By means of Green's functions methods, Gorkov derived the BCS theory on the basis of the Ansatz that the correlation part of the second-order Green's function could be factored in the form χχ* where χ is a two-particle function closely related to the Ginzburg–Landau order-parameter. Since the density matrix is an initial value of a Green's function, Gorkov's Ansatz is equivalent to an assumption about the 2-matrix. The present paper considers circumstances in which the Gorkov Ansatz is exactly satisfied by a system of a definite number of fermions.
The imaginary-time Green's function is a building block of various numerical methods for correlated electron systems. Recently, it was shown that a model-independent compact orthogonal representation of the Green's function can be constructed by decomposing its spectral representation. We investigate the performance of this so-called intermediate representation (IR) from several points of view. First, we develop an efficient algorithm for computing the IR basis functions of arbitrary high degree. Second, for two simple models, we study the number of coefficients required to represent the Green's function within a given tolerance. We show that the number of coefficients grows only as $O(ln\ensuremath{\beta})$ for fermions and converges to a constant for bosons as temperature $T=1/\ensuremath{\beta}$ decreases. Third, we show that this remarkable feature is ascribed to the properties of the physically constructed basis functions. The fermionic basis functions on the real-frequency axis have features whose width is scaled as
O(T), which is consistent with the low-
T properties of quasiparticles in a Fermi liquid state. On the other hand, the properties of the bosonic basis functions are consistent with those of spin/orbital susceptibilities at low
T. These results demonstrate the potential wide applications of the IR to calculations of correlated systems.
The disorder averaged single-particle Green's function of electrons subject to a time-dependent random potential with long-range spatial correlations is calculated by means of bosonization in arbitrary dimensions. For static disorder our method is equivalent with conventional perturbation theory based on the lowest order Born approximation. For dynamic disorder, however, we obtain a new non-perturbative expression for the average Green's function. Bosonization also provides a solid microscopic basis for the description of the quantum dynamics of an interacting many-body system via an effective stochastic model with Gaussian probability distribution.