Spectra of Hamiltonians with generalized single-site dynamical disorder

Abstract

Starting from the deformed commutation relationsa _q (t) a ^q _† (s)−q a ^q _† (s) a _q (t)=Γ(t−s)1, −1≦q≦1 with a covariance Γ(t−s) and a parameterq varying between −1 and 1, a stochastic process is constructed which continuously deforms the classical Gaussian and classical compound Poisson process. The moments of these distinguished stochastic processes are identified with the Hilbert space vacuum expectation values of products of$\hat \omega _q (t) = \gamma (a_q (t) + a_q^\dag (t)) + \xi a_q^\dag (t)a_q (t)$ with fixed parametersq, γ and ξ. Thereby we can interpolate between dichotomic, random matrix and classical Gaussian and compound Poisson processes. The spectra of Hamiltonians with single-site dynamical disorder are calculated for an exponential covariance (coloured noise) by means of the time convolution generalized master equation formalism (TC-GME) and the partial cumulants technique. The final result for the spectral function is given as aq-dependent infinite continued fraction. In the case of the random matrix processes the infinite continued fraction can be summed up yielding a self-consistent equation for the one-particle Green function.