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arXiv:1705.07915 · Physical review. B./Physical review. B

Superbosonization in disorder and chaos: Role of anomalies

Tigran Sedrakyan, K. B. Efetov

Published Aug 23, 2017• 2 citations
arXivHTMLPDF

Abstract

The superbosonization formula aims at rigorously calculating fermionic integrals via employing supersymmetry. We derive such a supermatrix representation of superfield integrals and specify integration contours for the supermatrices. The derivation is essentially based on the supersymmetric generalization of the Itzykson-Zuber integral in the presence of anomalies in the Berezinian and shows how an integral over supervectors is eventually reduced to an integral over commuting variables. The approach is tested by calculating both one and two point correlation functions in a class of random matrix models. It is argued that the approach is capable of producing nonperturbative results in various systems with disorder, including physics of many-body localization, and other situations hosting localization phenomena.

DOI
10.1103/physrevb.96.054208

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The superbosonization formula aims at rigorously calculating fermionic integrals via employing supersymmetry. We derive such a supermatrix representation of superfield integrals and specify integration contours for the supermatrices. The derivation is essentially based on the supersymmetric generalization of the Itzykson-Zuber integral in the presence of anomalies in the Berezinian and shows how an integral over supervectors is eventually reduced to an integral over commuting variables. The approach is tested by calculating both one and two point correlation functions in a class of random matrix models. It is argued that the approach is capable of producing nonperturbative results in various systems with disorder, including physics of many-body localization, and other situations hosting localization phenomena.
Considering disordered electron systems we suggest a scheme that allows us to include an electron-electron interaction into a supermatrix $\ensuremath{\sigma}$-model. The method is based on replacing the initial model of interacting electrons by a fully supersymmetric model. Although this replacement is not exact, it is a good approximation for a weak short range interaction and arbitrary disorder. The replacement makes the averaging over disorder and further manipulations straightforward and we come to a supermatrix $\ensuremath{\sigma}$-model containing an interaction term. The structure of the model is rather similar to the replica one, although the interaction term has a different form. We study the model making perturbation theory and renormalization group calculations. We check the renormalizability of the model in the first loop approximation and in the first order in the interaction. In this limit we reproduce the renormalization group equations known from earlier works. We hope that the new supermatrix $\ensuremath{\sigma}$-model may become a new tool for nonperturbative calculations for disordered systems with interaction.
The supersymmetry technique has proven to be an efficient method for studying disordered and chaotic systems as well as random matrix models. So far, it has been used for noninteracting systems only. Here, contrary to general belief, the robust methodology of supersymmetry to handle disordered and chaotic many-body systems is shown to work in effectively zero space dimensions. Within this framework, nonperturbative results are obtained for the celebrated Sachdev-Ye-Kitaev non-Fermi-liquid theory.
Generating functions with graded variables are mainly used for the description of disordered systems. Instead of averaging over many random samples, the averaging procedure is performed only once with a generating function containing anticommuting variables. An approach is described for the calculation of correlation functions by transfer matrix methods including the aspect of anticommuting variables. The transfer matrix of the generating function of a disordered many body system is constructed in coherent state representation and transformed to a differential eigenvalue equation. According to the presence of anticommuting numbers, this general eigenvalue equation separates into sets of differential eigenvalue systems of commuting variables. These characteristic eigenvalue systems are reduced with operators and their commutation relations to an ordinary Sturm Liouville eigenvalue problem whose lowest eigenvalues contain the information about the averaged, asymptotic critical values of the correlation function. The eigensolutions for the U(N) symmetry preserving problem can be obtained from a recursion process for eigenstates with an arbitrary number M of anticommuting pairs χ*χ, (1 ≤ j ≤ M) which is proved by induction. The given approach, avoiding statistical averaging techniques, can be generalized to problems with U(N) symmetry breaking terms.
We reformulate the nonperturbative functional renormalization group for the random field Ising model in a superfield formalism, extending the supersymmetric description of the critical behavior of the system first proposed by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)]. We show that the two crucial ingredients for this extension are the introduction of a weighting factor, which accounts for ground-state dominance when multiple metastable states are present, and of multiple copies of the original system, which allows one to access the full functional dependence of the cumulants of the renormalized disorder and to describe rare events. We then derive exact renormalization group equations for the flow of the renormalized cumulants associated with the effective average action.
We review the recent progress on the construction of the determinant representations of the correlation functions for the integrable supersymmetric fermion models. The factorizing FFF-matrices (or the so-called FFF-basis) play an important role in the construction. In the FFF-basis, the creation (and the annihilation) operators and the Bethe states of the integrable models are given in completely symmetric forms. This leads to the determinant representations of the scalar products of the Bethe states for the models. Based on the scalar products, the determinant representations of the correlation functions may be obtained. As an example, in this review, we give the determinant representations of the two-point correlation function for the gl\\glgl (i.e. q-deformed) supersymmetric t-J model. The determinant representations are useful for analysing physical properties of the integrable models in the thermodynamical limit.