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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.
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
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
Similarly, for the imaginary-time operators,
In real time, the -point Green function is defined by
In imaginary time, the corresponding definition is
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
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
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
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:
Secondly, is (anti)periodic under shifts of . Because of the small domain within which the function is defined, this means just
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
The imaginary-time propagator is then given by
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
This furthermore implies that obeys the following relationship between its real and imaginary parts:
The spectral density obeys a sum rule,
Hilbert transform
The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function
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
Since and are eigenstates of , the Heisenberg operators can be rewritten in terms of Schrödinger operators, giving
Momentum conservation allows the final term to be written as (up to possible factors of the volume)
The sum rule can be proved by considering the expectation value of the commutator,
Swapping the labels in the first term then gives
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,
The imaginary-time propagator is thus
Zero-temperature limit
As β → ∞, the spectral density becomes
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
Two-point functions
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,
Spectral representation
In this case,
The expressions for the Green functions are modified in the obvious ways:
Their analyticity properties are identical to those of and defined in the translationally invariant case. 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.
We therefore have
We then rewrite
Finally, the spectral density simplifies to give
See also
- Fluctuation theorem
- Green–Kubo relations
- Linear response function
- Lindblad equation
- Propagator
- Correlation function (quantum field theory)
- Numerical analytic continuation
References
Books
- 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.
Papers
- Bogolyubov N. N., Tyablikov S. V. Retarded and advanced Green functions in statistical physics, Soviet Physics Doklady, Vol. 4, p. 589 (1959).
- Zubarev D. N., Double-time Green functions in statistical physics, Soviet Physics Uspekhi 3(3), 320–345 (1960).
External links
- Linear Response Functions in Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum Jülich, 2014 ISBN 978-3-89336-953-9