Phase Transition of Topological Index driven by Dephasing

Abstract

We study topological insulators under dephasing noise. With examples of both a $2d$ Chern insulator and a $3d$ topological insulator protected by time-reversal symmetry, we demonstrate that there is a phase transition at finite dephasing strength between phases with nontrivial and trivial topological indices. Here the topological index is defined through the correlation matrix. The transition can be diagnosed through the spectrum of the whole correlation matrix or of a local subsystem. Interestingly, even if the topological insulator is very close to the topological-trivial critical point in its Hamiltonian, it still takes finite strength of dephasing to change the topological index, suggesting the robustness of topological insulators under dephasing. We further consider Chern insulators in the presence of real-space disorder, which exhibit a ground-state transition between topological and Anderson insulating phases. We find that even strongly-disordered Chern insulators, close to the critical disorder strength, exhibit robustness with respect to dephasing.

Figures (3)

• Figure 1: Dephasing-driven topological transition in a Chern insulator. (a) Evolution of the spectrum of $H_C=\mathbb{I}/2-C$ as a function of dephasing strength $g$ in the topological phase ($\gamma=1.5$). A gap closes in the spectrum at a critical dephasing strength $g_c$. (b) In the trivial phase ($\gamma=2.5$), the upper and lower bands of the correlation matrix remain separated throughout the evolution. (c) Phase diagram of the dephased Chern insulator on a $12\times12$ torus. Beige region denotes the topological phase with Chern number $C=1$. Navy region denotes the trivial phase ($C=0$). Red line is numerical solution for $g_c$ in the thermodynamic limit (see main text). The critical dephasing strength remains finite as $\gamma\to 2$ and diverges as $\gamma\to 0$. (d) Correlation sub-matrix gap $\Delta_{sub}$ on tori of various sizes. The gap is zero in the topological phase and finite in the trivial phase (see main text). Gray dashed line shows the critical dephasing strength in the thermodynamic limit , $g_c\approx0.918$. (e) The derivative of the sub-matrix gap with respect to $g$ for the same system sizes. The curves intersect at the critical dephasing strength.
• Figure 2: Dephasing-driven topological transition in a 3D TI. The model is given by Eq. (\ref{['eq:3dti_h']}) with $t=1$ and $\lambda=1/8$. The orange region is the strong TI phase with topological invariants $(\nu_0,\nu_1\nu_2\nu_3)=(1,111)$. The blue region is the weak TI phase with invariants $(0,111)$. They are separated by a gapless line at $\delta t=0$ (black). Both phases transition to a trivial phase (green) above some critical $g_c$, shown with a red dashed line. Note that $g_c$ is finite as $\delta t\to 2^-$ and diverges as $\delta t\to-1$.
• Figure 3: Dephased and disordered Chern insulator. (a) Average Chern number as a function of disorder width $\delta$ for tori of various sizes. Average is taken over 400 disorder realizations. The Chern-Anderson transition is smooth for finite system sizes but sharpens as the system size grows. (b-c) Distribution of critical dephasing strengths $g_c$ for different disorder configurations at fixed disorder widths $\delta$ and $\gamma=1.5$. Orange data is on a $6\times6$ torus and blue data is $8\times8$. Gray dashed line is $g_c$ for $\gamma=1.5$ and $\delta=0$. (a) For $\delta=1.5<\delta_c$, we see the modal $g_c>0$. (b) For $\delta=2.1>\delta_c$, the modal $g_c=0$. In both cases, the distribution is bimodal. This feature tells us that even highly disordered Chern insulators are quite robust to dephasing.