Entanglement Spectrum of a Disordered Topological Chern Insulator

We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level ($E_F$) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number ($C$) via a new, efficient real-space formula.

Figure 1

EnS and level statistics for a disordered (A) CI ($\zeta =0.3i$) and (B) trivial insulator ($\zeta =0.3$), at disorder strengths $W=$ (a) 3, (b) 5, (c) 8, and (d) 11. Panel  (A) also shows the phase diagram of the disordered CI inferred from panels (a)–(d), and a few histograms of the level spacings recorded at the indicated energies. The histograms are compared with $P_{\mathrm{GUE}}(s)$ [light grey (blue) line] and $P_{\mathrm{Poisson}}(s)$ [grey (red) lines] distributions. The light grey (blue) lines overlaying the EnS are the variances of the energy-spacing distributions plotted on a scale between 0 and 1. The dashed light grey (blue) line represents the variance ($\sim 0.178$) of $P_{\mathrm{GUE}}$.

Figure 2

(A) EtS and the variance of the level spacings (left panels), EnS (center panel) and $C$ number (right panels) for (A) $\zeta =0.3i$ CI and (B) $\zeta =0.3$ normal insulator both with $W=3$. The EtS and $C$ were computed for seven $E_F$’s as indicated. The light grey and grey (blue and red) data correspond to calculations on $30\times 30/40\times 40$ lattices. $C$ was identically zero for the normal insulator. The vertical dotted line in the EtS plots marks the 0.178 value. Panels (1–3) show the histograms of the level spacings collected at the entanglement energies marked by arrows. Overlaid is the Wigner-Dyson distribution $P_{\mathrm{GUE}}(s)$.

Figure 3

(a) Entanglement spectrum for a translationally invariant CI plotted versus momentum along the cut. (b) Level repulsion of the edge entanglement spectrum with disorder. Inset shows the full EtS.