A statistical field theory is developed to explore the density of states and spatial profile of 'tail states' at the edge of the spectral support of a general class of disordered non-Hermitian operators. These states, which are identified with symmetry broken, instanton field configurations of the theory, are closely related to localized sub-gap states recently identified in disordered superconductors. By focusing separately on the problems of a quantum particle propagating in a random imaginary scalar potential, and a random imaginary vector potential, we discuss the methodology of our approach and the universality of the results. Finally, we address potential physical applications of our findings.

We study spectral properties of a non-Hermitian Hamiltonian describing a quantum particle propagating in a random imaginary scalar potential. Cast in the form of an effective field theory, we obtain an analytical expression for the ensemble averaged one-particle Green function from which we obtain the density of complex eigenvalues. Based on the connection between non-Hermitian quantum mechanics and the statistical mechanics of polymer chains, we determine the distribution function of a self-interacting polymer in dimensions d > 4.

We study the critical properties of the weakly disordered two-dimensional Ising and Baxter models in terms of the renormalization group (RG) theory generalized to take into account the replica symmetry breaking (RSB) effects. Recently it has been shown that the traditional replica-symmetric RG flows in the dimensions D = 4−ǫ are unstable with respect to the RSB potentials and a new spin-glass type critical phenomena has been discovered [11],[12]. In contrast, here it is demonstrated that in the considered two-dimensional systems the renormalization-group flows are stable with respect to the RSB modes. It is shown that the solution of the renormalization group equations with arbitrary starting RSB coupling matrix exhibits asymptotic approach to the traditional replica-symmetric ones. Thus, in the leading order the non-perturbative RSB degrees of freedom does not effect the critical phenomena in the two-dimensional weakly disordered Ising and Baxter models studied earlier.The effects produced by weak quenched disorder on the critical phenomena near the phase transition point have been studied since many years ago [1]- [6]. According to the Harris criterion [1], the disorder effects the critical behaviour only if α, the specific heat exponent of the pure system, is positive. In this case a new universal critical behavior, with new critical exponents, is established sufficiently close to the phase transition point, where u ≪ 1 is the parameter which describes the strength of the disorder. In contrast, when α < 0, the disorder appears to be irrelevant for the critical behavior.Originally the modified critical behaviour has been derived for the classical φ 4 model near four dimensions [2], and later it has been studied for the two-dimensional Ising [3], Baxter [4] and Potts [5] models by various renormalization group (RG) techniques, and by numerical simulations [6].In dealing with the quenched disorder the traditional approach is the replica method, and in terms of replicas all the results obtained for the systems listed above correspond to the so-called replica-symmetric (RS) solutions. Physically it means that olny unique ground state is assumed to be relevant for the observable thermodynamics. The problem, however, is that in the presence of the quenched disorder there exist numerous local minimum configurations separated by finite barriers, and in this case the direct application of the traditional replica-symmetric RG scheme may be questioned.On the other hand, it is the Parisi Replica Symmetry Breaking (RSB) scheme which has been developed specifically for dealing with disordered systems which exhibit numerous local minima states (see e.g. [7]). Recent studies show that besides mean-field theory of spin-glasses the RSB approach can also be generalized for situations where one has to deal with fluctuations as well [8],[9],[10]. In the paper [11] qualitative arguments were presented demonstrating the mechanism how the summation over multiple local minima configurations could provide additional non-trivial RSB inte...

Using the supersymmetry approach, we study spectral statistical properties of a two-dimensional quantum particle subject to a non-uniform magnetic field. We focus mainly on the problem of regularisation of the field theory. Our analysis begins with an investigation of the spectral properties of the purely classical evolution operator. We show that, although the kinetic equation is formally time-reversible, density relaxation is controlled by irreversible classical dynamics. In the case of a weak magnetic field, the effective kinetic operator corresponds to diffusion in the angle space, the diffusion constant being determined by the spectral resolution of the inhomogeneous magnetic field. Applying these results to the quantum problem, we demonstrate that the low-lying modes of the field theory are related to the eigenmodes of the irreversible classical dynamics, and the higher modes are separated from the zero mode by a gap associated with the lowest density relaxation rate. As a consequence, we find that the long-time properties of the system are characterised by universal Wigner-Dyson statistics. For a weak magnetic field, we obtain a description in terms of the quasi one-dimensional non-linear σ-model.

Based on the analogy with the quantum mechanics of a particle propagating in a complex potential, we develop a field-theoretical description of the statistical properties of a self-avoiding polymer chain in a random environment. We show that the account of the non-Hermiticity of the quantum Hamiltonian results in a qualitatively different structure of the effective action, compared to previous studies. Applying the renormalisation group analysis, we find a transition between the weak-disorder regime, where the quenched randomness is irrelevant, and the strong-disorder regime, where the polymer chain collapses. However, the fact that the renormalised interaction constants and the chiral symmetry breaking regularisation parameter flow towards strong coupling raises questions about the applicability of the perturbative analysis.