Breaking supersymmetry in a one-dimensional random Hamiltonian

TL;DR

This paper investigates how adding a scalar random potential to a one-dimensional supersymmetric Hamiltonian affects its spectral and localization properties, revealing new behaviors such as state migration and finite Lyapunov exponents.

Contribution

It provides a detailed analysis of supersymmetry breaking effects on spectral properties using stochastic and replica methods, including Lifshits tail behavior and extreme eigenvalue statistics.

Findings

States migrate to negative spectrum as supersymmetry is broken.

Lyapunov exponent becomes finite at zero energy when supersymmetry is broken.

Lifshits tail involves competition between two types of randomness.

Abstract

The one-dimensional supersymmetric random Hamiltonian $H_{susy}=-\frac{d^2}{dx^2}+\phi^2+\phi'$, where $\phi(x)$ 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 $\gamma(E)\sim1/|\ln{}E|$ 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 $\sigma$. In the limit $\sigma\ll{g}^3$, a fraction of states $N(0)\sim{g}/\ln^2(g^3/\sigma)$ migrate to the negative spectrum and the Lyapunov exponent reaches a finite value $\gamma(0)\sim{g}/\ln(g^3/\sigma)$ 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 $\phi$ and $V$ whatever the larger is. This analysis relies on analytic results for $N(E)$ and $\gamma(E)$ 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.