Spin-charge rotating local reference frames: a unified U(2)=U(1)xSU(2) approach to the interacting electrons

0.09 MB·English

by  Tadeusz. K. Kopec

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Spin-charge rotating local reference frames: a unified
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U(2)=U(1)⊗SU(2) approach to the interacting electrons
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T.K.Kopec∗1

n
a 1 InstituteforLowTemperatureandStructureResearch,PolishAcademyofSciences,
J POB1410,50-950Wroclaw2,Poland

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Received6/2/2008
]
l
e
- Keywords Hubbardmodel,Coulombinteraction,gaugetheory
r
t PACS 74.20.Fg,74.72.h,71.10.Pm
s
.
t Aspin-chargeunifyingdescriptionfortheHubbardmodelbasedonthetimedependentlocalgaugetransfor-
a
mationsisdeveloped.Thecollectivevariablesforchargeandspinareisolatedintheformofthespace-time
m
fluctuatingU(1)phasefieldandrotatingspinquantizationaxisgovernedbytheSU(2)symmetry, respec-
- tively. As a result interacting electrons appear as a composite objects consisting of bare fermions with
d attachedU(1)andSU(2)gaugefields. Weelaborateonthemicroscopicoriginsoftheeffectiveactionwith
n theCoulombinteractionthatcontaintopologicalthetaterms.Furthermore,weunravelthelinkbetweennon-
o trivialmultiply-connected topological structureoftheU(2)=U(1)⊗SU(2)configurational spaceforgauge
c fieldsandtheinstantoncontributiontothestatisticalsum.
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1 1 Introduction
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0 Amongthe electronicHamiltoniansrelevantfor interactingsystems the Hubbardmodel[1, 2, 3] is con-
/ sideredasonethatcontainstheessentialingredientsforunderstandingthephysicsofcorrelatedelectrons.
t
a Inamany-bodysystemtherelevantphysicsisencodedinthesymmetriesandintheHubbardmodelthey
m
are represented by the charge U(1) gauge and spin rotational SU(2) groups relevant for the occurrence
- of the superconducting and magnetic orderings. Interestingly, the resulting U(2)=U(1)⊗SU(2) symme-
d
trygrouphasanontrivialmultiplyconnectedtopologicalstructure. Formultiplyconnectedconfiguration
n
o spacesnovelfeaturescanariseasforexampleintheAharonov-Bo¨hmeffect[4]governedbythemultiply
c connectedU(1)groupmanifold. Sincethehomotopyclassπ [U(2)]=Z alsoformsasetofintegerwind-
1
:
v ingnumbersthetopologicalstructureoftheconfigurationspaceisnontrivial,ambiguitiesmayarisewhen
i attemptsare madeto specify a value for the phase of a wave functionfor the whole configurationspace
X
[5]. Thusthe problemwe are facing is that of many-bodyquantummechanics on a multiply connected
r
configuration space. In the present work we develop a spin-charge unifying description for interacting
a
electrons given by the Hubbard model. It is based on the time dependentlocal U(2) gauge transforma-
tionstodisentangletheCoulombinteraction. Thecollectivevariablesforchargeandspinareisolatedina
formofthespace-timefluctuatingU(1)phasefieldandtherotatingspinquantizationaxisgovernedbythe
SU(2)symmetry,respectively. Asaresultinteractingelectronsappearascompositeobjectsconsistingof
barefermionswithattachedU(1)andSU(2)gaugefieldsandduenontrivialtopologyoftheU(2)groupan
instantoncontributiontothestatisticalsumemerges.
∗ Correspondingauthor:e-mail:[email protected]
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4 T.K.Kopec:Spin-chargerotatinglocalreferenceframes
2 Hubbard Hamiltonian
OurstartingpointisthepurelyfermionicHubbardHamiltonianH≡H +H :
t U
H=−t [c†(r)c (r′)+h.c.]+U n (r)n (r). (1)
α α ↑ ↓
hrr′i,α r
X X
Here, hr,r′i runs over the nearest-neighbor (n.n.) sites, t is the hopping amplitude, U stands for the
Coulomb repulsion, while the operator c†(r) creates an electron with spin α =↑,↓ at the lattice site r,
α
where n (r) = c†(r)c (r). Usually, working in the grand canonical ensemble a term is added to H
α α α
in Eq.(1)to control the average number of electrons, H → H −µ n(r) with µ being the chemical
r
potentialandn(r) = n (r)+n (r)thenumberoperator. ItiscustomarytointroduceGrassmannfields,
↑ ↓
P
c (rτ) dependingon the “imaginarytime” 0 ≤ τ ≤ β ≡ 1/k T, (with T being the temperature)that
α B
satisfytheanti–periodicconditionc (rτ)=−c (rτ+β),towritethepathintegralforthestatisticalsum
α α
Z = [Dc¯Dc]e−S[c¯,c]withthefermionicaction
R β
S[c¯,c]=S [c¯,c]+ dτH[c¯,c], (2)
B
Z0
thatcontainsthe fermionicBerryterm [6]: S [c¯,c] = βdτc¯ (rτ)∂ c (rτ) thatwill play an im-
B rα 0 α τ α
portantroleinourconsiderations.
P R
3 Spin-charge U(2)reference frames
Forstronglycorrelatedsystemitiscrucialtoconstructacovariantformulationofthetheorywhichnaturally
preserves the spin-rotational symmetry present in the Hubbard Hamiltonian. In order to maintain spin
rotationalinvariance,oneshouldconsiderthespin-quantizationaxistobea prioriarbitraryandintegrate
overallpossibledirectionsinthepartitionfunction.Forthispurposethedensity–densityproductinEq.(1)
wewrite,followingRef.[7],inaspin-rotationalinvariantway:
1
H =U n2(rτ)−[Ω(rτ)·S(rτ)]2 , (3)
U
4
r (cid:26) (cid:27)
X
whereSa(rτ)= 12 αα′c†α(rτ)σˆαaα′cα′(rτ)denotesthevectorspinoperator(a=x,y,z)withσˆabeing
thePaulimatrices. TheunitvectorΩ(rτ) = [sinϑ(rτ)cosϕ(rτ),sinϑ(rτ)sinϕ(rτ),cosϑ(rτ)]written
P
intermsofpolarangleslabelsvaryinginspace-timespinquantizationaxis.Thespin–rotationinvarianceis
madeexplicitbyperformingtheangularintegrationoverΩ(rτ)ateachsiteandtime. Bydecouplingspin
andchargedensitytermsinEq.(3)usingauxiliaryfields̺(rτ)andiV(rτ)respectively,wewritedownthe
partitionfunctionintheform[9]
Z = [DΩ] [DVD̺] [Dc¯Dc]e−S[Ω,V,̺,c¯,c]. (4)
Z Z Z
where[DΩ] ≡ sinϑ(rτk)dϑ(rτk)dϕ(rτk) isthespin-angularintegrationmeasure. Theeffectiveaction
rτk 4π
reads:
Q
β ̺2(rτ) V2(rτ)
S[Ω,V,̺,c¯,c] = dτ + + iV(rτ)n(rτ)+2̺(rτ)Ω(rτ)·S(rτ)]
U U
r Z0 (cid:20)
X
β
+ S [c¯,c]+ dτH [c¯,c]. (5)
B t
Z0
SimpleHartree-Focktheorywon’tworkforaHubbardmodelinwhichU isthelargestenergyintheprob-
lem. OnehastoisolatestronglyfluctuatingmodesgeneratedbytheHubbardtermaccordingtothecharge
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U(1) and spin SU(2) symmetries. To this end we write the fluctuating “imaginary chemical potential”
iV(rτ) as a sum of a static V (r) and periodic function V(rτ) = V (r) +V˜(rτ) using Fourier series
0 0
V˜(rτ) = 1 ∞ [V˜(rω )eiωnτ +c.c.]withω = 2πn/β (n = 0,±1,±2)beingthe(Bose)Matsubara
β n=1 n n
frequencies.Now,weintroducetheU(1)phase fieldφ(rτ)viatheFaraday–typerelation[8]
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∂φ(rτ) 1 ∂
φ˙(rτ)≡ =e−iφ(rτ) eiφ(rτ) =V˜(rτ). (6)
∂τ i ∂τ
Furthermore,byperformingthelocalgaugetransformationtothenewfermionicvariablesf (rτ):
α
c (rτ) z(rτ) 0 f (rτ)
α = α (7)
c¯ (rτ) 0 z¯(rτ) f¯ (rτ)
α α
(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
where the unimodularparameter|z(rτ)|2 = 1 satisfies z(rτ) = eiφ(rτ), we removethe imaginaryterm
i βdτV˜(rτ)n(rτ)foralltheFouriermodesoftheV(rτ)field,exceptforthezerofrequency.Subsequent
0
SU(2)transformationfromf (rτ)toh (rτ)operators,
α α
R
f (rτ) ζ (rτ) −ζ¯ (rτ) h (rτ) ζ (rτ) −ζ¯ (rτ)
↑ = ↑ ↓ 1 , R(rτ)≡ ↑ ↓ (8)
f (rτ) ζ (rτ) ζ¯ (rτ) h (rτ) ζ (rτ) ζ¯ (rτ)
↓ ↓ ↑ ↓ ↓ ↑
(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)
with the constraint |ζ (rτ)|2 + |ζ (rτ)|2 = 1 takes away the rotational dependence on Ω(rτ) in the
↑ ↓
spin sector. This is done by meansof the Hopf map R(rτ)σˆzR†(rτ) = σˆ ·Ω(rτ) that is based on the
enlargement from two-sphere S to the three-sphere S ∼ SU(2). The unimodular constraint can be
2 3
resolvedbyusingtheparametrization
ζ↑(rτ) = e−2i[ϕ(rτ)+χ(rτ)]cos ϑ(rτ)
2
(cid:20) (cid:21)
ζ↓(rτ) = e2i[ϕ(rτ)−χ(rτ)]sin ϑ(rτ) (9)
2
(cid:20) (cid:21)
with the Euler angular variables ϕ(rτ),ϑ(rτ) and χ(rτ), respectively. Here, the extra variable χ(rτ)
representstheU(1)gaugefreedomofthetheoryas a consequenceofS → S mapping. One cansum-
2 3
marizeEqs(7)and(8)bythesinglejointgaugetransformationexhibitingelectronoperatorfactorization
cα(rτ) = α′Uαα′(rτ)hα′(rτ), where U(rτ) = z(rτ)R(rτ) is a U(2) matrix which rotates the spin-
quantizationaxisatsiterandtimeτ. Thisreflectsthecompositenatureoftheinteractingelectronformed
P
frombosonicspinorialandchargedegreesoffreedomgivenbyRαα′(rτ)andz(rτ),respectivelyaswellas
remainingfermionicparth (rτ). Accordingly,theintegrationmeasureoverthegroupmanifoldbecomes
α
φ(rβ)=φ0(r)+2πm(r) Ω(rβ)=Ω0
2π
[DφDΩ]≡ dφ (r) dΩ (r) Dφ(rτ) DΩ(rτ) (10)
0 0
Z {mX(r)}Yr Z0 Z φ(r0)=Zφ0(r) Ω(r0Z)=Ω0
where dΩ··· = 1 πsinθdϑ 2πdϕ... and[DΩ(rτ)] = dΩ(rτ ). Here,m ∈ Z labelsequiva-
4π 0 0 k k
lenceclassesofhomotopicallyconnectedpaths[10].
R R R Q
4 Effective phase-angular action
AnticipatingthatspatialandtemporalfluctuationsofthefieldsV (r)and̺(rτ) willbeenergeticallype-
0
nalised, since they are gapedand decouplefrom the angularand phase variables. Therefore, in orderto
makefurtherprogresstowardswe nextsubjectthefunctionaltoasaddlepointanalysis. Theexpectation
valueofthestatic(zerofrequency)partofthefluctuatingelectrochemicalpotentialV (r)wecalculateby
0
the saddle point method to give V (r) = i µ− Un ≡ iµ¯ where µ¯ is the chemical potential with a
0 2 h
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6 T.K.Kopec:Spin-chargerotatinglocalreferenceframes
Hartreeshiftoriginatingfromthesaddle-pointvalueofthestaticvariableV (r)withn =n +n and
0 h h↑ h↓
n =hh¯ (rτ)h (rτ)i. Similarlyinthemagneticsector
hα α α
r
(−1) ∆
ρ(rτ)= c (11)
±∆
c
(cid:26)
where∆ =UhSz(rτisetsthemagnitudefortheMott-chargegap.ThetwochoicesdelineatedinEq.(11)
c
correspondto the saddle pointof the “antifferomagnetic”(with staggering ∆ ) or “ferromagnetictype”.
c
Notethatthenotionferromagnetic(antifferomagnetic)heredoesnotmeananactuallong–rangeordering-
forthistheangularspin-quantizationvariableshavetobeorderedaswell.Inthenewvariablestheactionin
Eq.(5)assumestheformS Ω,φ,̺,h¯,h =SB[h¯,h]+ 0βdτHΩ,φ[ρ,h¯,h]+S0[φ]+2 r 0βdτA(rτ)·
S (rτ),whereS (rτ)= 1 h¯ (rτ)σˆ h (rτ). Furthermore,
h h 2(cid:2) αγ α (cid:3) αγ γ R P R
β P φ˙2(rτ) 12µ
S [φ]= dτ + φ˙(rτ) (12)
0
U i U
r Z0 " #
X
standsforthekineticandBerrytermoftheU(1)phasefield inthechargesector.TheSU(2)gaugetrans-
formationinEq.(8)andthefermionicBerrytermgenerateSU(2)potentialsgivenbyR†(rτ)∂ R(rτ) =
τ
R† ϕ˙ ∂ +ϑ˙ ∂ +χ˙ ∂ R=−σˆ·A(rτ),where
∂ϕ ∂ϑ ∂χ
(cid:16) (cid:17)
i i
Ax(rτ) = ϑ˙(rτ)sinχ(rτ)− ϕ˙(rτ)sinθ(rτ)cosχ(rτ)
2 2
i i
Ay(rτ) = ϑ˙(rτ)cosχ(rτ)+ ϕ˙(rτ)sinθ(rτ)sinχ(rτ)
2 2
i i
Az(rτ) = ϕ˙(rτ)cosϑ(rτ)+ χ˙(rτ). (13)
2 2
aretheSU(2)gaugepotentials.Thefermionicsector,inturn,isgovernedbytheeffectiveHamiltonian
HΩ,φ = ̺(rτ)[h¯↑(rτ)h↑(rτ)−¯h↓(rτ)h↓(rτ)]
r
X
−t U†(rτ)U(r′τ) h¯ (rτ)h (r′τ)−µ¯ h¯ (rτ)h (rτ), (14)
αγ α γ α α
hXr,r′iXαγ (cid:2) (cid:3) Xrα
The resultofthe gaugetransformationsis thatwe havemanagedto cast the stronglycorrelatedproblem
intoasystemofmutuallynon-interactingfermions,submergedinthebathofstronglyfluctuatingU(1)and
SU(2)fieldswhosedynamicsisgovernedbytheenergyscalesetbytheCoulombinteractionU coupledto
fermionsviahoppingtermandwiththeZeeman-typecontributionwiththemassivefield̺(rτ).
In analogy to the chargeU(1) field the SU(2) spin system exhibitemergentdynamics. Integrationof
fermionswillgeneratethekinetictermfortheSU(2)rotors
1 β
S [Ω] = − ×4 dτdτ′ Aa(rτ)Ab(r′τ′)
0
2
rr′ Z0 ab
X X
β
× hSa(rτ)Sb(r′τ′)i−2 dτA(rτ)·hS (r′τ′)i (15)
h h h
r′ rr′ Z0
X X
with
1 1 ∆
hSa(rτ)Sb(r′τ′)i=− ×2δ , hSz(rτ)i= c (16)
h h 4 abE h U
s
In the AF phase, at the half-filling, it assumes the staggeredform ̺(rτ) = ∆ (−1)r with ∆ beingthe
c c
charge gap ∆c ∼ U/2 for U/t ≫ 1, with Ek = ε2k+∆2c, Q = (π,π) and εk = ǫk −µ¯. At zero
p
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temperature lim 1 = ∆2c so that, in the limit U/t ≫ 1 one obtains E ∼ 2∆ = U. However,
T→0 Es 2Ek3 s c
a nonzerovalue of ∆ does not imply the existence of anfiferromagneticlong–rangeorder. For this the
c
angulardegreesoffreedomΩ(rτ)havealsotobeordered,whoselow-lyingexcitationsareintheformof
spinwaves. ThereforethekineticterminthespinsectorbecomesS [Ω]=− 1 βdτA(rτ)·A(rτ)
0 Es r 0
sothatthetotalkineticenergyS[Ω,φ]=S [φ]+S [Ω]forU(1)andSU(2)rotorsbecomes
0 0 P R
β φ˙2(rτ) 12µ 1
S [Ω,φ] = dτ + φ˙(rτ) + ϑ˙2(rτ)+ϕ˙2(rτ)+χ˙2(rτ)
0
U i U 4E
Xr Z0 ( s h
∆
+ 2ϕ˙(rτ)χ˙(rτ)cosϑ(rτ)]+ c [ϕ˙(rτ)cosϑ(rτ)+χ˙(rτ)] . (17)
iU
(cid:27)
ThedistinctivefeatureofEq.(17)isthpresenceofthegeometricBerrycontributionsthatsignifytopolog-
icalfeaturesoftheunderlyingfieldtheory[11].
5 Summary
InthisworkwehaveobservedthattheimportantsymmetriesoftheHubbardmodelgivenbybythecharge
U(1)gaugeand spinrotationalSU(2)groupsimplythatthe quantumdynamicsis governednon-trivially
by the multiply connected U(2)=U(1)⊗SU(2)group manifold. Following this route we have developed
a spin-charge unifying description for interacting electrons given by the Hubbard model. It is based on
the time dependent local gauge transformations to disentangle the Coulomb interaction. The collective
variables for charge and spin are isolated in a form of the space-time fluctuating phase field and the ro-
tatingspinquantizationaxis. Asaresultinteractingelectronsappearasacompositeobjectsconsistingof
bare fermionswith attached gauge fields. Our results forma generalframeworktailored to describe eg.
bothmagneticandsuperconductingorderwhichisrootedinmicroscopicofstronglycorrelatedelectrons
and their basic symmetries and interactions [12]. In particular, it will enable to identify the degrees of
freedomofthehierarchyoflow-energyeffectivetheories,andthewaytheyareassociatedwiththeglobal
symmetries.
Acknowledgements ThisworkwassupportedbytheMinistryofEducationandScience(MEN)underGrantNo. 1
P03B10330intheyears2006–2008.
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