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physica status solidi (b)

Stochastic quantization approach to disordered fermions

E. Kolley, Winfried Kolley

Published Apr 1, 1987• 3 citations
Publisher

Abstract

Abstract The method of the stochastic quantization is applied to an electron system described by the random Hubbard model. Two different stochastic calculi are performed corresponding to Euclidean and non‐Euclidean versions of the quantum field theory in terms of Grassmann variables. In averaging over the quenched disorder a nontrivial normalization denominator can be avoided. An averaged, effective two‐time action is given as basis for treating localization.

DOI
10.1002/pssb.2221400223

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Abstract The method of the stochastic quantization is applied to an electron system described by the random Hubbard model. Two different stochastic calculi are performed corresponding to Euclidean and non‐Euclidean versions of the quantum field theory in terms of Grassmann variables. In averaging over the quenched disorder a nontrivial normalization denominator can be avoided. An averaged, effective two‐time action is given as basis for treating localization.
Abstract On the basis of the attractive Hubbard model with a Gaussian random potential we describe superconductivity in the presence of quenched disorder. A dynamic path‐integral formalism with an 8‐component Keldysh‐Nambu spinor is shown to be convenient for solving the averaging problem without the recourse to the replica trick. The effective action is bosonized in terms of collective variables, namely local Q‐matrices and real space pair fields. Saddle point values are calculated in two simplified limits.
Following a proposal by Aronov and Ioselevich, we express the Green functions (GF) of a noninteracting disordered Fermi system as a functional integral on a real time/frequency lattice. The normalizing denominator of this functional integral is equal to unity, because of identities satisfied by the GF. The GF can then be simply averaged with respect to the random disorder potential. We describe the fermionic fields not belonging to the external frequency by means of a bosonic auxiliary field g. The Hubbard-Stratonovich field Q is introduced only with respect to the fermionic fields for the external frequency.
Abstract A real‐time finite‐temperature field theory on the Keldysh contour is used to describe the dynamics of an electron system within a random Hubbard model. The effective, i.e. quenched‐averaged, fermionic action under the functional integral is bosonized by introducing both a matrix field and an exchange field via Hubbard‐Stratonovich transformations. The resulting collective action permits the loop expansion, giving rise to a systematic approximation scheme. The saddle point and Gaussian fluctuations around it are calculated.
We consider two strategies for stochastic quantization. With the first, one posits an additional time dimension (fictitious time) and describes the evolution of classical fields by means of the Langevin equation. One then evaluates stochastic averages of the field functions. In the limit that the fictitious time goes to infinity, these approach the time ordered correlation functions of canonically quantized field theory. We conclude that, while this strategy successfully describes QED and other quantum field theories, it is contrived and probably lacks deep physical significance. With the second strategy for field quantization, one begins with a classical action in either one or two extra dimensions coupled to an external random source. We review a method of quantizing bosonic fields which uses this strategy. Furthermore, we present an analogous method for quantizing fermion fields and a possible new way of quantizing interacting fermion and boson fields. Finally, we discuss applications to quantum mechanics and stochastically quantize the simple harmonic oscillator.
Abstract We review the nonlinear σ‐model approach to the localization problem. However, different from usual, we do not make use of replicas or superfields in order to perform the averaging over the realizations of the disorder. Instead, we pursue the derivation of the nonlinear σ‐model as an effective theory for the disordered electron system within the framework of the so‐called time‐path formalism, which is a real‐time finite‐temperature quantum field theory. This formalism is especially suitable for the description of the disordered electron system due to the automatic normalization of real‐time quantities, thereby avoiding the denominator problem. We shall compare this dynamic method with the replica‐ and superfield‐methods as the perturbative level of the calculation of physical quantities.
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.