Green's function (many-body theory)


Green's function (many-body theory)

In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point 'Green's functions' are Green's functions in the mathematical sense; the linear operator that they invert is the part of the Hamiltonian operator that is quadratic in the fields.)

patially uniform case

Basic definitions

Throughout, zeta is +1 for bosons and -1 for fermions and [ldots,ldots] denotes either a commutator or anticommutator as appropriate.

The signs of the Green functions have been chosen so that the thermal Green function for a free particle is:mathcal{G}(mathbf{k},omega_n) = frac{1}{-mathrm{i}omega_n + xi_mathbf{k,and the retarded Green function is:G^{mathrm{R(mathbf{k},omega) = frac{1}{-(omega+mathrm{i}eta) + xi_mathbf{k.In this section, functions will have their arguments abbreviated as single numbers: mathbf{x}_n t_n and mathbf{x}_n au_n will be replaced by n.

The imaginary-time Heisenberg operators can be written in terms of Schrödinger operators as:psi(mathbf{x}, au) = mathrm{e}^{K au} psi(mathbf{x}) mathrm{e}^{-K au}:arpsi(mathbf{x}, au) = mathrm{e}^{K au} psi^dagger(mathbf{x}) mathrm{e}^{-K au}where K = H - mu N is the grand-canonical Hamiltonian. Similarly, for the real-time operators,:psi(mathbf{x},t) = mathrm{e}^{mathrm{i} K t} psi(mathbf{x}) mathrm{e}^{-mathrm{i} K t},and arpsi(mathbf{x},t) = [psi(mathbf{x},t)] ^*.

In imaginary time, the general Green function is defined by:mathcal{G}^{(n)}(1 ldots n | 1' ldots n')= langle Tpsi(1)ldotspsi(n)arpsi(n')ldotsarpsi(1') angle,where n signifies mathbf{x}_n, au_n. (The imaginary-time variables au_n are restricted to the range 0 to eta.)

In real time, the definition is:G^{(n)}(1 ldots n | 1' ldots n')= mathrm{i}^n langle Tpsi(1)ldotspsi(n)arpsi(n')ldotsarpsi(1') angle,where n signifies mathbf{x}_n, t_n.

Two-point functions

The Green function with a single pair of arguments (n=1) is referred to as the two-point function, or propagator. In the presence of both spatial and temporal translational symmetry, it depends only on the difference of its arguments. Taking the Fourier transform with respect to both space and time gives:mathcal{G}(mathbf{x} au|mathbf{x}' au') = int_mathbf{k} frac{1}{eta}sum_{omega_n} mathcal{G}(mathbf{k},omega_n) mathrm{e}^{mathrm{i} mathbf{k}cdot(mathbf{x}-mathbf{x}')-mathrm{i}omega_n ( au- au')},where the sum is over the appropriate Matsubara frequencies (and the integral involves an implicit factor of (2pi)^{-d}, as usual).

In real time, we will explicitly indicate the time-ordered function with a superscript T::G^{mathrm{T(mathbf{x} t|mathbf{x}' t') = int_mathbf{k} int frac{mathrm{d}omega}{2pi} G(mathbf{k},omega) mathrm{e}^{mathrm{i} mathbf{k}cdot(mathbf{x} -mathbf{x} ')-mathrm{i}omega(t-t')}.

The real-time two-point Green function can be written in terms of `retarded' and `advanced' Green functions, which will turn out to have simpler analyticity properties. The retarded and advanced Green functions are defined by:G^{mathrm{R(mathbf{x} t|mathbf{x} 't') = mathrm{i}langle [psi(mathbf{x} ,t),arpsi(mathbf{x} ',t')] angleTheta(t-t')and:G^{mathrm{A(mathbf{x} t|mathbf{x} 't') = -mathrm{i}langle [psi(mathbf{x} ,t),arpsi(mathbf{x} ',t')] angleTheta(t'-t),respectively.

They are related to the time-ordered Green function by:G^{mathrm{T(mathbf{k},omega) = [1+zeta n(omega)] G^{mathrm{R(mathbf{k},omega) - zeta n(omega) G^{mathrm{A(mathbf{k},omega),where:n(x) = frac{1}{mathrm{e}^{eta x}-zeta}is the Bose-Einstein or Fermi-Dirac distribution function.

Imaginary-time ordering and eta-periodicity

The thermal Green functions are defined only when both imaginary-time arguments are within the range 0 to eta. The two-point Green function has the following properties. (The position or momentum arguments are suppressed in this section.)

Firstly, it depends only on the difference of the imaginary times::mathcal{G}( au, au') = mathcal{G}( au - au').The argument au - au' is allowed to run from -eta to eta.

Secondly, mathcal{G}( au) is periodic under shifts of eta. Because of the small domain within which the function is defined, this means just:mathcal{G}( au - eta) = zeta mathcal{G}( au),for 0 < au < eta. (Note that the function is antiperiodic for fermions.) Time ordering is crucial for this property, which can be proved straightforwardly, using the cyclicity of the trace operation.

These two properties allow for the Fourier transform representation and its inverse,:mathcal{G}(omega_n) = int_0^eta mathrm{d} au , mathcal{G}( au), mathrm{e}^{mathrm{i}omega_n au}.

Finally, note that mathcal{G}( au) has a discontinuity at au = 0; this is consistent with a long-distance behaviour of mathcal{G}(omega_n) sim 1/|omega_n|.

pectral representation

The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by: ho(mathbf{k},omega) = frac{1}{mathcal{Zsum_{alpha,alpha'} 2pi delta(E_alpha-E_{alpha'}-omega);
langlealpha|psi_mathbf{k}^dagger|alpha' angle|^2left(mathrm{e}^{-eta E_{alpha'-zetamathrm{e}^{-eta E_{alpha ight),where |alpha angle refers to a (many-body) eigenstate of the grand-canonical Hamiltonian H-mu N, with eigenvalue E_alpha.

The imaginary-time propagator is then given by:mathcal{G}(mathbf{k},omega_n) = int_{-infty}^{infty} frac{mathrm{d}omega'}{2pi}frac{ ho(mathbf{k},omega')}{-mathrm{i}omega_n+omega'}.and the retarded propagator by:G^{mathrm{R(mathbf{k},omega) = int_{-infty}^{infty} frac{mathrm{d}omega'}{2pi}frac{ ho(mathbf{k},omega')}{-(omega+mathrm{i}eta)+omega'},where the limit as eta ightarrow 0^+ is implied.

The advanced propagator is given by the same expression, but with -mathrm{i}eta in the denominator. The time-ordered function can be found in terms of G^{mathrm{R and G^{mathrm{A. As claimed above, G^{mathrm{R(omega) and G^{mathrm{A(omega) have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) half-plane. The thermal propagator mathcal{G}(omega_n) has all its poles and discontinuities on the imaginary omega_n axis.

The spectral density can be found very straightforwardly from G^{mathrm{R, using the Sokhatsky-Weierstrass theorem:lim_{eta ightarrow 0^+}frac{1}{x+mathrm{i}eta} = {P}frac{1}{x}-ipidelta(x),where P denotes the Cauchy principal part.This gives: ho(mathbf{k},omega) = 2mathrm{Im}, G^{mathrm{R(mathbf{k},omega).

This furthermore implies that G^{mathrm{R(mathbf{k},omega) obeys the following relationship between its real and imaginary parts::mathrm{Re}, G^{mathrm{R(mathbf{k},omega) = -2 P int_{-infty}^{infty} frac{mathrm{d}omega'}{2pi}frac{mathrm{Im}, G^{mathrm{R(mathbf{k},omega')}{omega-omega'},where P denotes the principal value of the integral.

The spectral density obeys a sum rule::int_{-infty}^{infty} frac{mathrm{d}omega}{2pi} ho(mathbf{k},omega) = 1,which gives:G^{mathrm{R(omega)simfrac{1}as |omega| ightarrow infty.

Hilbert transform

The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function:G(mathbf{k},z) = int_{-infty}^infty frac{mathrm{d} x}{2pi} frac{ ho(mathbf{k},x)}{-z+x},which is related to mathcal{G} and G^{mathrm{R by:mathcal{G}(mathbf{k},omega_n) = G(mathbf{k},mathrm{i}omega_n)and:G^{mathrm{R(mathbf{k},omega) = G(mathbf{k},omega + mathrm{i}eta).A similar expression obviously holds for G^{mathrm{A.

The relation between G(mathbf{k},z) and ho(mathbf{k},x) is referred to as a Hilbert transform.

Proof of spectral representation

We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as:mathcal{G}(mathbf{x} , au|mathbf{x} ', au') = langle Tpsi(mathbf{x} , au)arpsi(mathbf{x} ', au') angle.

Due to translational symmetry, it is only necessary to consider mathcal{G}(mathbf{x} , au|mathbf{0},0) for au > 0, given by:mathcal{G}(mathbf{x}, au|mathbf{0},0) = frac{1}{mathcal{Zsum_{alpha'} mathrm{e}^{-eta E_{alpha'langlealpha' | psi(mathbf{x}, au)arpsi(mathbf{0},0) |alpha' angle.Inserting a complete set of eigenstates gives:mathcal{G}(mathbf{x} , au|mathbf{0},0) = frac{1}{mathcal{Zsum_{alpha,alpha'} mathrm{e}^{-eta E_{alpha'langlealpha' | psi(mathbf{x} , au)|alpha anglelanglealpha |arpsi(mathbf{0},0) |alpha' angle.

Since |alpha angle and |alpha' angle are eigenstates of H-mu N, the Heisenberg operators can be rewritten in terms of Schr"odinger operators, giving:mathcal{G}(mathbf{x} , au|mathbf{0},0) = frac{1}{mathcal{Zsum_{alpha,alpha'} mathrm{e}^{-eta E_{alpha'mathrm{e}^{ au(E_{alpha'} - E_alpha)}langlealpha' | psi(mathbf{x} )|alpha anglelanglealpha |psi^dagger(mathbf{0}) |alpha' angle.Performing the Fourier transform then gives:mathcal{G}(mathbf{k},omega_n) = frac{1}{mathcal{Z sum_{alpha,alpha'} mathrm{e}^{-eta E_{alpha'frac{1-zeta mathrm{e}^{eta(E_{alpha'} - E_alpha){-mathrm{i}omega_n + E_alpha - E_{alpha' int_{mathbf{k}'} langlealpha |psi(mathbf{k})|alpha' anglelanglealpha' |psi^dagger(mathbf{k}')|alpha angle.

Momentum conservation allows the final term to be written as (up to possible factors of the volume):
langlealpha' |psi^dagger(mathbf{k})|alpha angle|^2,which confirms the expressions for the Green functions in the spectral representation.

The sum rule can be proved by considering the expectation value of the commutator,:1 = frac{1}{mathcal{Z sum_alpha langlealpha |mathrm{e}^{-eta(H-mu N)} [psi_mathbf{k},psi_mathbf{k}^dagger] |alpha angle,and then inserting a complete set of eigenstates into both terms of the commutator::1 = frac{1}{mathcal{Z sum_{alpha,alpha'} mathrm{e}^{-eta E_alpha} left(langlealpha |psi_mathbf{k} |alpha' anglelanglealpha' | psi_mathbf{k}^dagger|alpha angle - zeta langlealpha |psi_mathbf{k}^dagger |alpha' anglelanglealpha' | psi_mathbf{k}|alpha angle ight).

Swapping the labels in the first term then gives:1 = frac{1}{mathcal{Z sum_{alpha,alpha'}left(mathrm{e}^{-eta E_{alpha' - zeta mathrm{e}^{-eta E_alpha} ight)
langlealpha | psi_mathbf{k}^dagger|alpha' angle|^2,which is exactly the result of the integration of ho.

Non-interacting case

In the non-interacting case, psi_mathbf{k}^dagger|alpha' angle is an eigenstate with (grand-canonical) energy E_{alpha'} + xi_mathbf{k}, where xi_mathbf{k} = epsilon_mathbf{k} - mu is the single-particle dispersion relation measured with respect to the chemical potential. The spectral density therefore becomes: ho_0(mathbf{k},omega) = frac{1}{mathcal{Z,2pidelta(xi_mathbf{k} - omega) sum_{alpha'}langlealpha' |psi_mathbf{k}psi_mathbf{k}^dagger|alpha' angle(1-zeta mathrm{e}^{-etaxi_mathbf{k)mathrm{e}^{-eta E_{alpha'.

From the commutation relations,:langlealpha' |psi_mathbf{k}psi_mathbf{k}^dagger|alpha' angle =langlealpha' |(1+zetapsi_mathbf{k}^daggerpsi_mathbf{k})|alpha' angle,with possible factors of the volume again. The sum, which involves the thermal average of the number operator, then gives simply [1 + zeta n(xi_mathbf{k})] mathcal{Z}, leaving: ho_0(mathbf{k},omega) = 2pidelta(xi_mathbf{k} - omega).

The imaginary-time propagator is thus:mathcal{G}_0(mathbf{k},omega) = frac{1}{-mathrm{i}omega_n + xi_mathbf{kand the retarded propagator is:G_0^{mathrm{R(mathbf{k},omega) = frac{1}{-(omega+mathrm{i} eta) + xi_mathbf{k.

Zero-temperature limit

As eta ightarrowinfty, the spectral density becomes: ho(mathbf{k},omega) = 2pisum_{alpha} left [ delta(E_alpha - E_0 - omega)
langlealpha |psi_mathbf{k}^dagger|0 angle|^2- zeta delta(E_0 - E_{alpha} - omega)
langle0 |psi_mathbf{k}^dagger|alpha angle|^2 ight] where alpha = 0 corresponds to the ground state. Note that only the first (second) term contributes when omega is positive (negative).

General case

Basic definitions

We can use `field operators' as above, or creation and annihilation operators associated with other single-particle states, perhaps eigenstates of the (noninteracting) kinetic energy. We then use:psi(mathbf{x} , au) = varphi_alpha(mathbf{x} ) psi_alpha( au),where psi_alpha is the annihilation operator for the single-particle state alpha and varphi_alpha(mathbf{x} ) is that state's wavefunction in the position basis. This gives:mathcal{G}^{(n)}_{alpha_1ldotsalpha_n|eta_1ldotseta_n}( au_1 ldots au_n | au_1' ldots au_n')= langle Tpsi_{alpha_1}( au_1)ldotspsi_{alpha_n}( au_n)arpsi_{eta_n}( au_n')ldotsarpsi_{eta_1}( au_1') anglewith a similar expression for G^{(n)}.

Two-point functions

These depend only on the difference of their time arguments, so that:mathcal{G}_{alphaeta}( au| au') = frac{1}{eta}sum_{omega_n}mathcal{G}_{alphaeta}(omega_n),mathrm{e}^{-mathrm{i}omega_n ( au- au')}and:G_{alphaeta}(t|t') = int_{-infty}^{infty}frac{mathrm{d}omega}{2pi},G_{alphaeta}(omega),mathrm{e}^{-mathrm{i}omega(t-t')}.

We can again define retarded and advanced functions in the obvious way; these are related to the time-ordered function in the same way as above.

The same periodicity properties as described in above apply to mathcal{G}_{alphaeta}. Specifically,:mathcal{G}_{alphaeta}( au| au') = mathcal{G}_{alphaeta}( au- au')and:mathcal{G}_{alphaeta}( au) = mathcal{G}_{alphaeta}( au + eta),for au < 0.

pectral representation

In this case,: ho_{alphaeta}(omega) = frac{1}{mathcal{Zsum_{m,n} 2pi delta(E_n-E_m-omega);langle m |psi_alpha|n anglelangle n |psi_eta^dagger|m angleleft(mathrm{e}^{-eta E_m} - zeta mathrm{e}^{-eta E_n} ight) ,where m and n are many-body states.

The expressions for the Green functions are modified in the obvious ways::mathcal{G}_{alphaeta}(omega_n) = int_{-infty}^{infty} frac{mathrm{d}omega'}{2pi}frac{ ho_{alphaeta}(omega')}{-mathrm{i}omega_n+omega'}and:G^{mathrm{R_{alphaeta}(omega) = int_{-infty}^{infty} frac{mathrm{d}omega'}{2pi}frac{ ho_{alphaeta}(omega')}{-(omega+mathrm{i}eta)+omega'}.

Their analyticity properties are identical. The proof follows exactly the same steps, except that the two matrix elements are no longer complex conjugates.

Noninteracting case

If the particular single-particle states that are chosen are `single-particle energy eigenstates', ie,: [H-mu N,psi_alpha^dagger] = xi_alphapsi_alpha^dagger,then for |n angle an eigenstate::(H-mu N)|n angle = E_n |n angle,so is psi_alpha |n angle::(H-mu N)psi_alpha|n angle = (E_n - xi_alpha) psi_alpha |n angle,and so is psi_alpha^dagger|n angle::(H-mu N)psi_alpha^dagger|n angle = (E_n + xi_alpha) psi_alpha^dagger |n angle.

We therefore have:langle m |psi_alpha|n anglelangle n |psi_eta^dagger|m angle =delta_{alphaeta}delta_{n m}langle ilde{m} |psi_alphapsi_alpha^dagger| ilde{m} angle,where:
ilde{m angle} = frac{psi_alpha^dagger|m angle}.

We then rewrite:psi_alpha psi_alpha^dagger = zeta psi_alpha^dagger psi_alpha + 1,and use the fact that the thermal average of the number operator gives the Bose-Einstein or Fermi-Dirac distribution function.

Finally, the spectral density simplifies to give: ho_{alphaeta} = 2pi delta(xi_eta - omega)delta_{alphaeta},so that the thermal Green function is:mathcal{G}_{alphaeta}(omega_n) = frac{delta_{alphaeta{-mathrm{i}omega_n + xi_eta}and the retarded Green function is:G_{alphaeta}(omega) = frac{delta_{alphaeta{-(omega+mathrm{i}eta) + xi_eta}.Note that the noninteracting Green function is diagonal, but this will not be true in the interacting case.

References

* Negele, J.W. and Orland, H.: "Quantum Many-Particle Systems" AddisonWesley (1988).

* Abrikosov, A.A., Gorkov, L.P. and Dzyaloshinski, I.E.: "Methods of Quantum Field Theory in Statistical Physics" Englewood Cliffs: Prentice-Hall (1963).

ee also

*Fluctuation theorem
*Green-Kubo_relations
*Linear response function
*Lindblad equation


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