Density-matrix Chern insulators: Finite-temperature generalization of topological insulators

Thermal noise can destroy topological insulators (TI). However, we demonstrate how TIs can be made stable in dissipative systems. To that aim, we introduce the notion of band Liouvillian as the dissipative counterpart of band Hamiltonian, and show a method to evaluate the topological order of its steady state. This is based on a generalization of the Chern number valid for general mixed states (referred to as density-matrix Chern value), which witnesses topological order in a system coupled to external noise. Additionally, we study its relation with the electrical conductivity at finite temperature, which is not a topological property. Nonetheless, the density-matrix Chern value represents the part of the conductivity which is topological due to the presence of quantum mixed edge states at finite temperature. To make our formalism concrete, we apply these concepts to the two-dimensional Haldane model in the presence of thermal dissipation, but our results hold for arbitrary dimensions and density matrices.

Figure 1

Pictorial image of the action of a band Liouvillian $\mathcal{L}={\sum }_{\mathit{k}}{\mathcal{L}}_{\mathit{k}}$. The vertical lines denote the only possible processes involving the (initially empty) conduction band and (initially filled) valence bands, i.e., those where the momentum $\mathit{k}$ is preserved. The violet fog represents some bath at a certain temperature $T$ which mediates such a process [see Eq. (22)] and the plane indicates the Fermi energy $E_F$.

Figure 2

Color map depicting the Chern value in the Haldane model with dissipation, for different values of $\varphi$, $M$, and bath temperature $T$ (in units of $t_2=1$). As $T$ increases, the Chern value decreases (in absolute value), and for $T=0$ we recover the phase diagram obtained by Haldane (Ref. 27). The dashed black lines enclose the region displaying topological order at $T=0$, so that all nonvanishing Chern values are inside of this region for any $T$. Approximately $T=1$ and 5 correspond to less than $10\%$ and $50\%$ of the gap, respectively.

Figure 3

Occupation along the direction ${\mathit{a}}_1$ for all particles with momentum $k_2=\frac{4}{5}\frac{\pi }{|{\mathit{a}}_2|}$, $M=0$, and $\varphi =\pi /2$, for different values of $T$ (in units of $t_2=1$). Note the presence of edge states at finite temperature [Eq. (33)] in the positions 1 and 50 along the direction ${\mathit{a}}_1$. However, as the temperature $T$ significantly increases, the population of the edge modes becomes similar to the population of the bulk modes.

Figure 4

Color map depicting the conductivity Eq. (34) for different values of $\varphi$, $M$, and the bath temperature $T$ (in units of $t_2=1$). As $T$ increases, the conductivity decreases (in absolute value), and for $T=0$ we recover the Chern number result (Fig. 2). The dashed black lines enclose the region displaying topological order at $T=0$. Thus, contrarily to the Chern value, the conductivity is not a topological property for finite $T$, as it does not vanish for every point outside this region for any $T$.

Figure 5

System of coordinates $\{{\mathit{a}}_1,{\mathit{a}}_2\}$ taken to write the Haldane Hamiltonian in real space (B1). The solid white and black circles denote fermions $a$ and $b$, respectively, and the green enclosures highlight the two-site unit cell.

Figure 6

Energy bands of the Hamiltonian (B17) as a function of the momentum $k_2$. The red lines correspond to the edge-state modes. The rest of the parameters are $M=0$, $\varphi =\pi /2$, and $t_1=4$ (in units of $t_2=1$).