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arXiv:math-ph/0409014 · Journal of Physics Condensed Matter

On Hubbard–Stratonovich transformations over hyperbolic domains

Yan V Fyodorov

Published May 6, 2005• 7 citations
arXivHTMLPDF

Abstract

We discuss and prove the validity of the Hubbard–Stratonovich (HS) identities over hyperbolic domains which are used frequently in studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with 'chiral' symmetry. Apart from this we outline a way of deriving the nonlinear σ-model from the gauge-invariant Wegner k-orbital model avoiding the use of the HS transformations.

DOI
10.1088/0953-8984/17/20/018

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We discuss and prove the validity of the Hubbard–Stratonovich (HS) identities over hyperbolic domains which are used frequently in studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with 'chiral' symmetry. Apart from this we outline a way of deriving the nonlinear σ-model from the gauge-invariant Wegner k-orbital model avoiding the use of the HS transformations.
The present work consists of two parts. The first part deals with hyperbolic Hubbard-Stratonovich transformations. Such transformations are used to derive non-linear sigma models that describe the low energy behaviour of disordered electron systems. For a long time the mathematical status of hyperbolic Hubbard-Stratonovich transformations of Pruisken-Schäfer type remained unclear. Only recently the two special cases of pseudounitary and pseudoorthogonal symmetry were proven . In this thesis we prove the transformation for a general (essentially semisimple) symmetry group. The proof is descriptive and shows explicitly the connection to the standard Gaussian integrals. In the second part we develop a novel method to bosonise granular fermionic systems. The method is related to the method of coherent states. In particular it is not based on the well known bosonisation of (1+1)-dimensional systems. One approach is to use the colour-flavour transformation to transform the Grassmann path integral representation of a grand canonical partition function in a way that allows to eliminate the Grassmann variables. The result is a path integral in generalised coherent states with special boundary conditions.
We analyze a variety of Gaussian integrals with the aim of revisiting the derivation of the Hubbard–Stratonovich transformation as given in standard graduate-level textbooks and provide an overview of its applications. We pinpoint problematic steps in the usual discussions and propose careful derivations of the Hubbard–Stratonovich identity pertinent to a variety of situations relevant to statistical physics and quantum field theory. These derivations are based on direct use of either a resolution identity or a series expansion. A few homework problems for students are suggested.
The application of functional integral methods and the Hubbard--Stratonovich transformation to the Hubbard model is discussed. For the attractive case, using a simple gauge transformation of the superconducting order parameter field, the effective action for the low-energy phase excitations is derived. Inclusion of the electromagnetic field is straightforward. For the repulsive case at half--filling use of a fluctuating spin reference frame allows a derivation of the effective nonlinear sigma model for arbitrary coupling strength U. Results for the spin-wave velocity and the spin stiffness are obtained for arbitrary U. Away from half--filling, at large U an effective action for doped fermions and the spins is found which is related to the slave fermion formulation of the correlated fermion problem. The formalism is generalized to include the full SU(2)xSU(2) symmetry of the Hubbard model. This allows the derivation of a Ginzburg-Landau like theory that includes both particle-hole and particle-particle excitations. (paper at the NATO workshop on the Hubbard model, San Sebastian, Spain, oct. 1993)
Non-Hermitian disordered systems have emerged as a central arena in modern physics, with ramifications spanning condensed matter, quantum, statistical, and high energy contexts. The same principles also underlie phenomena beyond physics, such as network science, complex systems, and biophysics, where dissipation, nonreciprocity, and stochasticity are ubiquitous. Here, we review the physics and mathematics of non-Hermitian disordered systems, with particular emphasis on non-Hermitian random matrix theory. We begin by presenting the 38-fold symmetry classification of non-Hermitian systems, contrasting it with the 10-fold way for Hermitian systems. After introducing the classic Ginibre ensembles of non-Hermitian random matrices, we survey various diagnostics for complex-spectral statistics and distinct universality classes realized by symmetry. As a key application to physics, we discuss how non-Hermitian random matrix theory characterizes chaos and integrability in open quantum systems. We then turn to the criticality due to the interplay of disorder and non-Hermiticity, including Anderson transitions in the Hatano-Nelson model and its higher-dimensional extensions. We also discuss the effective field theory description of non-Hermitian disordered systems in terms of nonlinear sigma models.
The Hubbard - Stratonovich transformation of many - body theory is reconsidered in various versions. Learning from its failures in coping with competing channels of collective phenomena, a combined Hubbard - Stratonovich transformation is proposed to overcome these difficulties. That yields a variety of fluctuating bosonic quantum fields describing the competition of different phases in the system under consideration. As a consequence of results, the validity of the Hubbard - Stratonovich transformation and its relation to a given order parameter are discussed.