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北京大学：量子多体的理论与计算（讲义）Quantum Theory of Many Particle Systems

Chapter 1 Second quantization and coherent states Chapter 2 Green’s functions Chapter 3 Functional integrals Chapter 4 Perturbation theory Chapter 5 Effective action theory and energy functionals Chapter 6 Theory of electron liquid Chapter 7 Phase transitions and spontaneous symmetry breaking

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Response of a wave-function δ |ψ(t)⟩ = Z t ti dt1U(t1) δUˆ(t, ti) δU(t1) U→0 |ψ(ti)⟩, (2.38) δUˆ(t, ti) δU(t1) U→0 = − i ℏ Uˆ(t, t1)OˆUˆ(t1, ti) U→0 = − i ℏ e − i ℏ Hˆ (t−ti)Oˆ(H)(t1), (2.39) δ |ψ(t)⟩ = − i ℏ e − i ℏ Hˆ (t−ti) Z t ti dt1Oˆ(H)(t1)|ψ (ti)⟩U(t1). (2.40) where Oˆ(H) (t) ≡ e i ℏ Hˆ (t−ti)Oeˆ − i ℏ Hˆ (t−ti) . (2.41) Expectation value of an observable Rˆ: δR(t) = X n ρn hDψn(t) Rˆ δψn(t) E + D δψn(t) Rˆ ψn(t) Ei (2.42) = − i ℏ Z ∞ −∞ dt1θ(t − t1) DhRˆ(H)(t), Oˆ(H)(t1) iEU(t1), (2.43) Dr RO(t, t′ ) ≡ δR(t) δU(t ′) = − i ℏ θ(t − t ′ ) DhRˆ(H)(t), Oˆ(H)(t ′ ) iE . (2.44) We see that: • The response function is a retarded one. • It is related to a correlation function between operators. Scattering experiments • An external particle interacts weakly with the constituents of a many-body system through an interaction v(r − r ′ ). Examples include electron energy loss spectroscopy (EELS) and neutron scattering. The scattering cross section is related to the density correlation function: σ(q, ω) = |vq| 2 Z dteiωt ρˆ † q (t) ˆρq , (2.45) where ρˆq = P i exp (−iq · rˆi) is the Fourier transform of the density operator, and vq is the Fourier transform of the interaction. Proof – We determine the transition matrix element of an external particle scattered from wave-vector k to k + q, and the probed system from |α⟩to |β⟩: Tαβ(q) = Z dr * β e −i(k+q)·r X i v (r − ri) e ik·r α + = vq * β X i e −iq·ri α + = vq ⟨β | ρˆq | α⟩. (2.46) – The scattering cross-section is determined by Fermi’s Golden rule: σα (q, ω) = 2π ℏ |vq| 2X β δ (Eβ − Eα − ℏω)|⟨β | ρˆq | α⟩|2 (2.47) = |vq| 2X β Z dt e−i(Eβ−Eα)t/ℏ+iωt |⟨β | ρˆq | α⟩|2 (2.48) = |vq| 2X β Z dt eiωt D α e iHt/ ˆ ℏ ρˆ † q e −iHt/ ˆ ℏ β E ⟨β | ρˆq | α⟩ (2.49) = |vq| 2 Z dt eiωt α ρˆ † q (t)ˆρq α . (2.50) 19

as we have done for the corresponding energies. This, however, is far from trivial because during the photoemission process itself the system will relax. The problem simplifies within the sudden approximation, which is extensively used in many-body calculations of photoemission spectra from interacting electron systems and which is in principle applicable only to electrons with high kinetic energy. In this limit, the photoemission process is assumed to be sudden, with no post-collisional interaction between the photoelectron and the system left behind (in other words, an electron is instantaneously removed and the effective potential of the system changes discontinuously at that instant). The N-particle final state !f N can then be written as !f N!A "f k !f N"1 , (6) where A is an antisymmetric operator that properly antisymmetrizes the N-electron wave function so that the Pauli principle is satisfied, "f k is the wave function of the photoelectron with momentum k, and !f N"1 is the final state wave function of the (N"1)-electron system left behind, which can be chosen as an excited state with eigenfunction !m N"1 and energy Em N"1 . The total transition probability is then given by the sum over all possible excited states m. Note, however, that the sudden approximation is inappropriate for photoelectrons with low kinetic energy, which may need longer than the system response time to escape into vacuum. In this case, the so-called adiabatic limit, one can no longer factorize !f N into two independent parts and the detailed screening of photoelectron and photohole has to be taken into account (Gadzuk and S˘unjic´, 1975). In this regard, it is important to mention that there is evidence that the sudden approximation is justified for the cuprate hightemperature superconductors even at photon energies as low as 20 eV (Randeria et al., 1995; Sec. II.C). For the initial state, let us assume for simplicity that !i N is a single Slater determinant (i.e., Hartree-Fock formalism), so that we can write it as the product of a oneelectron orbital "i k and an (N"1)-particle term: !i N!A "i k !i N"1 . (7) More generally, however, !i N"1 should be expressed as !i N"1 !ck!i N , where ck is the annihilation operator for an electron with momentum k. This also shows that !i N"1 is not an eigenstate of the (N"1) particle Hamiltonian, but is just what remains of the N-particle wave function after having pulled out one electron. At this point, we can write the matrix elements in Eq. (4) as #!f N!Hint!!i N$!#"f k !Hint!"i k $#!m N"1 !!i N"1 $, (8) where #"f k !Hint!"i k $%Mf,i k is the one-electron dipole matrix element, and the second term is the (N"1)-electron overlap integral. Note that here we replaced !f N"1 with an eigenstate !m N"1 , as discussed above. The total photoemission intensity measured as a function of Ekin at a momentum k, namely, I(k,Ekin)!&f,iwf,i , is then proportional to & f,i !Mf,i k ! 2 &m !cm,i! 2'(Ekin#Em N"1 "Ei N"h )*, (9) where !cm,i! 2!"#!m N"1 !!i N"1 $" 2 is the probability that the removal of an electron from state i will leave the (N"1)-particle system in the excited state m. From this we can see that, if !i N"1 !!m0 N"1 for one particular state m!m0 , then the corresponding !cm0 ,i! 2 will be unity and all the other cm,i zero; in this case, if Mf,i k +0, the ARPES spectra will be given by a delta function at the Hartree-Fock orbital energy EB k !", k , as shown in Fig. 3(b) (i.e., the noninteracting particle picture). In strongly correlated systems, however, many of the !cm,i! 2 will be different from zero because the removal of the photoelectron results in a strong change of the systems effective potential and, in turn, !i N"1 will overlap with many of the eigenstates !m N"1 . Thus the ARPES spectra will not consist of single delta functions but will show a main line and several satellites according to the number of excited states m created in the process [Fig. 3(c)]. This is very similar to the situation encountered in photoemission from molecular hydrogen (Siegbahn et al., 1969) in which not simply a single peak but many lines separated by a few tenths of eV from each other FIG. 3. Angle-resolved photoemission spetroscopy: (a) geometry of an ARPES experiment in which the emission direction of the photoelectron is specified by the polar (- ) and azimuthal (.) angles; (b) momentum-resolved one-electron removal and addition spectra for a noninteracting electron system with a single energy band dispersing across EF ; (c) the same spectra for an interacting Fermi-liquid system (Sawatzky, 1989; Meinders, 1994). For both noninteracting and interacting systems the corresponding groundstate (T!0 K) momentum distribution function n(k) is also shown. (c) Lower right, photoelectron spectrum of gaseous hydrogen and the ARPES spectrum of solid hydrogen developed from the gaseous one (Sawatzky, 1989). 478 Damascelli, Hussain, and Shen: Photoemission studies of the cuprate superconductors Rev. Mod. Phys., Vol. 75, No. 2, April 2003 Figure 2.1: The spectral function can be measured by the angle-resolved photoemission spectroscopy (ARPES) technique. The measured intensity is proportional to f(ω)A(k, ω), where f(ω) is the Fermi distribution function. (b) and (c) illustrate the spectral functions of non-interacting systems and interacting Fermi-liquid system, respectively. Ref. [7] – Average over the initial state α. • Angle-resolved photoemission spectroscopy (ARPES): a photon excites a photoelectron out of a many-body system (see Fig. 2.1) [7]: Ik(ω) ∝ Z dteiωt D aˆ † k (0) ˆak(t) E . (2.51) 2.1.4 Species of Green’s functions HJ§3.2 Time-ordered Green’s function G(rt; r ′ t ′ ) = −i D T h ψˆ(rt)ψˆ† (r ′ t ′ ) iE (2.52) Retarded Green’s function G r (rt; r ′ t ′ ) = −iθ(t − t ′ ) h ψˆ(rt), ψˆ† (r ′ t ′ ) i −ζ . (2.53) Advanced Green’s function G a (rt; r ′ t ′ ) = iθ(t ′ − t) h ψˆ(rt), ψˆ† (r ′ t ′ ) i −ζ . (2.54) Lesser Green’s function G <(rt; r ′ t ′ ) = −iζ D ψˆ† (r ′ t ′ )ψˆ(rt) E . (2.55) Greater Green’s function G >(rt; r ′ t ′ ) = −i D ψˆ(rt)ψˆ† (r ′ t ′ ) E . (2.56) Relations G r − G a = G > − G <, (2.57) G = θ(t − t ′ )G > + θ(t ′ − t)G < = G < + G r = G > + G a , (2.58) G r = θ (t − t ′ ) G > − G < , (2.59) G a = −θ (t ′ − t) G > − G < , (2.60) where the Green’s functions all have the argument (rt; r ′ t ′ ). Conjugate relations G a (rt; r ′ t ′ ) = [G r (r ′ t ′ ; rt)]∗ , (2.61) G <(rt; r ′ t ′ ) = − G <(r ′ t ′ ; rt) ∗ , (2.62) G >(rt; r ′ t ′ ) = − G >(r ′ t ′ ; rt) ∗ . (2.63) 20

• Making use of the definition Eq. (2.65), we have G <(k, ω) = −iζnζ (ω)A(k, ω), (2.74) G >(k, ω) = −i [1 + ζnζ (ω)] A(k, ω), (2.75) nζ (ω) ≡ 1 e βℏω − ζ . (2.76) • Making use of Eqs. (2.58-2.60), we have    G(k, ω) Gr (k, ω) Ga (k, ω)    = Z dω1 2π A(k, ω1)    − ζnζ(ω1) ω−ω1−iη + 1+ζnζ(ω1) ω−ω1+iη 1 ω−ω1+iη 1 ω−ω1−iη    . (2.77) • It follows: Re    G(k, ω) Gr (k, ω) Ga (k, ω)    = P Z dω1 2π A(k, ω1) ω − ω1 , (2.78) Im    G(k, ω) Gr (k, ω) Ga (k, ω)    =    − h tanh βℏω 2 i−ζ − +    1 2 A(k, ω). (2.79) 2.2.2 Thermal Green’s function and analytic continuation FW§31 The thermal Green’s function can also be related to the spectral function by: G(k, ωn) = Z ∞ −∞ dω1 2π A(k, ω1) iωn − ω1 (2.80) The derivation is similar to that for the real-time Green’s functions. Analytic continuation: The real-time Green’s functions at the finite temperature can be obtained from the thermal Green’s function via the process of the ANALYTIC CONTINUATION: 1. We have: (2.77) G r (k, ω) = G(k, ωn)| iωn→ω+iη , (2.81) G a (k, ω) = G(k, ωn)| iωn→ω−iη . (2.82) 2. With an analytic form of the thermal Green’s function, the spectral function can be determined by (2.64) A(k, ω) = i h G(k, ωn)| iωn→ω+iη − G(k, ωn)| iωn→ω−iη i . (2.83) 3. Other real-time Green’s function can then be obtained by applying the fluctuation-dissipation relations Eqs. (2.74–2.77). 2.3 Non-equilibrium Green’s function HJ§4 Motivation We introduce the non-equilibrium (Keldysh’s, or time contour-ordered) Green’s function because: • The real-time Green’s function are directly related to physical observables. Unfortunately, they are difficult to calculate. • The analytic continuation is only useful when we have an analytic expression for the thermal Green’s function. If determined numerically, the thermal Green’s function is only defined for a discrete set of the Matsubara frequencies. 22

Ret Imt −T0/2 C1 + Cβ C2 − −T0/2 − iℏβ Figure 2.2: Time contour C for defining the non-equilibrium Green’s functions. Note that Green’s functions in the vertical part of the contour have the (anti-) periodicity Eq. (2.13). Adapted from HJ§4.3. • When the system is not an equilibrium one (e.g., a system driven by a strong external field beyond the linear response regime), the analytic continuation cannot apply. • By introducing the non-equilibrium Green’s function, all real-time Green’s functions can be unified into a single Green’s function. • It is motivated by the observation that all real-time Green’s function (e.g., G>) can be converted to a contour-ordered trace: G >(rt; r ′ t ′ ) = −i D ψˆ(H)(rt)ψˆ(H)† (r ′ t ′ ) E = −i Tr h e −βHˆ Uˆ − T0 2 , t ψˆ(r)Uˆ (t, t′ ) ψˆ† (r ′ )Uˆ t ′ , − T0 2 i Tre−βHˆ = −i Tr h Uˆ − T − 0 2 − iℏβ, − T − 0 2 Uˆ − T − 0 2 , t− ψˆ(r)Uˆ (t −, t′+) ψˆ† (r ′ )Uˆ t ′+, − T + 0 2 i Tre−βHˆ = −i Tr h TˆCe − i ℏ ∫ C dtHˆ (t)ψˆ(rt −)ψˆ† (r ′ t ′+) i Tr h TˆCe − i ℏ ∫ C dtHˆ (t) i , (2.84) where ψˆ(rt −) and ψˆ† (r ′ t ′+) in the last line are not Heisenberg operators: their time arguments just indicate where they should appear. A time (contour) ordered trace can be conveniently evaluated by using functional integrals. Time contour is defined in Fig. 2.2. Contour-ordered Green’s function is defined as GC(rt, r ′ t ′ ) = −i 1 Z Tr h TˆCe − i ℏ ∫ C dtHˆ (t)ψˆ(rt)ψˆ† (r ′ t ′ ) i . (2.85) • There are two-branches of the real time: C1 and C2. The real-time Green’s functions can be obtained by assigning appropriate branches to their time arguments: G(rt, r ′ t ′ ) = G 11 C (rt, r ′ t ′ ), (2.86) G >(rt, r ′ t ′ ) = G 21 C (rt, r ′ t ′ ), (2.87) G <(rt, r ′ t ′ ) = G 12 C (rt, r ′ t ′ ). (2.88) • The four components of GC are not independent: G 11 C + G 22 C = G 12 C + G 21 C . (2.89) 23

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