Edge-Selective Extremal Damping from Topological Heritage of Dissipative Chern Insulators

One of the most important practical hallmarks of topological matter is the presence of topologically protected, exponentially localized edge states at interfaces of regions characterized by unequal topological invariants. Here, we show that, even when driven far from their equilibrium ground state, Chern insulators can inherit topological edge features from their parent Hamiltonian. In particular, we show that the asymptotic long-time approach of the nonequilibrium steady state, governed by a Lindblad master equation, can exhibit edge-selective extremal damping. This phenomenon derives from edge states of non-Hermitian extensions of the parent Chern insulator Hamiltonian. The combination of (non-Hermitian) topology and dissipation hence allows one to design topologically robust, spatially localized damping patterns.

Figure 1

A Chern insulator coupled to a bath inducing loss or gain on every site. The topology of the Chern insulator can stabilize edge-selective extremal damping, i.e., maximal damping on one edge and minimal damping on the opposite edge, as indicated by the size of the lattice site symbols.

Figure 2

(a) Independent nonzero components of the momentum-resolved NESS density matrix for $m=3$, $\alpha =\beta =1$, $\mu =0$, ${\mathrm{\Gamma }}_0=0$, and ${\mathit{k}}^T=(0.38302,-0.05893)$. The density matrix is ${\rho }_{\mathit{k}}^{\mathrm{NESS}}={\sum }_{a,b,c,d}(x_{ab;cd}+i{y}_{ab;cd})|ab⟩⟨cd|$ with $x_{ab;cd}=x_{cd;ab}$ and $y_{ab;cd}=-y_{cd;ab}$. (b) Corresponding purity $\mathrm{tr}[({\rho }_{\mathit{k}}^{\mathrm{NESS}}{)}^2]$. The purity of a completely mixed state is indicated by the dashed line.

Figure 3

The steady-state current resulting from the mixed-state character. (a) $⟨j_y{⟩}_{\mathrm{NESS}}$ as function of ${\mathrm{\Gamma }}_0$ and ${\mathrm{\Gamma }}_z$ in units of its maximal amplitude $j_y^{\max }$. (b) Cuts along fixed ${\mathrm{\Gamma }}_0$ as indicated in (a). We use $m=3$, $\alpha =\beta =1$, and $\mu =0$, as well as PBCs along $x$ and $y$.

Figure 4

Momentum-resolved purity $\mathrm{tr}[({\rho }_{\mathit{k}}^{\mathrm{NESS}}{)}^2]$ for $\alpha =\beta =1$, ${\mathrm{\Gamma }}_0=0.08$, and ${\mathrm{\Gamma }}_z=1.2$ for $m=2$ in (a) and $m=3$ in (b). The blue dot highlights the momentum associated with a pure state.

Figure 5

Eigenvalues of the damping matrix $X$ for 15 sites and OBC along $x$, combining 50 equidistant values of $k_y$. The dot color indicates the localization of the corresponding eigenstates: bulk states are black, states at the left (right) edge are red (blue). We set $m=0$ in (a), $m=1$ in (b), and use ${\mathrm{\Gamma }}_0=0.08$, ${\mathrm{\Gamma }}_z=1.2$, $\alpha =\beta =1$, and $\mu =0$.

Figure 6

Deviation of the electronic density from its steady-state value as function of time $t$ and site number $j$ for spin $s$, $\mathrm{\Delta }{n}_{js}(t)$, in a system with 15 sites, OBCs along $x$, and PBCs along $y$ after initialization in a homogeneous half filled state at $t=0$. (a) Shows $s=\uparrow$; (b) corresponds to $s=\downarrow$. We set all parameters as in Fig. 5 and fix $k_y=1.629$. The reference density deviation $\mathrm{\Delta }{n}_{\mathrm{ref}}(t)$ is the geometric mean of the largest and smallest density deviations from the steady-state value at time $t$ over all sites and spins.