Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model

Abstract

We establish the connection between a multichannel disordered model --the 1D Dirac equation with $N\times N$ matricial random mass-- and a random matrix model corresponding to a deformation of the Laguerre ensemble. This allows us to derive exact determinantal representations for the density of states and identify its low energy ($\varepsilon\to0$) behaviour $ρ(\varepsilon)\sim|\varepsilon|^{α-1}$. The vanishing of the exponent $α$ for $N$ specific values of the averaged mass over disorder ratio corresponds to $N$ phase transitions of topological nature characterised by the change of a quantum number (Witten index) which is deduced straightforwardly in the matrix model.

7+4 pages, 9+1 pdf figures ; v2: paper reorganised, discussion of non-isotropic case added