whereweintroducethesymbol(Bosons)(1.39)(Fermions)Note that P is a projection operator:p2 = p.(1.40)Overlap1(1.41)(Bi... BN|Q.QN) S (B:lai)),Veng!lan!where s is a permanent or determinantfor Bosons and Fermions,respectively,definedby(1.42)Per(M)=Mi,PiM2.P2...MN,PN,Pdet (M)=(-1)PMi,P1M2,P2 ..MN,PN.(1.43)Closure relation can be obtained from Eq. (1.33) by applying the projection p: Pai...an)(a1...anlP=p.(1.44)Iana!,7(1.45)Q1...QN) (a1...aN|=1pN!01...0Nwhere lp=p is the identityoperatoroftheprojected space.1.4Creationand annihilationoperatorsNO51.41.4.1BasicsCreation operator adds a particle(1.46)atJa1...αn)=Vna +1aQ1...an),(Boson)Jaa...an)α$ [a,...,an],(Fermion)(1.47)ata1...an):10QE(a,..-,an]VacuumstateJo):a statewithnoparticle.Notethatitisnotazerostate!Asymmetrized statecanbecreatedfromthevacuumstateby(1.48)aaat...aan10)[Q1...Qn)-Vllana!Commutation relations: the symmetry or antisymmetry properties ofthe many-particle statesimposecommutationoranticommutationrelationsbetweenthecreationoperators:atat - atat = [at, al]- = o(Bosons)(1.49)(1.50)atag + agat = [at, atl+ = o(Fermions)Fock space: the creation operator changes the number of particles. Therefore, the space ofall statesshould includeall Hilbert spaces with different numbers of particles:B = Bo @ Bi @...(1.51)(1.52)F=Fo④Fi@....4
where we introduce the symbol ζ = ( 1 (Bosons) −1 (Fermions) . (1.39) Note that Pˆ is a projection operator: Pˆ2 = Pˆ. (1.40) Overlap ⟨β1 · · · βN |α1 · · · αN ⟩ = 1 qQ β nβ! Q α nα! S (⟨βi |αj ⟩), (1.41) where S is a permanent or determinant for Bosons and Fermions, respectively, defined by Per (M) = X P M1,P 1M2,P 2 . . . MN,P N , (1.42) det (M) = X P (−1)PM1,P 1M2,P 2 . . . MN,P N . (1.43) Closure relation can be obtained from Eq. (1.33) by applying the projection Pˆ: X α1.αN P | ˆ α1 . . . αN ) (α1 . . . αN |Pˆ = Pˆ. (1.44) X α1.αN Q α nα! N! |α1 . . . αN ⟩ ⟨α1 . . . αN | = 1P, (1.45) where 1P ≡ Pˆ is the identity operator of the projected space. 1.4 Creation and annihilation operators 1.4.1 Basics NO§1.4 Creation operator adds a particle a † α |α1 . . . αN ⟩ = √ nα + 1 |αα1 . . . αN ⟩, (Boson), (1.46) a † α |α1 . . . αN ⟩ = ( |αα1 . . . αN ⟩ α /∈ {α1, . . . , αN } 0 α ∈ {α1, . . . , αN } , (Fermion). (1.47) Vacuum state |0⟩: a state with no particle. Note that it is not a zero state! A symmetrized state can be created from the vacuum state by |α1 . . . αN ⟩ = 1 pQ α nα! a † α1 a † α2 . . . a† αN |0⟩. (1.48) Commutation relations: the symmetry or antisymmetry properties of the many-particle states impose commutation or anticommutation relations between the creation operators: aˆ † αaˆ † β − aˆ † β aˆ † α ≡ [ˆa † α, aˆ † β ]− = 0 (Bosons) (1.49) aˆ † αaˆ † β + ˆa † β aˆ † α ≡ [ˆa † α, aˆ † β ]+ = 0 (Fermions) (1.50) Fock space: the creation operator changes the number of particles. Therefore, the space of all states should include all Hilbert spaces with different numbers of particles: B = B0 ⊕ B1 ⊕ . . . , (1.51) F = F0 ⊕ F1 ⊕ . . . . (1.52) 4
(1.45)Closurerelations:IIa na!10) (0I +7(1.53)[Q1....QN)(Q1...QN|=1N!N=1Q1...0NAnnihilation operator ag is the adjoint of at,and removes a particle:N1(1.54)a[Q1...QN) -Eci-a,a ai..Qi-iQi+1....an)na台NO p.14Commutator between the creation and annihilation operators is(1.55)[,]-=g= a,(Bosons)(1.56)[aa,al+=aaat+agaa=ag.(Fermions)Number representation labels states with the numbers of particles occupying single-particle states:(1.57)[a1...Qn)→[naina2),:Bosons:(1.58)aanaina2...)=Vna,nainaa...(na-1)...),(1.59)at,nana...)=na,+inanaa...(na,+1)...),.Fermions[(-1)/=i na [na,na.(na→0)...) na =1(1.60)aa,naina2...)0na,=0[(-1)j=i na, nan a..(na 1)...) Nar =0(1.61)at,(naina..) -nay=10Basis transformation: The creation/annihilation operators with respect to two bases Ja) and Ja)are related by:at =(ala)at(1.62)(1.63)a=-al)aa.aField operators are creation/annihilation operators in the position basis:d(r) =a(r)aa(1.64)aIt is obtained by setting [a) = [r)(1.65)[(r),(r')]-c= [t(r), t(r')]-c = 0,(1.66)[(r), t(r')]-c = 8(r -r')Second quantization expresses a physical quantity in terms of the creation and annihilation oper-ators:Number operator:na = afaa.(1.67)One-body operator U =-E, i:=(αlalB)atag.(1.68)ap5
Closure relations: (1.45) |0⟩ ⟨0| + X∞ N=1 X α1.αN Q α nα! N! |α1 . . . αN ⟩ ⟨α1 . . . αN | = 1. (1.53) Annihilation operator aα is the adjoint of a † α, and removes a particle: aα |α1 . . . αN ⟩ = 1 √ nα X N i=1 ζ i−1 δα,αi |α1 . . . αi−1αi+1 . . . αN ⟩. (1.54) Commutator between the creation and annihilation operators is NO p. 14 [aα, a † β ]− ≡ aαa † β − a † β aα = δαβ, (Bosons) (1.55) [aα, a † β ]+ ≡ aαa † β + a † β aα = δαβ.(Fermions) (1.56) Number representation labels states with the numbers of particles occupying single-particle states: |α1 . . . αN ⟩ ⇒ |nα1 nα2 . . .⟩, (1.57) • Bosons: aαi |nα1 nα2 . . .⟩ = √ nαi |nα1 nα2 . . .(nαi − 1). . .⟩, (1.58) a † αi |nα1 nα2 . . .⟩ = p nαi + 1 |nα1 nα2 . . .(nαi + 1). . .⟩. (1.59) • Fermions aαi |nα1 nα2 . . .⟩ = ( (−1) ∑i−1 j=1 nαj |nα1 nα2 . . .(nαi → 0). . .⟩ nαi = 1 0 nαi = 0 , (1.60) a † αi |nα1 nα2 . . .⟩ = ( (−1) ∑i−1 j=1 nαj |nα1 nα2 . . .(nαi → 1). . .⟩ nαi = 0 0 nαi = 1 . (1.61) Basis transformation: The creation/annihilation operators with respect to two bases |α⟩ and |α˜⟩ are related by: aˆ † α˜ = X α ⟨α | α˜⟩ aˆ † α, (1.62) aˆα˜ = X α ⟨α˜ | α⟩ aˆα. (1.63) Field operators are creation/annihilation operators in the position basis: ψˆ(r) = X α ϕα(r)ˆaα. (1.64) It is obtained by setting |α˜⟩ = |r⟩. [ψˆ(r), ψˆ(r ′ )]−ζ = [ψˆ† (r), ψˆ† (r ′ )]−ζ = 0, (1.65) [ψˆ(r), ψˆ† (r ′ )]−ζ = δ(r − r ′ ). (1.66) Second quantization expresses a physical quantity in terms of the creation and annihilation operators: Number operator: nˆα = a † αaα. (1.67) One-body operator Uˆ = P i uˆi : Uˆ = X αβ ⟨α | uˆ | β⟩ a † αaβ. (1.68) 5
Proof.We choosetheeigenstates ofi,as the single-particlebasis:(1.69)i[u) =u[u]: We can calculate a general matrix element:(u.u...n) - (u (u.n |u... n).(1.70)) (ui...u'nlui...un).(1.71)nuu.ThereforeU-(ulalu)nu=(ulalu)atau(1.72): Transform from the diagonal basis to a general basis by applying Eqs. (1.62, 1.63).Kinetic energy and single-body potentialh2Tdrit(r)V2(r),(1.73)2mU=[ drU (r)st(r)(r).(1.74)Two-body operator = (1/2), oig:V=- aip oa(1.75)Notetheorderofthe state indexes,and thatthematrixelementiscalculatedwithproductstates instead ofsymmetrized states.Proof. We assume that a two-body operator V may be diagonalized in product states:(1.76)[αβ) = Vαβ[β) We can calculatea general matrix element:(1.77)(α1...Q'n|0|Qi...Qn)(a1...QNlQ1...QN)aa详. The number of times that uag appears in the summation is nang for α+β andna(na-1)for α=β.We can thus define a operatorto countthe number:(1.78)Pap= nang - Sana=atagagda.andZapP..V=:(1.79)αβ: Transform from the diagonal basis to a general basis by applying Eqs. (1.62, 1.63).InteractionV=/drdr'v(r-r)t(r)t(r)b(r')(r)(1.80)Normal ordering: all creation operators are to the left of annihilation operators.6
Proof • We choose the eigenstates of uˆi as the single-particle basis: uˆ |u⟩ = u |u⟩. (1.69) • We can calculate a general matrix element: D u ′ 1 . . . u′ N Uˆ u1 . . . uN E = X i ui ! ⟨u ′ 1 . . . u′ N | u1 . . . uN ⟩ (1.70) = X u nuu ! ⟨u ′ 1 . . . u′ N | u1 . . . uN ⟩. (1.71) • Therefore Uˆ = X u ⟨u | uˆ | u⟩ nˆu = X u ⟨u | uˆ | u⟩ aˆ † uau. (1.72) • Transform from the diagonal basis to a general basis by applying Eqs. (1.62, 1.63). Kinetic energy and single-body potential Tˆ = − ℏ 2 2m Z drψˆ† (r)∇2ψˆ(r), (1.73) Uˆ = Z drU (r) ψˆ† (r)ψˆ(r). (1.74) Two-body operator Vˆ = (1/2)P ij vˆij : Vˆ = 1 2 X αβγδ (αβ |vˆ| γδ) a † αa † β aδaγ. (1.75) Note the order of the state indexes, and that the matrix element is calculated with product states instead of symmetrized states. Proof • We assume that a two-body operator Vˆ may be diagonalized in product states: vˆ |αβ) = vαβ |αβ). (1.76) • We can calculate a general matrix element: ⟨α ′ 1 . . . α′ N | vˆ | α1 . . . αN ⟩ = 1 2 X i̸=j vαiαj ⟨α ′ 1 . . . α′ N | α1 . . . αN ⟩. (1.77) • The number of times that vαβ appears in the summation is nαnβ for α ̸= β and nα(nα − 1) for α = β. We can thus define a operator to count the number: Pˆ αβ = ˆnαnˆβ − δαβnˆα = ˆa † αaˆ † β aˆβaˆα. (1.78) and Vˆ = 1 2 X αβ vαβPˆ αβ. (1.79) • Transform from the diagonal basis to a general basis by applying Eqs. (1.62, 1.63). Interaction Vˆ = 1 2 Z drdr ′ v(r − r ′ )ψˆ† (r)ψˆ† (r ′ )ψˆ(r ′ )ψˆ(r). (1.80) Normal ordering: all creation operators are to the left of annihilation operators. 6
1.4.2Secondquantized HamiltoniansFrom the“Hamiltonian ofeverything",one can deduce variousmodel Hamiltonians appropriate fordifferentphysical circumstances.Thehierarchyofthesemodels is showninthefollowingdiagram:Hamiltonian of"everything".2h2K=drit(r)()((r)()wa(r)+drd2meKeN,Zie2h2Z,Z,e2t(r)bo(r)-VR+r-Rl2MR-RiiVafrozenionsvibratingionsignore lattice potentialeffectof latticepotentialElectron-phonon coupling(1.83)JelliummodelofthedegenerateBlochwavefunctions,Wannierfunctionselectrongas(1.84, 1.85)(1.82)secondquantizedinthe Wannier basissecond quantized in Bloch basisignore the Umklapp processesTight-binding model (1.87)oneband,intra-site interaction onlyTHubbard model (1.90)interaction in a single siteAnderson impurity model (1.92)isolated spin limitKondo model (1.93)FW53Degenerate Electron gashH=dr(r)?(r)+/drV(r)(r)a(r)2mdrdr(r)(r)(r)(r),(1.81)2r-r/where V(r) denotes the potential exerted by a uniform positive charge background (jelliummodel). In the momentum basis pk= v-1/2eik-r, the Hamiltonian can be written as:Th2k2ea+a-aipoikomH=(1.82)ataaka+2V2.2m.02kagt0ka.poNotethattheg=ocomponentoftheinteractioniscanceledbythepotentialofthepositivecharge background.Electron-phononcouplingdescribestheinteractionbetweenelectronsandthevibrationsofionsHJ53.1in a solid:Hel-ph=EMgae+qakcg + h.c.(1.83)VVkq7
1.4.2 Second quantized Hamiltonians From the“Hamiltonian of everything”, one can deduce various model Hamiltonians appropriate for different physical circumstances. The hierarchy of these models is shown in the following diagram: Hamiltonian of “everything” Kˆ = X σ Z drψˆ† σ (r) − ℏ 2 2me ∇2 r − µ ψˆ σ(r) + 1 2 X σσ′ Z drdr ′ e 2 |r − r ′ | ψˆ† σ (r)ψˆ† σ′ (r ′ )ψˆ σ′ (r ′ )ψˆ σ(r) | {z } Kˆe − X Ni i=1 X σ Z dr Zie 2 |r − Ri | ψˆ† σ (r)ψˆ σ(r) | {z } Vˆei − X i ℏ 2 2Mi ∇2 Ri + 1 2 X ij ZiZj e 2 |Ri − Rj | | {z } Hˆi frozen ions vibrating ions ignore lattice potential Electron-phonon coupling Jellium model of the degenerate effect of lattice potential Bloch wave functions, Wannier functions second quantized in second quantized in Bloch basis Tight-binding model (1.87) ignore the Umklapp processes one band, intra-site interaction only Hubbard model (1.90) interaction in a single site Anderson impurity model (1.92) isolated spin limit Kondo model (1.93) electron gas the Wannier basis (1.82) (1.83) (1.84, 1.85) Degenerate Electron gas FW§3 Hˆ = − ℏ 2 2m X σ Z drψˆ† σ (r)∇2ψˆ σ(r) +X σ Z drVb(r)ψˆ† σ (r)ψˆ σ(r) + 1 2 X σσ′ Z drdr ′ e 2 |r − r ′ | ψˆ† σ (r)ψˆ† σ′ (r ′ )ψˆ σ′ (r ′ )ψˆ σ(r), (1.81) where Vb(r) denotes the potential exerted by a uniform positive charge background (jellium model). In the momentum basis φk = V −1/2 e ik·r , the Hamiltonian can be written as: Hˆ = X kσ ℏ 2k 2 2m aˆ † kσ aˆkσ + e 2 2V X q̸=0 X kσ,pσ′ 4π q 2 aˆ † k+qσ aˆ † p−qσ′aˆpσ′aˆkσ. (1.82) Note that the q = 0 component of the interaction is canceled by the potential of the positive charge background. Electron-phonon coupling describes the interaction between electrons and the vibrations of ions in a solid: HJ§3.1 Hˆ el−ph = 1 √ V X kq Mqaˆ † k+q aˆkcˆq + h.c. (1.83) 7
with M, being the matrix element of the electron-phonon couplingElectrons in periodic potential are relevant for solids.The counterparts of themomentum basisand the position basis are the Bloch states pnk(r) and the Wannier states wn (r - R), respec-tively.They are related by:1Z e-ik-Rupnk(r),(1.84)Wn(r-R)=NKEB.Z.1eRua(r - R),(1.85)Pnk(r) =RiNote thatthemomentum conservation is modified to:(1.86)ki+k2=k++Kwith a reciprocal wave-vectorK.It leads to the UMKLAPP scattering process when K|+ 0.Tight-binding models are Hamiltonians second quantized in the Wannier function basis. A gen-AS52.2eral single-bandtight-bindingHamiltoniancanbewrittenas:H=-tijataaja+Uujjralaataajaajo.(1.87)ij,aijInparticular,interactingtermsareclassifiedas:Direct coupling: Uin= Vu,and Hu-E Vun,ngExchange coupling: Jg = Uji,and Hu =-2, Ju (S..S, + Ining),where S, is the“spinoperator" :S; =Eat,taadia',(1.88)2GOwithPaulimatrices=[],=[ ] =[](1.89)Hubbard model The single band tight-binding model with well localized atomic orbits could beFR52approximatedas:H =-t [at.ajg +h.c.] +unitniu,(1.90)(5).0where (ij)denotes thatiand j are nearest neighbors.TheinteractingpartcanbealternativelywrittenasNU2UHu=Unitni=IS.(1.91)232Andersonimpuritymodeldescribestheinteraction betweenconductionband electronsand animpurity:H-eftfa+Unftnf+vftaka+h.c.+enat.aka(1.92)kaakaKondomodel describes the interaction between conduction band electrons andalocal spin:H=ekataak-Js(i).S(e)(1.93)kowhere S(e) denotes an electron-spin operator Eq. (1.88) at the origin. The Anderson impuritymodel isreducedtotheKondomodel inthelimitU→ooforadeeplyburied impurity.Thephysicsof theKondomodel is theorigin of correlation effects in heavy-Fermion systems.SeeRef. [10].8
with Mq being the matrix element of the electron-phonon coupling. Electrons in periodic potential are relevant for solids. The counterparts of the momentum basis and the position basis are the Bloch states φnk(r) and the Wannier states wn (r − Ri), respectively. They are related by: wn(r − Ri) = 1 √ N X k∈B.Z. e −ik·Riφnk(r), (1.84) φnk(r) = 1 √ N X Ri e ik·Riwn(r − Ri). (1.85) Note that the momentum conservation is modified to: k1 + k2 = k ′ 1 + k ′ 2 + K, (1.86) with a reciprocal wave-vector K. It leads to the UMKLAPP scattering process when |K| ̸= 0. Tight-binding models are Hamiltonians second quantized in the Wannier function basis. A general single-band tight-binding Hamiltonian can be written as: AS§2.2 Hˆ = − X ij,σ tijaˆ † iσaˆjσ + X ii′jj′ Uii′jj′aˆ † iσaˆ † i ′σ′aˆj ′σ′aˆjσ. (1.87) In particular, interacting terms are classified as: Direct coupling: Uii′ii′ = Vii′ , and HU = P ii′ Vii′nˆinˆi ′ ; Exchange coupling: Jij ≡ Uijji, and HˆU = −2 P ij Jij Sˆ i · Sˆ j + 1 4 nˆinˆj , where Sˆ i is the “spin operator”: Sˆ i = 1 2 X σσ′ aˆ † iστˆσσ′aˆiσ′ , (1.88) with Pauli matrices τˆx = 0 1 1 0 , τˆy = 0 −i i 0 , τˆz = 1 0 0 −1 . (1.89) Hubbard model The single band tight-binding model with well localized atomic orbits could be approximated as: FR§2 Hˆ = −t X ⟨ij⟩,σ h aˆ † iσaˆjσ + h.c. i + U X i nˆi↑nˆi↓, (1.90) where ⟨ij⟩ denotes that i and j are nearest neighbors. The interacting part can be alternatively written as: HˆU ≡ U X i nˆi↑nˆi↓ = − 2U 3 X i Sˆ i 2 + NeU 2 , (1.91) Anderson impurity model describes the interaction between conduction band electrons and an impurity: Hˆ = X σ ϵf ˆf † σ ˆfσ + Unˆf↑nˆf↓ + X kσ h Vk ˆf † σaˆkσ + h.c. i + X kσ ϵkaˆ † kσ aˆkσ. (1.92) Kondo model describes the interaction between conduction band electrons and a local spin: Hˆ = X kσ ϵkaˆ † kσ aˆkσ − JSˆ(i) · Sˆ (e) 0 , (1.93) where Sˆ (e) 0 denotes an electron-spin operator Eq. (1.88) at the origin. The Anderson impurity model is reduced to the Kondo model in the limit U → ∞ for a deeply buried impurity. The physics of the Kondo model is the origin of correlation effects in heavy-Fermion systems. See Ref. [10]. 8