Breaking supersymmetry in a one-dimensional random Hamiltonian

Abstract

The one-dimensional supersymmetric random Hamiltonian , 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) ∼ 1/ln2E and a delocalization transition related to the behavior of the Lyapunov exponent (inverse localization length) vanishing like γ(E) ∼ 1/|ln E| as E → 0. We study how this picture is affected by breaking supersymmetry with a scalar random potential: H = Hsusy + V(x), where V(x) is a Gaussian white noise of variance σ. In the limit σ ≪ g3, a fraction of states N(0) ∼ g/ln2(g3/σ) migrate to the negative spectrum and the Lyapunov exponent reaches a finite value γ(0) ∼ g/ln(g3/σ) at E = 0. The exponential (Lifshits) tail of the IDoS for E → −∞ is studied in detail and is shown to involve a competition between the two noises ϕ(x) and V(x), 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 nth 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.