Non-Anderson critical scaling of the Thouless conductance in 1D

Abstract: We propose and investigate numerically a one-dimensional model which exhibits a non-Anderson disorder-driven transition. Such transitions have recently been attracting a great deal of attention in the context of Weyl semimetals, one-dimensional systems with long-range hopping and high-dimensional semiconductors. Our model hosts quasiparticles with the dispersion $\pm |k|^\alpha \mathrm{sign} k$ with $\alpha<1/2$ near two points (nodes) in momentum space and includes short-range-correlated random potential which allows for scattering between the nodes and near each node. In contrast with the previously studied models in dimensions $d<3$, the model considered here exhibits a critical scaling of the Thouless conductance which allows for {an accurate} determination of the critical properties of the non-Anderson transition, with a precision significantly exceeding the results obtained from the critical scaling of the density of states, usually simulated at such transitions. We find that in the limit of the vanishing parameter $\varepsilon=2\alpha-1$ the correlation-length exponent $\nu=2/(3|\varepsilon|)$ at the transition is inconsistent with the prediction $\nu_{RG}=1/|\varepsilon|$ of the perturbative renormalisation-group analysis. Our results allow for a numerical verification of the convergence of $\varepsilon$-expansions for non-Anderson disorder-driven transitions and, in general, interacting field theories near critical dimensions.