superconductivity of Metals and Alloys / Ben-0

jamin,1

New York, 19662 .�44¼

L. N. Oliveira,E. K. U. Gross,andW. Kohn,Phys.Rev.Lett. 60,24303 4

19885 .�45E.6

K. U. Gross,S. Kurth, K. Capelle,andM. Luders,7

in Density-þ

Functional Theory, Vol. 337of NATOÿ

Advanced Study Institute,Series�

B: Physics,° editedby E. K. U. GrossandR. M. Dreizler8Plenum,,

New York, 19959 .46O.-J.:

Wackeret al., Phys.Rev.Lett. 73Ù

,° 2915 ; 1994< .�47¼

M. B. Suvasinietè al.,° Phys.Rev.B 48,° 1202 = 1993> .�48¼

M. B. SuvasiniandB. L. Gyorffy, PhysicaB 195,° 109 ? 1992@ .49E.6

K. U. GrossandS. Kurth, Int. J. QuantumChem.,Symp.25á

,2893 A

1991B .�50¿

J.³

D. Bjorken and S. D. Drell, Relativistic Quantum MechanicsandC Relativistic Quantum Fields D McGraw-Hill, New York,1964E .

51¿

C.F

Itzykson and J.-B. Zuber, Quantum Field Theory G McGraw-Hill, New York, 1980H .�

52¿

The matrix Iˆ in Eq. J 19K is twice the matrix called Lˆ in Ref. 29.Thedefinitionusedhereis moreconvenientfor thepresentpur-

pose.M Defining Nˆ ,° andconsequentlyOˆ 0P,° ..., Qˆ 3

R, with a factor 1

2SasC in Ref.29T wouldU leadto unpleasantfactors1

8V and 1

32W in Eqs.X

24Y – Z 34[ \

andC in TableI.53¿

J.³

J. Sakurai,Advanced�

Quantum Mechanics ] Addison-Wesley,Reading,

MA, 1976 .54¿

J.³

M. Jauchand F. Rohrlich, The Theory of Photons and Elec-trons_ , 2nd ed. ` Springer-Verlag,Berlin, 1980a .�

55¿

P.,

W. Anderson,Basicb

Notions of Condensed Matter PhysicscBenjamin/Cummings,Menlo Park,California,1984d ,° p. 311.

56¿

G.e

E. Volovik andL. P. Gorkov, JETP61f

,° 843 g 1985h .�57¿

S.!

Yip andA. Garg,Phys.Rev.B 48,° 3304 i 1993j .58¿

M.ù

Luders,7

S. Kurth, andE. K. U. Gross k unpublishedl .59¿

Thepossibility of forming bilinearcovariantswith m Tn

insteadofo¯ prq †s t

ˆ 0P

isu

alsopointedout in Sec.3.4 of Ref. 54, whereit isstatedv that thesealternativecovariants‘‘ do not seem to have anyphysicalw significance in radiation theory.’’� As follows from theaboveC considerationit is superconductivity wheretheanomalousbilinearx

covariantsaquirephysicalsignificanceastherelativisticgeneralizationy of the orderparameter.

60Â

G. J. Ljubarski, Anwendungen der Gruppentheorie in der PhysikzDeutscherVerlag der Wissenschaften,Berlin, 1962{ .�

61Â

J. S. Lomont, Applications�

of Finite Groups | Academic,NewYork, 1959} .

62Â

J. P. Elliot andP. G. Dawber,Symmetry in Physics ~ MacMillan,London,1979� .

63Â

Whetherthe states� 51� �

areC interpretedas positronsor negative-energyelectronsdependson whetherthey areoccupiedor not.For the constructionof a basisset, this is immaterial,becausethe occupationof the statesof the basisset is not a propertyof

thebasissetitself. Theinterpretationof thestatesinvolving C isdiscussedin detail in Sec. IV B and the secondpaper in thisseries.

64Â

From the discussionin Sec.II A it follows that the sameconnec-tion existsalsoin thenonrelativistictheoryof superconductivity:Linear combinationsof pair statesconstructedfrom the discrete

symmetriesof the Schrodinger equation(T� ˆ and P

� ˆ ) are repre-sentedin spin spaceby combinationsof matriceswhich reflectthe behaviorof the OP underspatial rotations � singlet: scalar,triplet: vector� .�

65Â

It follows that thereare no electron-positronpairs containedin

our theory,asthesewould requiretermslike bp� (�1)�

( bp� (�3R

)�)†s.� Note

that electron-positronpairs would yield neither supercurrentsnor a Meissnereffect.

66Â

N. N. Bogolubov,Nuovo Cimento7, 794 � 1958� .�67Â

J. G. Valatin, Nuovo Cimento7Ù,° 843 � 1958� .

68Â

M. Tinkham, Introduction�

to Superconductivity � McGraw-Hill,New York, 1975� .�

69Â

W. N. MathewsJr., Phys.StatusSolidi B 90�

,° 327 � 1978� .�70Æ

J.-P. Blaizot and G. Ripka, Quantum�

Theory of Finite Systems�MITù

Press,Cambridge,1986� .71Æ

H. KucharekandP. Ring, Z. Phys.A 339´

, 2° 3 � 1997� .72Æ

W. Zimdahl,Czech.J. Phys.Sect.B 29, 713 � 1979� .�73Æ

J. Bardeenet al.,° Phys.Rev. 187'

,° 556 � 1969� .�74Æ

H. Plehnetè al.,° J. Low Temp.Phys.83�

, 71 � 1991� .�75Æ

H. Plehnetè al.,° Phys.Rev.B 49, 1° 2 140 � 1994 .�76Æ

F. Gygi andM. Schluter, Phys.Rev.Lett. 65, 1820 ¡ 1990¢ .77Æ

F. Gygi andM. Schluter, Phys.Rev.B 43£

,° 7609 ¤ 1991¥ .�78Æ

J. D. Shoreetè al.,° Phys.Rev.Lett. 62, 3089 ¦ 1989§ .79Æ

M. RasoltandZ. Tesanovic,Rev.Mod. Phys.64f

,° 709 ¨ 1992© .�80É

P. Ring and P. Schuck, The Nuclear Many-Body ProblemªSpringer-Verlag,!

New York, 1980« .�81É

D. PinesandA. Alpar, Nature ¬ London 316´

, 27 ® 1985 .�82É

J. A. Sauls,D. L. Stein,andJ. W. Serene,Phys.Rev.D 25,° 967°1982± .


Documents

Relativistic framework for microscopic theories of ... · Relativistic framework for microscopic theories of superconductivity. I. The Dirac equation for superconductors K. Capelle