Regularized universal topological local markers for Dirac systems

TL;DR

This work addresses the limitations of previous universal local markers, notably boundary irregularities and weak encoding of global invariants in disordered settings, by introducing a regularized marker built from the wrapping number $deg[n]$ using periodic-boundary-compatible operators. The local marker $C(r)$, defined through a topological operator, yields a translation-symmetric real-space proxy that reproduces the global invariant when averaged and remains uniformly quantized in clean Dirac-type models. The authors establish explicit equivalences to the Bott index in 2D for symmetry classes A, D, and C, and to the spin Chern number in DIII and AII, unifying these indices within a single framework. They also demonstrate robustness to disorder, with the variance of the local marker peaking at phase boundaries to signal disorder-induced topological transitions, and validate the approach on representative 1D and 2D models. The results offer a practical, translation-invariant real-space diagnostic for topological phases in Dirac systems and suggest pathways to extend beyond Dirac descriptions using local symmetry operators and related constructions.

Abstract

Local markers provide an efficient and powerful characterization of topological features of many systems, especially when the translation symmetry is broken. Recently, a universal topological local marker applicable in different symmetry classes of topological systems is proposed. However, it suffers from irregular behaviors at the boundary and its connection to other topological indexes remains elusive. In this work, we construct regularized universal topological local markers that apply to Dirac systems by utilizing position operators that are compatible with periodic boundary conditions. The regularized local markers eliminate the obstructive boundary irregularities successfully, and give rise to the desired global topological invariants such as the Chern number consistently when integrated over all the lattice sites. Furthermore, the regularized form allows us to establish an explicit connection between the local markers and some other known topological indices in two dimensions. For instance, it turns out to be equivalent to the Bott index in classes A, D, and C, and equivalent to the spin Chern number in classes DIII and AII. We further examine the utility and stability of this new marker in disordered scenarios. We find that its variance shows peaks at the phase boundaries, which promotes it as a useful indicator for detecting disorder-induced topological phase transitions.

Figures (4)

• Figure 1: Regularized universal topological local markers for various models: (a) for 1D SSH model; (b) for 1D Kitaev Chain; (c) for 2D Chern Insulator; and (d) for 2D BHZ model. In (a) and (b), we use a lattice of 100 unite cells; in (c) and (d), we use a lattice of $40\times 40$ unite cells. The parameters used in each model are detailed in Appendix \ref{['sec:app3']}. In all cases, the local markers distribute uniformly and converge to the expected topological invariant representing the corresponding phase.
• Figure 2: Comparison for topological local markers and the Bott index formula to verify Eq. \ref{['equ:barc-bott']}. We use the two-dimensional system of symmetry classes D in this numerical simulation. The parameters are $t=-1,\Delta=0.5$. The inset provides a close-up for partial detail.
• Figure 3: (a)-(b) Phase diagrams of the extended SSH model: (a) Clean system as a function of $m$ and $t_2$ with $t_1=1.0$ and $\Omega=0$, the dashed purple line represent the case $t_2=-2.0$; (b) Disordered phase diagram as a function of $m$ and disorder strength $\Omega$ with $t_1=1.0$ and $t_2=-2.0$. We have averaged over 10 disorder realizations. The two dashed purple lines indicate the cases $m=0.5$ and $m=-3.0$, respectively. $A_1,B_1,C_1$ are three points with specific disorder strength on the line $m=0.5$. The corresponding values of local markers at these representative points are presented in (c); Similarly, $A_2,B_2,C_2$ are three points for the case $m=-3$. (c) and (e): Local markers $C(r)$ at specific disorder strengths $\Omega=3,10$, and $25$ for a representative disorder realization (central 60 sites shown), for $m=0.5$ and $m=-3.0$, respectively. (d) and (f): Global topological invariant $\bar{C}$ (blue curve, obtained by averaging all the local markers) and corresponding variance $\mathrm{Var}(C)$ (orange curve) are plotted as functions of the disorder strength $\Omega$ for $m=0.5$ and $m=-3.0$, respectively. Each data point is averaged over 100 disorder realizations with system size of 1000 unit cells.
• Figure 4: (a) and (b): Phase diagrams of one dimensional D class Kitaev chain. (a) Clean system as a function of $\mu$ and $t$ with $\Delta=2.5$ and $\Omega=0$. The dashed purple line represent the case $t=1.0$; (b) Disordered phase diagram as a function of $\mu$ and disorder strength $\Omega$ with $t=1.0$ and $\Delta=2.5$. We have averaged 10 disorder realizations. The two dashed purple lines indicate the cases $\mu=-0.5$ and $\mu=-2.5$, respectively. $A_1,B_1,C_1$ are three points with specific disorder strength on the line $\mu=-0.5$, and the corresponding values of local markers at these representative points are presented in (c); Similarly, $A_2,B_2,C_2$ are three points for the case $m=-2.5$. (c) and (e): Local markers $C(r)$ at specific disorder strengths $\Omega=3,10,25$ for a representative disorder configuration (central 60 sites shown), for $\mu=-0.5$ and $\mu=-2.5$, respectively. (d) and (f): Global topological invariant $\bar{C}$ (blue curve, obtained by averaging all local markers) and variance $\mathrm{Var}(C)$ (orange curve) are plotted as functions of the disorder strength $\Omega$ for $\mu=-0.5$ and $\mu=-2.5$, respectively. Each data point is averaged over 100 disorder realizations with system size of 1000 unit cells.