Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii-Moriya interaction

The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the 'relative' Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii-Moriya-like (DM) interaction parameterized by phi that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at phi = 0 which belongs to the BDI class is indexed by an integer winding number while the phi not equal 0 daughter Hamiltonian which belongs to class D is specified by a Z(2) index nu = +/- 1. We show that the localization length, in addition to determining Z(2), can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian-which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.