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 × 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 (ε → 0) behaviour ρ(ε) ∼ |ε| 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.

Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

“…Using the connection with the MSDE studied in Ref. [48], we generalize in this paragraph Rider and Valkó's result for β = 1 to both symmetry classes (β = 1 and 2). The matricial process studied in Ref.…”

“…The matricial process studied in Ref. [48], which arises in a different multichannel localization model, is…”

“…The distribution was obtained from the analysis of the related matrix Fokker-Planck equation (the case where the noise η is not isotropic was also considered in [48]). The mapping to our problem can be easily realised : we set…”

Wigner–Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

“…Using the connection with the MSDE studied in reference [61], we generalize in this paragraph Rider and Valkó's result for β = 1 to both symmetry classes (β = 1 and 2). The matricial process studied in reference [61], which arises in a different multichannel localization model, is…”

Wigner time delay and related concepts: Application to transport in coherent conductors

“…The case m(x) = µg = 0 : time delay as a probe for zero mode We now analyse the case m(x) = µg = 0 which was not considered in the literature. Although the spectral properties of the model are invariant under the change of the sign of the mass, this is not the case for the scattering properties : neither for the phase distribution [104] nor for the Wigner time delay distribution, as we demonstrate here. The distribution P L (τ ) has an interesting property : the existence of a limit law is correlated with the choice of the boundary condition at x = 0 and the sign of the average mass m(x) = µg.…”