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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' in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields.)

Contents

Spatially uniform caseEdit

Basic definitionsEdit

We consider a many-body theory with field operator (annihilation operator written in the position basis)  .

The Heisenberg operators can be written in terms of Schrödinger operators as

 

and  , where   is the grand-canonical Hamiltonian.

Similarly, for the imaginary-time operators,

 
 

[Note that the imaginary-time creation operator   is not the Hermitian conjugate of the annihilation operator  .]

In real time, the  -point Green function is defined by

 

where we have used a condensed notation in which   signifies   and   signifies  . The operator   denotes time ordering, and indicates that the field operators that follow it are to be ordered so that their time arguments increase from right to left.

In imaginary time, the corresponding definition is

 

where   signifies  . (The imaginary-time variables   are restricted to the range from   to the inverse temperature  .)

Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform of the two-point ( ) thermal Green function for a free particle is

 

and the retarded Green function is

 

where

 

is the Matsubara frequency.

Throughout,   is   for bosons and   for fermions and   denotes either a commutator or anticommutator as appropriate.

(See below for details.)

Two-point functionsEdit

The Green function with a single pair of arguments ( ) 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

 

where the sum is over the appropriate Matsubara frequencies (and the integral involves an implicit factor of  , as usual).

In real time, we will explicitly indicate the time-ordered function with a superscript 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

 

and

 

respectively.

They are related to the time-ordered Green function by

 

where

 

is the Bose–Einstein or Fermi–Dirac distribution function.

Imaginary-time ordering and β-periodicityEdit

The thermal Green functions are defined only when both imaginary-time arguments are within the range   to  . 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:

 

The argument   is allowed to run from   to  .

Secondly,   is (anti)periodic under shifts of  . Because of the small domain within which the function is defined, this means just

 

for  . 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,

 

Finally, note that   has a discontinuity at  ; this is consistent with a long-distance behaviour of  .

Spectral representationEdit

The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by

 

where |α⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian H − μN, with eigenvalue Eα.

The imaginary-time propagator is then given by

 

and the retarded propagator by

 

where the limit as   is implied.

The advanced propagator is given by the same expression, but with   in the denominator.

The time-ordered function can be found in terms of   and  . As claimed above,   and   have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) half-plane.

The thermal propagator   has all its poles and discontinuities on the imaginary   axis.

The spectral density can be found very straightforwardly from  , using the Sokhatsky–Weierstrass theorem

 

where P denotes the Cauchy principal part. This gives

 

This furthermore implies that   obeys the following relationship between its real and imaginary parts:

 

where   denotes the principal value of the integral.

The spectral density obeys a sum rule,

 

which gives

 

as  .

Hilbert transformEdit

The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function

 

which is related to   and   by

 

and

 

A similar expression obviously holds for  .

The relation between   and   is referred to as a Hilbert transform.

Proof of spectral representationEdit

We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as

 

Due to translational symmetry, it is only necessary to consider   for  , given by

 

Inserting a complete set of eigenstates gives

 

Since   and   are eigenstates of  , the Heisenberg operators can be rewritten in terms of Schrödinger operators, giving

 

Performing the Fourier transform then gives

 

Momentum conservation allows the final term to be written as (up to possible factors of the volume)

 

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,

 

and then inserting a complete set of eigenstates into both terms of the commutator:

 

Swapping the labels in the first term then gives

 

which is exactly the result of the integration of ρ.

Non-interacting caseEdit

In the non-interacting case,   is an eigenstate with (grand-canonical) energy  , where   is the single-particle dispersion relation measured with respect to the chemical potential. The spectral density therefore becomes

 

From the commutation relations,

 

with possible factors of the volume again. The sum, which involves the thermal average of the number operator, then gives simply  , leaving

 

The imaginary-time propagator is thus

 

and the retarded propagator is

 

Zero-temperature limitEdit

As β→∞, the spectral density becomes

 

where α = 0 corresponds to the ground state. Note that only the first (second) term contributes when ω is positive (negative).

General caseEdit

Basic definitionsEdit

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

 

where   is the annihilation operator for the single-particle state   and   is that state's wavefunction in the position basis. This gives

 

with a similar expression for  .

Two-point functionsEdit

These depend only on the difference of their time arguments, so that

 

and

 

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  . Specifically,

 

and

 

for  .

Spectral representationEdit

In this case,

 

where   and   are many-body states.

The expressions for the Green functions are modified in the obvious ways:

 

and

 

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

Noninteracting caseEdit

If the particular single-particle states that are chosen are `single-particle energy eigenstates', i.e.

 

then for   an eigenstate:

 

so is  :

 

and so is  :

 

We therefore have

 

We then rewrite

 

therefore

 

use

 

and 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

 

so that the thermal Green function is

 

and the retarded Green function is

 

Note that the noninteracting Green function is diagonal, but this will not be true in the interacting case.

See alsoEdit

ReferencesEdit

BooksEdit

  • Bonch-Bruevich V. L., Tyablikov S. V. (1962): The Green Function Method in Statistical Mechanics. North Holland Publishing Co.
  • Abrikosov, A. A., Gorkov, L. P. and Dzyaloshinski, I. E. (1963): Methods of Quantum Field Theory in Statistical Physics Englewood Cliffs: Prentice-Hall.
  • Negele, J. W. and Orland, H. (1988): Quantum Many-Particle Systems AddisonWesley.
  • Zubarev D. N., Morozov V., Ropke G. (1996): Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory (Vol. 1). John Wiley & Sons. ISBN 3-05-501708-0.
  • Mattuck Richard D. (1992), A Guide to Feynman Diagrams in the Many-Body Problem, Dover Publications, ISBN 0-486-67047-3.

PapersEdit

External linksEdit