Symmetry-protected control of Liouvillian topological phases via Hamiltonian band topology

Abstract

We establish a symmetry-protected correspondence between band topology of coherent Hamiltonians and Liouvillian spectral winding in Lindblad descriptions of open quantum systems. This allows the Hamiltonian topology to act as a knob for controlling Liouvillian topology and corresponding non-equilibrium dynamics, rather than being passively manipulated by system-environment exchanges. In particular, by exactly solving the Liouvillian spectrum in a class of one-dimensional dissipative lattices, we find that the Hamiltonian band topology constrains the Liouvillian spectral winding and determines the Liouvillian skin effect, provided the Hamiltonian and quantum jump operators respect the same chiral symmetry. We further demonstrate that lattice parity controls the associated bulk-boundary correspondence and the coherence properties of the steady state. Our results unveil a symmetry-enforced topological control of spectral and spatial organization in open quantum systems, providing a unified perspective on topology in Hamiltonian and dissipative dynamics.

Figures (6)

• Figure 1: Schematic of dissipative SSH model. $J_0$ (black single lines) and $J_1$ (black double lines) denote coherent hopping amplitudes. The blue and orange arrows represent two opposite dissipative couplings, $\gamma_0$ and $\gamma_1$. Under OBC, their corresponding non-reciprocal dissipation channels are spatially balanced [i.e., satisfying the symmetry of Eq. \ref{['eq:symmetry_T']}] in an edge-defect lattice with an edge site removed (e.g., $|b,L\rangle$ in the dashed box).
• Figure 2: (a) Topological transition of LSE characterized by the band winding number $W_H$ [red dots in (a1)] and Liouvillian spectrum winding number $W_0$ [blue squares in (a1)]. Under OBC, the transition manifests as the closing of the Liouvillian gap $\Delta\lambda$ (a2) and the jump of normalized average position $\bar{n}/(L-0.5)$ of the steady state between $-1$ and $1$ (a3), both are asymptotically achieved as the system size increases. (b) and (c) show the Liouvillian spectrum under OBC (gray) and PBC (colored). Colors represent different $K$ values of the PBC eigenvalues. (d) and (e) the density matrix of the OBC steady-state $\hat{\rho}_{0}$ in (b) and (c), respectively. (f) and (g) time evolution of diagonal elements of the density matrix, $\rho_{nn}$, for an initial state $|\psi_{\rm ini}\rangle=|a,L/2\rangle$. Green solid lines indicate the average position $\langle n\rangle=\sum_n n\rho_{nn}$. (b), (d), and (f) have $J_0=0.5$, and (c), (e), and (g) have $J_0=2$. Other parameters are $J_1=1$, $\gamma_0=\gamma_1=1.2$, and $L=20$.
• Figure 3: (a) and (b) $\overline n$ and $\xi_c$ in logarithmic coordinates with $J_1=2$ and $\gamma_0=2$ under OBC. Insets show the same quantities for the edge-defect lattice with $|b,L\rangle$ removed [the system center in Eq. \ref{['eq:ave_n']} becomes $n_0=L$]. (c) and (d) the PBC spectra near $\lambda=0$ for the two crosses marked in (a). The spectra show the same winding direction (clockwise), despite that the system possesses opposite edge localization in (a). Insets in (c) and (d) are the full Liouvillian spectra.
• Figure 4: Phase diagram and LSE with chiral-asymmetric dissipation. (a) The average position $\overline{n}$ for different values of $J_0$ and $\gamma_1^{a,a}$. White dash line marks the transition of the band topology characterized by $W_H$. (b) Band winding number $W_H$ (red dots) and Liouvillian spectrum winding number $W_0$ (blue squares) for different values of $\gamma_1^{a,a}$. (c) and (d) the density matrix of the OBC steady-state $\hat{\rho}_{0}$ with $J_0=0.6$ and $J_0=1.4$, respectively, marked by the white crosses in (a). The same localization direction is observed despite that they possess different $W_H$. Other parameters are $J_1=1$, $\gamma_{-1}^{b,b}=1.2$, $L=20$; and $\gamma_1^{a,a}=2.25$ in (b) and (c).
• Figure S1: (a) Topological transition of LSE characterized by the band winding number $W_H$ [red dots in (a1)] and Liouvillian spectrum winding number $W_0$ [blue squares in (a1)]. Under OBC, the transition manifests as the closing of the Liouvillian gap $\Delta\lambda$ (a2) and the jump of average position $\bar{n}$ of the steady state from one end to the other end of the system. (b) and (c) show the Liouvillian spectrum under OBC (gray) and PBC (colored). Colors represent different $K$ values of the OBC eigenvalues. (d) and (e) the density matrix of the OBC steady-state $\hat{\rho}_{0}$ in (b) and (c), respectively. (b), (d) have $J_0=0.5$, and (c), (e) have $J_0=2.5$. The other parameters are $J_{1,+}=1$, $J_{1,-}=0.5$, $\gamma_0=\gamma_1=1.2$, and $L=20$ for OBC.