Green's function (many-body theory)

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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.

Many-body theory is an area of physics which provides the framework for understanding the collective behavior of large numbers of interacting particles, often on the order of Avogadro's number. In general terms, many-body theory deals with effects that manifest themselves only in systems containing large numbers of constituents. While the underlying physical laws that govern the motion of each individual particle may be simple, the study of the collection of particles can be extremely complex. In some cases emergent phenomena may arise which bear little resemblance to the underlying elementary laws.

Correlation function (quantum field theory)

In quantum field theory, the n-point correlation function is defined as the functional average of a product of field operators at different positions

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. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.

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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.)

In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

Spatially uniform case

Basic definitions

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

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, an arbitrary fixed basis rigidly underlying the theory.

In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant with respect to time. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures.

and the creation operator is , where is the grand-canonical Hamiltonian.

In statistical mechanics, a grand canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

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 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 β-periodicity

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 representation

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 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 by

and

A similar expression obviously holds for .

The relation between and 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

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 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

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 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

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', 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 also

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