Dissipative preparation of Chern insulators

Engineered dissipation can be employed to prepare interesting quantum many-body states in a nonequilibrium fashion. The basic idea is to obtain the state of interest as the unique steady state of a quantum master equation, irrespective of the initial state. Due to a fundamental competition of topology and locality, the dissipative preparation of gapped topological phases with a nonvanishing Chern number has so far remained elusive. Here, we study the open quantum system dynamics of fermions on a two-dimensional lattice in the framework of a Lindblad master equation. In particular, we discover a mechanism to dissipatively prepare a topological steady state with nonzero Chern number by means of short-range system bath interaction. Quite remarkably, this gives rise to a stable topological phase in a nonequilibrium phase diagram. We demonstrate how our theoretical construction can be implemented in a microscopic model that is experimentally feasible with cold atoms in optical lattices.

Figure 1

Density plot of the Berry curvature $\mathcal{F}$ [half of integrand in Eq. (7)] as a function of momentum for $\beta =-4.0,d=g=0$ (left), $\beta =-3.0,d=g=0$ (center), and $\beta =-3.0,d=0.7,g=2.0$ (right). Performing the integral in Eq. (7) for the plotted function gives $\mathcal{C}=-1$ (left), $\mathcal{C}=0$ (center), and $\mathcal{C}=-1$ (right), respectively. In the central plot, the peak (dark region) around $k=0$ compensates the smooth negative curvature away from the center. In the right plot the dissipative hole-plugging mechanism depletes the central peak thus maintaining the nonvanishing Chern number.

Figure 2

Left panel: ${\widehat{n}}_k^3$ as a function of $k_x$ at $k_y=0$ for $\delta =0,d=g=0$ (green dotted), for $\delta =0.5,d=g=0$ (red dashed), and for $\delta =0.5,d=0.7,g=1.0$ (blue solid). Right panel: purity gap $p=|{\vec{n}}_k{|}^2$ of the steady state ${\rho }_k^s$ as a function of $k_x$ at $k_y=0,\delta =d=0.2$. Gap at $g=0.1$ (blue dotted) in the topologically trivial phase, purity critical point at $g=0.2$ (red dashed), and gap at $g=1.0$ (green solid) in the nontrivial phase. Inset: phase diagram of the steady state as a function of $\delta =-4-\beta$ and $g$. $d=1.0$ is fixed. The purple region has Chern number $\mathcal{C}=-1$, while $\mathcal{C}=0$ in the bright region. The purity gap closes at the transition lines. The damping gap is finite everywhere except at the critical point $\delta =g=0$.

Figure 3

${\widehat{n}}_k^3=-\text{Tr}\left\{{\rho }_k^s{\tau }_3\right\}$ as a function of $k_x$ at $k_y=0$. Red dashed plot for $\delta =-4-\beta =0.5$ in the presence of ${\ell }_j^C$ only. Blue solid plot shows the self-consistent solution of Eq. (15) for $\delta =0.5,g_u=g_v=5.0,d_u=0.5,d_v=1.5$ in the presence of both ${\ell }_j^C$ and ${\ell }_j^A$ on a lattice of $501\times 501$ sites.