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

Abstract

Topological index of a system specifies gross features of the system. However, in situations such as strong disorder where by level repulsion mechanism the spectral gap is closed, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined from appropriate use of 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 related Hamiltonians. As a case study, we study a generalized XY model (2XY model) plus a generalized Dziyaloshinskii-Moriya-like (DM) interaction that after fermionization breaks the time-reversal invariance and is parameterized by $\phi$. The {\em parent} Hamiltonian at $\phi=0$ which belongs to BDI class is indexed by integer winding number while the $\phi\ne 0$ {\em daughter} Hamiltonian which belongs to class D is specified by a $Z_2$ index $\nu=\pm 1$. We show that the localization length in addition to determining the $Z_2$ can count the number of Majorana zero modes left over at the boundary of the daughter Hamiltonian -- which are not protected by 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.