Breaking supersymmetry in a one-dimensional random Hamiltonian

Abstract

The one-dimensional supersymmetric random Hamiltonian $H_{susy}=-\frac{d^2}{dx^2}+ϕ^2+ϕ'$, where $ϕ(x)$ is a Gaussian white noise of zero mean and variance $g$, presents particular spectral and localization properties at low energy: a Dyson singularity in the integrated density of states (IDoS) $N(E)\sim1/\ln^2E$ and a delocalization transition related to the behaviour of the Lyapunov exponent (inverse localization length) vanishing like $γ(E)\sim1/|\ln{}E|$ as $E\to0$. We study how this picture is affected by breaking supersymmetry with a scalar random potential: $H=H_{susy}+V(x)$ where $V(x)$ is a Gaussian white noise of variance $σ$. In the limit $σ\ll{g}^3$, a fraction of states $N(0)\sim{g}/\ln^2(g^3/σ)$ migrate to the negative spectrum and the Lyapunov exponent reaches a finite value $γ(0)\sim{g}/\ln(g^3/σ)$ at E=0. Exponential (Lifshits) tail of the IDoS for $E\to-\infty$ is studied in detail and is shown to involve a competition between the two noises $ϕ$ and $V$ whatever the larger is. This analysis relies on analytic results for $N(E)$ and $γ(E)$ obtained by two different methods: a stochastic method and the replica method. The problem of extreme value statistics of eigenvalues is also considered (distribution of the n-th excited state energy). The results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation/creation local rates.

33 pages, LaTeX, 13 eps figures ; 2nd version : refs. added