Fluctuations of random matrix products and 1D Dirac equation with random mass

Abstract

We study the fluctuations of certain random matrix products ${\Pi }_N=M_N\cdots M_2{M}_1$ of $\mathrm{SL}(2,\mathbb{R})$, describing localisation properties of the one-dimensional Dirac equation with random mass. In the continuum limit, i.e. when matrices $M_n$'s are close to the identity matrix, we obtain convenient integral representations for the variance ${\Gamma }_2={\lim }_{N\to \infty }\mathrm{Var}(\ln ||{\Pi }_N||)/N$. The case studied exhibits a saturation of the variance at low energy $\epsilon$ along with a vanishing Lyapunov exponent ${\Gamma }_1={\lim }_{N\to \infty }\ln ||{\Pi }_N||/N$, leading to the behaviour ${\Gamma }_2/{\Gamma }_1\sim \ln (1/|\epsilon |)\to \infty$ as $\epsilon \to 0$. Our continuum description sheds new light on the Kappus-Wegner (band center) anomaly.

LaTeX, 19 pages, 6 pdf figures ; v2: 2 new figures

References in corpus (6)

• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Products of random matrices and generalised quantum point scatterers
• One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
• Localization for one-dimensional random potentials with large local fluctuations

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