Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator

TL;DR

Addresses the problem of defining topological invariants in inhomogeneous systems lacking translational symmetry. Proposes a real-space Chern number based on a Wilson loop computed from corner overlaps in a large supercell, showing equivalence to the Bott index and computational efficiency. Demonstrates this framework by studying a disordered Rice-Mele–derived Chern insulator, revealing that normal disorder can drive a nontrivial-to-trivial transition while polarized disorder largely preserves nontrivial topology; these findings are supported by linear conductance and density-of-states analysis. This work provides a practical, scalable toolkit for identifying and characterizing topological phases in disordered materials.

Abstract

In this paper, we formulate the real-space Chern number in a supercell framework. In this framework, the overlap matrix between two corners of the Brillouin zone (BZ) is derived from diagonalizing the real-space Hamiltonian with periodic boundary conditions. The pass-ordered product of overlap matrices around the BZ boundary forms a Wilson loop, and defines the Chern number in real space. It is analytically shown that the real-space Chern number is quantized at integers for large enough systems and coincides with the Bott index used in the previous studies. The formulation is greatly simplified for the former so that it makes numerical computations more efficient. The real-space formula is used to numerically elucidate topological phases in a disordered Chern insulator. The Chern insulator is modeled by dimensional extension of the Rice-Mele (RM) model consisting of two sublattices, and is disordered by including a random onsite potential. As disorder strength increases, the nontrivial-to-trivial phase transition takes place for normal disorder with no sublattice polarization. By contrast, the phase diagram is almost unaffected by polarized disorder, indicating that nontrivial topology persists against disorder. These observations are supported by the linear conductance and the density of bulk states.

Figures (8)

• Figure 1: Phase diagrams of Chern numbers ${C_ \pm }$ in the parameter space $(v,w)$. The upper two panels show the momentum-space invariants for (a) lower and (b) upper bands. The lower two panels display the real-space invariants for (c) occupied and (d) unoccupied states in the absence of disorder. The parameters used in the calculation are $(t,m,\mu ) = (1,0,0)$. The system size is $N = 40$.
• Figure 2: The upper two panels show energy eigenvalues of (a) $H({k_y})$ and (b) $H({k_x})$ for cylindrical models with open ends. As shown in each figure, two chiral edge modes traverse the gap. The lower two panels display the probability densities of edge modes at momenta (c) ${k_y} = 4\pi /7$ and (d) ${k_x} = 4\pi /7$. In each panel, dots and solid lines represent numerical and analytical results, respectively. The parameters used in the calculation are $(v,w,t,m) = (1,1,1,0)$. The cylinder lengths are ${N_x} = 100$ for (a) and (c), and ${N_y} = 100$ for (b) and (d).
• Figure 3: Real-space Chern numbers ${C_ \pm }$ as a function of system size $N$ in the absence of disorder. The parameters used in the calculation are $(v,w,t,m,\mu ) = (1,1,1,0,0)$.
• Figure 4: Real-space Chern number ${C_ - }$ in the parameter space $(v,w)$ in the presence of disorder. The left two panels show ${C_ - }$ for normal disorder at (a) $W = 1$ and (b) $W = 10$. The right two panels display ${C_ - }$ for polarized disorder at (c) ${W_A} = 1$ and (d) ${W_A} = 10$. The parameters used in the calculation are $(t,m,\mu ) = (1,0,0)$. The system size is $N = 40$.
• Figure 5: Real-space Chern numbers ${C_ \pm }$ as a function of disorder strength. Two panels show the numerical results for (a) normal disorder and (b) polarized disorder, where the disorder strengths are denoted by $W$ and ${W_A}$, respectively. The parameters used in the calculation are $(v,w,t,m,\mu ) = (1,1,1,0,0)$. The system size is $N = 100$.