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

`and 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

 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

`and 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 ˆ

V3I

,� Ã 24b Ä

ňA0`

1Æ 1Ç 1

1

ÈEɈ ʈ 5� Ë

ˆV0`

,� Ì 25b Í

ΈAÏ1

1 Ð1 Ñ

1

1

ÒEÓˆ Ôˆ 5� Õ

ˆV1 ,� Ö 26×

؈AÏ2

i°

i°

Ù i°

Ú i°

ÛE܈ ݈ 5� Þ

ˆV2 ,� ß 27à

áˆA3I

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

�ˆA� �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 ˆ 3l

0à Ĉ

0 ň V0g Æ

ˆV1 Lj

V0g È

ˆV2 Ɉ

V0g Ê

ˆV3l

Ë�̈V0g Í

ˆV1 0

ΠψV1 Ј

V2j Ñ

ˆV1 Òˆ

V3l

Ó�ÔˆV0g Õ

ˆV2 Ö�׈

V1 ؈

V2 0

Î ÙˆV2 Úˆ

V3l

ÛÝ܈V0g 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

7144 PRB 59K. CAPELLE AND E. K. U. GROSS

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 ň ÆÂLj 0] ÈmÉ�ÊZÈ

T ˈ Ì Í 42Î Ï

is invariant under this operationas well. We concludethatÐ T ш 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, namely܈ 0],ü ݈ 1,ü Þˆ 2,ü ߈ 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 ^E_`Iaˆ

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� .

7146 PRB 59K. CAPELLE AND E. K. U. GROSS

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 ñˆi 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

7148 PRB 59K. CAPELLE AND E. K. U. GROSS

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, ˈ 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.

7150 PRB 59K. CAPELLE AND E. K. U. GROSS

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

³µ´i jk ¶ r· u@ i j ¸ k0 ¹»º r¼=½ u@ i jk ¾ r¿�À i j Á k0 ®à 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-

7152 PRB 59K. CAPELLE AND E. K. U. GROSS

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.�

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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

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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.

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7154 PRB 59K. CAPELLE AND E. K. U. GROSS