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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.)
Spatially uniform case
We consider a many-body theory with field operator (annihilation operator written in the position basis) .
[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
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
They are related to the time-ordered Green function by
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 .
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 H - ?N, with eigenvalue E?.
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.
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 ?.
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
As ?->?, the spectral density becomes
where ? = 0 corresponds to the ground state. Note that only the first (second) term contributes when ? is positive (negative).
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 .
These depend only on the difference of their time arguments, so that
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,
In this case,
where and are many-body states.
The expressions for the Green functions are modified in the obvious ways:
Their analyticity properties are identical. The proof follows exactly the same steps, except that the two matrix elements are no longer complex conjugates.
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
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.