Problems1.Asetof Nparticlesareinsingle-particlestatesBi)...IBn).Thesingle-particlestateshavethe coordinate representation (r|β:) = p.(r). Show the coordinate representations of thenormalizedsymmetrizedmany-bodystatesforBosonsandFermionsbyusingtheoverlapformula Eq. (1.41).2.Secondquantization:(a) Derive Eq. (1.82).(b) Determine the commutation relations of s, defined in (1.88). Are they the same as thoseforangularmomentumoperators?(c) Prove Eq. (1.91) by making use of the identity Tab -Ted = 26adobe - Sabcd3.Derive the closurerelation Eq. (1.98)by(a) showingdu(o) [0) (al =In) (nl .(1.149)(b)andtheclosurerelationEq.(1.45)canbewrittenas(1.150)ZInana2...)(naina2.../=1.(na)4.Boson coherent states in thelarge Nlimit could be interpreted as classical fields.For example,a classical electromagneticfield canbeviewed asa coherentstateofphotons.Thiscan alsobeseen inanother system,ie.,the harmonic oscilator with021H=2m+jmw222(1.151)(a) ShowH=h(1.152)withh(at +a)(1.153)=2mwhum (at - a)(1.154)p=iy2andaandatsatisfythecommutationrelationsof Bosonannihilationandcreationoper-ators.Therefore,a harmonic oscillatorcan beviewed as aphonon system(b) Assume that the system is in the coherent state |po) at t = 0. Show that the state at thefinite time t is[b(t) = e-int/2 [poe-iwt) .(1.155)(c) Determine the expectation value (), (), and () with respect to |(t). Compare the re-sults with those for a classical harmonic oscillator with initial values r(t = 0) = 2h/mw|poland p(t = 0) = 0.5.Provetheidentityof integralbypartforGrassmannvariablesde'dee-"(s-) A(, ") = 0,(1.156)acfor any A.14
Problems 1. A set of N particles are in single-particle states |β1⟩. . . |βN ⟩. The single-particle states have the coordinate representation ⟨r|βi⟩ = ψβi (r). Show the coordinate representations of the normalized symmetrized many-body states for Bosons and Fermions by using the overlap formula Eq. (1.41). 2. Second quantization: (a) Derive Eq. (1.82). (b) Determine the commutation relations of Sˆ i defined in (1.88). Are they the same as those for angular momentum operators? (c) Prove Eq. (1.91) by making use of the identity τab · τcd = 2δadδbc − δabδcd. 3. Derive the closure relation Eq. (1.98) by (a) showing Z dµ(ϕ)|ϕ⟩ ⟨ϕ| = X n |n⟩ ⟨n| . (1.149) (b) and the closure relation Eq. (1.45) can be written as X {nα} |nα1 nα2 . . .⟩ ⟨nα1 nα2 . . . | = 1. (1.150) 4. Boson coherent states in the large N limit could be interpreted as classical fields. For example, a classical electromagnetic field can be viewed as a coherent state of photons. This can also be seen in another system, i.e., the harmonic oscillator with Hˆ = pˆ 2 2m + 1 2 mω2xˆ 2 (1.151) (a) Show Hˆ = ℏω aˆ † aˆ + 1 2 (1.152) with xˆ = r ℏ 2mω aˆ † + ˆa (1.153) pˆ = ir ℏωm 2 aˆ † − aˆ (1.154) and aˆ and aˆ † satisfy the commutation relations of Boson annihilation and creation operators. Therefore, a harmonic oscillator can be viewed as a phonon system. (b) Assume that the system is in the coherent state |ϕ0⟩ at t = 0. Show that the state at the finite time t is |ψ(t)⟩ = e −iωt/2 |ϕ0e −iωt⟩. (1.155) (c) Determine the expectation value ⟨xˆ⟩, ⟨pˆ⟩, and D Hˆ E with respect to |ψ(t)⟩. Compare the results with those for a classical harmonic oscillator with initial values x(t = 0) = p 2ℏ/mω|ϕ0| and p(t = 0) = 0. 5. Prove the identity of integral by part for Grassmann variables: Z dξ ∗dξe−ξ ∗ ξ ξ − ∂ ∂ξ∗ A(ξ, ξ∗ ) = 0, (1.156) for any A. 14
Chapter 2Green's functionsWe define a set of quantities called Green's functions:. Properties of a many-particle system, e. g., quasi-excitations and spectral weights, can be ex-tractedfromtheGreen'sfunctions.Awiderangeofobservablesofdirectexperimental interestmayalsoberelatedtothemWe first introduce time-ordered Green's functions, which are special because they could be conve-nientlyevaluatedbyusingthefunctional integrals(sees3).Forgeneralcircumstances,onehastointroducemorespeciesofGreen'sfunctions.In equilibrium,thedifferentspeciesoftheGreen'sfunc-tionscanberelatedbythefluctuation-dissipation theorem.Moregenerally,theycanbeunified intoa singleGreen'sfunctiondefinedonatimecontour.2.1Green'sfunctionsand observables2.1.1Time-ordered Green's functionsReal-time Green's functionG(n) (aiti....antn;aiti...dh.th)= (-i)" ([aa(ti)...a()(tn)a()t(th)...a)(t))(2.1)wherethe superscript (H)denotestheHeisenbergrepresentation.TheGreen'sfunction could(1.18)be evaluated by using thefunctional integrals for an equilibrium system at the zero temperature.(See $4.3)Time-ordered product rearranges operators in descending order of time:T[O(H)(t)O(H)(t2) ...O(H)(tn)| =(PO(H)(tp1)O(H)(tp2) ...O(H)(tpn).(2.2)P is a permutation which orders the time such thattpi > tp2 > .. .tpn, and yields normalorder at equaltimes.Single-particleGreen's function(2.3)G(at;α't") = -i(↑[aa(t)aa)t(t)])It can be interpreted as the propagation amplitude of an added particle/hole:(aa(t)aa)t(t)=En Pn (atn(t)le-i(t-t)/at,n(t)t>t'G(at;α't') =(a)(t)a(t)=(Pn(an()e-(t-)/an(t)t<t(2.4)15
Chapter 2 Green’s functions We define a set of quantities called Green’s functions: • Properties of a many-particle system, e. g., quasi-excitations and spectral weights, can be extracted from the Green’s functions. • A wide range of observables of direct experimental interest may also be related to them. We first introduce time-ordered Green’s functions, which are special because they could be conveniently evaluated by using the functional integrals (see §3). For general circumstances, one has to introduce more species of Green’s functions. In equilibrium, the different species of the Green’s functions can be related by the fluctuation-dissipation theorem. More generally, they can be unified into a single Green’s function defined on a time contour. 2.1 Green’s functions and observables 2.1.1 Time-ordered Green’s functions Real-time Green’s function G (n) (α1t1, . . . αntn; α ′ 1 t ′ 1 , . . . α′ n t ′ n ) = (−i)n D Tˆ h aˆ (H) α1 (t1). . . aˆ (H) αn (tn)ˆa (H)† α′ n (t ′ n ). . . aˆ (H)† α′ 1 (t ′ 1 ) iE , (2.1) where the superscript (H) denotes the Heisenberg representation. The Green’s function could (1.18) be evaluated by using the functional integrals for an equilibrium system at the zero temperature. (See §4.3) Time-ordered product rearranges operators in descending order of time: Tˆ h Oˆ(H)(t1)Oˆ(H)(t2). . . Oˆ(H)(tn) i = ζ P Oˆ(H)(tP 1)Oˆ(H)(tP 2). . . Oˆ(H)(tP n). (2.2) P is a permutation which orders the time such that tP 1 > tP 2 > . . . tP n, and yields normal order at equal times. Single-particle Green’s function G (αt; α ′ t ′ ) = −i D Tˆ h aˆ (H) α (t)ˆa (H)† α′ (t ′ ) iE . (2.3) It can be interpreted as the propagation amplitude of an added particle/hole: G (αt; α ′ t ′ ) = −i D aˆ (H) α (t)ˆa (H)† α′ (t ′ ) E = P n ρn D aˆ † αΨn(t)|e −iHˆ (t−t ′ )/ℏ |aˆ † α′Ψn(t ′ ) E t > t′ ζ D aˆ (H)† α′ (t ′ )ˆa (H) α (t) E = ζ P n ρn D aˆα′Ψn(t ′ )|e −iHˆ (t ′−t)/ℏ |aˆαΨn(t) E t ≤ t ′ . (2.4) 15
Equation of motion of a non-interacting system: By assuming the Hamiltonian Ko = . (ea- μ)ataawehavedGo(at;d't)=h<aa,at]s(t - t) + (ca - μ) Go (at;a't)hot=hoa/8(t-t')+ (a-μ)Go(αt;α't'), (2.5)wherewemakeuseoftherelation-[ -(c)a(2.6)otThe equation of motion can be solved by imposing the boundary conditions:i+(na t→t+o+(2.7)Go(αt,a't")=-ioaat→tCnawhere na=(ataa)is the occupation number of the stateα.The solution isGo(at; d/t') = -iaave-t(ca-μ)(t-t) [(t -t' - n)(1+(na) + 0(t' -t+ n)(na] ,(2.8)wheren=o+inthe-functionistofixthevalueoftheGreen'sfunctionattheequaltimeTheFouriertransformisdtGo(at; a't)eia(t-t)Goa (w)=(na1+(na(2.9)-(eα-μ)/h+inw-(ea-μ)/h-inFor interacting systems, the equation of motion is related to higher order Green's func-tions and not closed by itself.Thermal Green's function is the Green's function for the imaginary time t =-ir, r e [0, hp)1g(n) (a1TI,...anTn;aiTi,...nTh)= (-1)" (T()...a((n)aa)t()...)(i))(2.10)wherea()(T)=ere-套T(2.11)a(H)t(t)=efrate-套r(2.12)Note thata(H)t(t)anda (+)are not Hermitian adjoints.The thermal-Green's function can be evaluated by using the functional integrals for an equilib-riumsystematthefinitetemperature.:Itis introducedforfacilitatingthecalculationsoffinite-temperatureequilibriumsystems. It exploits the property that the equilibrium density matrix can be regarded as a time-(1.22)evolution operator fora time-interval t=-ihBFW524. It displays the (anti-)periodicity:(2.13)GlT=0 = ( gl,=hp,wheretdenotesoneofthetimeargumentsof theGreen'sfunction1The definition has an extra factor (-1)",to be consistent with Fetter-Walecka's definition. See FW Eq. (23.6)16
Equation of motion of a non-interacting system: By assuming the Hamiltonian Kˆ 0 = P α (ϵα − µ) ˆa † αaˆα, we have iℏ ∂ ∂tG0 (αt; α ′ t ′ ) = ℏ h aˆα, aˆ † α′ i −ζ δ(t − t ′ ) + (ϵα − µ) G0 (αt; α ′ t ′ ) = ℏδαα′δ(t − t ′ ) + (ϵα − µ) G0 (αt; α ′ t ′ ), (2.5) where we make use of the relation: iℏ ∂aˆ (H) α (t) ∂t = h aˆ (H) α (t), Kˆ 0 i = (ϵα − µ) ˆa (H) α (t). (2.6) The equation of motion can be solved by imposing the boundary conditions: G0(αt, α′ t ′ ) = −iδαα′ ( 1 + ζnα t → t ′ + 0+ ζnα t → t ′ , (2.7) where nα ≡ aˆ † αaˆα is the occupation number of the state α. The solution is G0(αt; α ′ t ′ ) = −iδαα′e − i ℏ (ϵα−µ)(t−t ′ ) [θ(t − t ′ − η) (1 + ζnα) + θ(t ′ − t + η)ζnα] , (2.8) where η ≡ 0 + in the θ-function is to fix the value of the Green’s function at the equal time. The Fourier transform is G˜ 0αα′ (ω) ≡ Z ∞ −∞ dtG0(αt; α ′ t ′ )e iω(t−t ′ ) = δαα′ 1 + ζnα ω − (ϵα − µ) /ℏ + iη − ζnα ω − (ϵα − µ) /ℏ − iη e iωη . (2.9) For interacting systems, the equation of motion is related to higher order Green’s functions and not closed by itself. Thermal Green’s function is the Green’s function for the imaginary time t = −iτ , τ ∈ [0, ℏβ): 1 G (n) (α1τ1, . . . αnτn; α ′ 1 τ ′ 1 , . . . α′ n τ ′ n ) = (−1)n D T h aˆ (H) α1 (τ1). . . aˆ (H) αn (τn)ˆa (H)† α′ n (τ ′ n ). . . aˆ (H)† α′ 1 (τ ′ 1 ) iE , (2.10) where aˆ (H) α (τ ) ≡ e Kˆ ℏ τ aˆαe − Kˆ ℏ τ , (2.11) aˆ (H)† α (τ ) ≡ e Kˆ ℏ τ aˆ † αe − Kˆ ℏ τ . (2.12) Note that aˆ (H)† α (τ ) and aˆ (H) α (τ ) are not Hermitian adjoints. The thermal-Green’s function can be evaluated by using the functional integrals for an equilibrium system at the finite temperature. • It is introduced for facilitating the calculations of finite-temperature equilibrium systems. • It exploits the property that the equilibrium density matrix can be regarded as a timeevolution operator for a time-interval t = −iℏβ. (1.22) • It displays the (anti-)periodicity: FW§24 G|τi=0 = ζ G|τi=ℏβ , (2.13) where τi denotes one of the time arguments of the Green’s function. 1The definition has an extra factor (−1)n, to be consistent with Fetter-Walecka’s definition. See FW Eq. (23.6). 16
ProofSTr [e-BKa(H)t(T')aa| =-Tr [e-BR (eBRae-BR)a()t()]g (α0;αT) =Tr [e-BRa() (hB) a()t(t)] =(g (ahB;d'"). (2.14)Matsubara freguency: Because of the periodicity Eq. (2.13),the thermal Green's function canberelatedtoitsFouriertransformbyTe-iwn(r-r')gaa' (wn),(2.15)g(qT,Q'T') =h4hβdreiwn(r-t")g(aT,a't'),(2.16)Gaa' (wn):where we assume that the system is time-independent, and henceg (α,q't')= g (αT-t',q'o).Thediscretesetoffrequenciesisdefinedby[Boson(2.17),nezWn(2n+1)Fermion(2.5)Equation of motion: For a non-interacting system, we can establish:-% - 二") go(ar,d't) = 0a(r - T),(2.18)OTh(2.7).The solution isGo(aT,d't)=-aa'e-(ca-m)(t-")/n[0(T-T/ -n)(1+(na)+C0(t/ -T+n)na]. (2.19). The periodicity Eq. (2.13) yields(na =(e-B(a-m) (1 +(na),(2.20)1(2.21)na= eB(ca-}) -(which is the Bose-Einstein (C = 1) or Fermi-Dirac (C =-1) distribution function.:TheFourier transform iseiwnn(2.22)Goa(an)= Da iwn -(ca) /hNO55.12.1.2EvaluationofObservablesKinetic energy[ar1-v(rt)(2.23)(1.73)T)=ic[ ekda P c(k.u)eln=icv(2.24)(2元)4 2mwhere t+ = t+ 0+, andG (rt; r't+) = -i(↑[(H)(rt)(H)t(r't+)])(2.25)G(k,w)= / dr / dtG(rt;r't)e-ik(r--')+io(tt).(2.26)17
Proof G (α0; α ′ τ ′ ) = − ζ Z Tr h e −βKˆ aˆ (H)† α′ (τ ′ )ˆaα i = − ζ Z Tr h e −βKˆ e βKˆ aˆαe −βKˆ aˆ (H)† α′ (τ ′ ) i ≡ − ζ Z Tr h e −βKˆ aˆ (H) α (ℏβ) ˆa (H)† α′ (τ ′ ) i = ζG (α ℏβ; α ′ τ ′ ). (2.14) Matsubara frequency: Because of the periodicity Eq. (2.13), the thermal Green’s function can be related to its Fourier transform by G (ατ, α′ τ ′ ) = 1 ℏβ X n e −iωn(τ−τ ′ )Gαα′ (ωn), (2.15) Gαα′ (ωn) = Z ℏβ 0 dτeiωn(τ−τ ′ )G (ατ, α′ τ ′ ), (2.16) where we assume that the system is time-independent, and hence G (ατ, α′ τ ′ ) = G (α τ − τ ′ , α′0). The discrete set of frequencies is defined by ωn = (2πn ℏβ Boson (2n+1)π ℏβ Fermion , n ∈ Z (2.17) Equation of motion: For a non-interacting system, we can establish: (2.5) − ∂ ∂τ − ϵα − µ ℏ G0(ατ, α′ τ ′ ) = δαα′δ(τ − τ ′ ). (2.18) • The solution is (2.7) G0(ατ, α′ τ ′ ) = −δαα′e −(ϵα−µ)(τ−τ ′ )/ℏ [θ(τ − τ ′ − η) (1 + ζnα) + ζθ(τ ′ − τ + η)nα] . (2.19) • The periodicity Eq. (2.13) yields ζnα = ζe−β(ϵα−µ) (1 + ζnα), (2.20) nα = 1 e β(ϵα−µ) − ζ , (2.21) which is the Bose-Einstein (ζ = 1) or Fermi-Dirac (ζ = −1) distribution function. • The Fourier transform is G0αα′ (ωn) = δαα′ e iωnη iωn − (ϵα − µ) /ℏ . (2.22) 2.1.2 Evaluation of Observables NO§5.1 Kinetic energy D Tˆ E = iζ Z d 3 r − ℏ 2 2m ∇2 rG rt; r ′ t + r′=r (1.73) (2.23) = iζV Z d 3kdω (2π) 4 ℏ 2k 2 2m G˜(k, ω)e iωη , (2.24) where t + ≡ t + 0+, and G rt; r ′ t + = −i D Tˆ h ψˆ(H)(rt)ψˆ(H)† (r ′ t +) iE , (2.25) G˜(k, ω) = Z dr Z dtG (rt; r ′ t ′ ) e −ik·(r−r ′ )+iω(t−t ′ ) . (2.26) 17
Interactionenergyh++mG(rt;(2.27)V)dr+h212d3kdwpiwrhiw+μ) G(k,w)(2.28)2m(2元)4Toderivetheformula,wemakeuseoftheidentitysit(r)ni) = it(r) (hV2-) (rt)+ / dr'st(rt)t(r't)(r-r')(r't)(rt) (2.29)2mOtTotal energy[a[(-+Eo=(↑+V)=G(rt;r't(2.30)Ot2mh21:2d3kdw1CG(k,w)(2.31)2(2元）2mh21:2d3k(ihwn+2mu)G(k,wn)(2.32)L(2元）3Note: The use of the one-particle Green's function for evaluating the total energy could be dan-gerous: a seemingly innocuous approximation having little effect on one-particle propertiesmay have a large uncontrolled effect on the energy.NO52.12.1.3 Response functionsLinearResponsesConductivityTocalculatetheconductivityofasystem,weintroduceanexternalelectricfield,andseehowmuchtheelectriccurrentisgenerated:. The external electric field induces a modification to the HamiltonianAH= -e(2.33)/drp(r)o(r),where (r) is the electric potential, and p(r) = it(r)(r) is the density operator.:We need to calculate the expectation value of the electric current density operator3(n) -端(([i] [() ()],(2.34)to the linear order of the electric field, or (r).: A linear response has a useful property: the total response to multiple fields is the sum oftheresponsestoeachfield.Linear responseformula Weconsider a time-dependent infinitesimally small external field:Hu(t)= H +OU(t)(2.35)Evolution operator ((tr)=u(tr,ti) (b(t:)):h品u(t) = Hu()u(t),(2.36)dtHu(t)dt= Jlim e-iehu(tm)/he-icHu(tM-1)/h..-icHu(ti)/hu(t,t) =Tex(2.37)wherewesplitthetimeinterval [ti,t] intoMinfinitesimallysmalltimeintervalse(t-t)/M,andt=t;+(k-1)e,k=1...M.Theevolutionoperatorforthefull timeinterval is obtained by accumulating the action of the evolution operator for eachsmalltime interval.18
Interaction energy D Vˆ E = iζ 2 Z d 3 r iℏ ∂ ∂t + ℏ 2 2m ∇2 r + µ G (rt; r ′ t ′ ) r′=r,t′=t+ (2.27) = iζ 2 V Z d 3kdω (2π) 4 e iωη ℏω − ℏ 2k 2 2m + µ G˜(k, ω). (2.28) To derive the formula, we make use of the identity: ψˆ† (rt)iℏ ∂ψˆ(rt) ∂t = ψˆ† (rt) − ℏ 2 2m ∇2 − µ ψˆ(rt)+Z dr ′ψˆ† (rt)ψˆ† (r ′ t)v(r−r ′ )ψˆ(r ′ t)ψˆ(rt). (2.29) Total energy E0 = D Tˆ + Vˆ E = iζ 2 Z d 3 r iℏ ∂ ∂t − ℏ 2 2m ∇2 r + µ G (rt; r ′ t ′ ) r′=r,t′=t+ (2.30) = iζ 2 V Z d 3kdω (2π) 4 e iωη ℏω + ℏ 2k 2 2m + µ G˜(k, ω) (2.31) = 1 2 ζV 1 ℏβ X ωn Z d 3k (2π) 3 e iωnη iℏωn + ℏ 2k 2 2m + µ G˜(k, ωn) (2.32) Note: The use of the one-particle Green’s function for evaluating the total energy could be dangerous: a seemingly innocuous approximation having little effect on one-particle properties may have a large uncontrolled effect on the energy. 2.1.3 Response functions NO§2.1 Linear Responses Conductivity To calculate the conductivity of a system, we introduce an external electric field, and see how much the electric current is generated: • The external electric field induces a modification to the Hamiltonian ∆Hˆ = −e Z drρˆ(r)ϕ(r), (2.33) where ϕ(r) is the electric potential, and ρˆ(r) ≡ ψˆ† (r)ψˆ(r) is the density operator. • We need to calculate the expectation value of the electric current density operator ˆj(r) = ieℏ 2m n ψˆ† (r) h ∇ψˆ(r) i − h ∇ψˆ† (r) i ψˆ(r) o , (2.34) to the linear order of the electric field, or ϕ(r). • A linear response has a useful property: the total response to multiple fields is the sum of the responses to each field. Linear response formula We consider a time-dependent infinitesimally small external field: HˆU (t) = Hˆ + OUˆ (t). (2.35) Evolution operator |ψ(tf)⟩ = Uˆ(tf , ti)|ψ(ti)⟩: iℏ d dt Uˆ(t, ti) = HˆU (t)Uˆ(t, ti), (2.36) Uˆ(tf , ti) = Tˆ exp − i ℏ Z tf ti HˆU (t)dt ≡ lim M→∞ e −iϵHˆU (tM)/ℏ e −iϵHˆU (tM−1)/ℏ . . . e−iϵHˆU (t1)/ℏ , (2.37) where we split the time interval [ti , tf ] into M infinitesimally small time intervals ϵ ≡ (tf − ti)/M, and tk = ti + (k − 1)ϵ, k = 1 . . . M. The evolution operator for the full time interval is obtained by accumulating the action of the evolution operator for each small time interval. 18