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Journal of Physics C Solid State Physics

Correlation functions and Green functions: zero-frequency anomalies

P C Kwok, T D Schultz

Published Jul 1, 1969• 43 citations
Publisher

Abstract

The zero-frequency anomaly that can arise in the Fourier transform of certain quantum mechanical correlation functions is investigated. Two physical interpretations are given to the anomalous term. It is shown that any method for calculating the thermal Green function (though not the retarded or advanced commutator Green functions) should also yield the strength of the anomaly. An example is given. A critical discussion is given of recent contributions by other authors.

DOI
10.1088/0022-3719/2/7/312

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The zero-frequency anomaly that can arise in the Fourier transform of certain quantum mechanical correlation functions is investigated. Two physical interpretations are given to the anomalous term. It is shown that any method for calculating the thermal Green function (though not the retarded or advanced commutator Green functions) should also yield the strength of the anomaly. An example is given. A critical discussion is given of recent contributions by other authors.
In this work we will establish the definitions of the Green’s functions in statistical mechanics and their basic properties. These functions depend on double-time and temperature. It can be observed by mean of theier definitions, where we can find the medium values of the operators product, this average is done over the grand canonical ensemble, for instance, (FAB(t − t0) ≡ hAˆ(t)Bˆ(t0)i ou FBA(t − t0) ≡ hBˆ(t0)Aˆ(t)i), with β = 1/kBT, and T is the absolute temperature. The operators involved in theses functions satisfy the Heisenberg equation of motion, which permit us describe the evolution equations for Green’s functions. By mean of spectral representation of the time correlation functions(FAB(t−t0) ≡ hAˆ(t)Bˆ(t0 )i or FBA(t−t0) ≡ hBˆ(t0)Aˆ(t)i), that is done by the introduction of a Fourier transform to change the system from time-space to frequency-space, we can obtain the spectral representations for the retarded Gr, advanced Gr e causal Gc green’s functions. Lastly, we will make the use of retarded green’s function to describe the electrical conductivity of a system of electrons under a external time-dependent electric field, in others words, we will write the electrical conductivity tensor in terms of Gr and, at last, we will calculate the electrical conductivity of a system of electrons and phonons.
Recent attempts to develop double-time Green's-function theories for the transverse Ising model (TIM) have been subject to certain difficulties; the appearance of zero-frequency poles in the commutator Green's function (CGF) led to the avoidance of CGF and an incomplete treatment based only upon anticommutator Green's functions (AGF), and ambiguities in the determination of some thermal averages. Upon decoupling the equations-of-motion hierarchy, we develop a consistent Green's-function theory of the TIM. The AGF and CGF are shown to provide consistent results, the problem of zero-frequency poles in CGF being treated appropriately. We resolve the ambiguity problem and obtain reasonable values for the Weiss critical field and the ground-state magnetization.
Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.