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Green's function (many-body theory)

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In many-body theory

Many-body theory

The many-body theory is an area of physics which provides the framework for understanding the collective behavior of vast assemblies of interacting particles. In general terms, the many-body theory deals with effects that manifest themselves only in systems containing large numbers of constituents...

, the term Green's function (or Green function) is sometimes used interchangeably with correlation function

Correlation function (quantum field theory)

In quantum field theory, the matrix element computed by inserting a product of operators between two states, usually the vacuum states, is called a correlation function....

, but refers specifically to correlators of field operators or creation and annihilation operators

Creation and annihilation operators

Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one...

.

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

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

that is quadratic in the fields.)

Basic definitions

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

.

The Heisenberg operators

Heisenberg picture

In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...

can be written in terms of Schrödinger operators

Schrödinger picture

In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...

and

, where

is the grand-canonical

Grand canonical ensemble

In statistical mechanics, a grand canonical ensemble is a theoretical collection of model systems put together to mirror the calculated probability distribution of microscopic states of a given physical system which is being maintained in a given macroscopic state...

Hamiltonian.

Similarly, for the imaginary-time

Imaginary time

Imaginary time is a concept derived from quantum mechanics and is essential in connecting quantum mechanics with statistical mechanics.- In quantum mechanics :...

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 the 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

.)

Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of the two-point (

) thermal Green function for a free particle is

and the retarded Green function is

(See below for details.)

Throughout,

for boson

Boson

In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s and

for fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s and

denotes either a commutator

Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

or anticommutator as appropriate.

Two-point functions

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 -periodicity

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

. 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

.

Secondly,

is periodic under shifts of

. Because of the small domain within which the function is defined, this means just

for

. (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,

Finally, note that

has a discontinuity at

; this is consistent with a long-distance behaviour of

Spectral representation

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

where

refers to a (many-body) eigenstate of the grand-canonical Hamiltonian

, with eigenvalue

.

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

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

Hilbert transform

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

A similar expression obviously holds for

.

The relation between

and

is referred to as a Hilbert transform

Hilbert transform

In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...

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

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

Non-interacting case

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 limit

, the spectral density becomes

where

corresponds to the ground state. Note that only the first (second) term contributes when

is positive (negative).

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

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 functions

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 representation

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 case

If the particular single-particle states that are chosen are `single-particle energy eigenstates', ie,

then for

an eigenstate:

so is

and so is

We therefore have

where

We then rewrite

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

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.

Green's function (many-body theory)

Basic definitions

Two-point functions

Imaginary-time ordering and -periodicity

Spectral representation

Hilbert transform

Proof of spectral representation

Non-interacting case

Zero-temperature limit

Basic definitions

Two-point functions

Spectral representation

Noninteracting case

See also

Books

Papers