Topological response in open quantum systems with weak symmetries

Abstract

In open quantum systems, the interaction of the system with its environment gives rise to two types of symmetry: a strong one, where the system's symmetry charge is conserved exactly, and a weak one, where the system can exchange symmetry charge with the environment but still preserve symmetry at the ensemble level. While generic open quantum systems feature weak symmetries only, the symmetry protected topological response for bosonic/spin systems has only been considered in the stricter setup with additional strong symmetries. Here, we address the generic case and demonstrate that weak symmetries alone can protect topological responses that distinguish different phases of matter. For bosonic systems, focusing on one-dimensional mixed states described by locally purifiable density operators, we propose a quantized response characterizing qualitatively distinct phases. It is detectable via the decay behavior of different string order parameters. We illustrate our general results through a noisy Affleck-Kennedy-Lieb-Tasaki model. In particular, we show that the coupling to the environment can induce a phase transition to a state protected by weak symmetries, without a pure-state or strong-symmetry analog.

Figures (2)

• Figure 1: Phase detection for the decohered AKLT state at different noise rate $p$ via string order parameters. Panel (a) shows the normalized string order parameters in a 200-site periodic chain with string length 50. The dashed line is analytically computed in the limit of $N\gg l\gg 1$. The quantized responses $(e^{i\mathcal{Q}(R_x, R_z)},e^{i\mathcal{Q}(R_y, R_z)})$ identify two phases: $(-1,\ -1)$, representing the AKLT phase, and $(-1,\ +1)$, an intrinsic weak-symmetry protected phase (WSPP). These phases manifest as distinct patterns in the normalized string order parameter, with $\mathcal{S}_y^{(n)}=0$ for the former, and $\mathcal{O}(1)$ for the latter. Panel (b) presents the analytically computed decay exponents $\xi_x$ and $\xi_y$ (cf. Eq. \ref{['eq: string']}); their crossing point marks the transition fn_fig2.
• Figure S1: Illustration of symmetry flux insertion. The modular Hamiltonian $K$ is decomposed into four terms: $K_L$ and $K_R$, supported on the left and right intervals, together with $K_{LR}$ and $K_{RL}$, which couple them. The twisted boundary condition is implemented through conjugating $K_{LR}$ by $U_{g_1}^{(R)}$, supported on the right interval. Panel (b) shows how an $N$-site periodic chain can be obtained from an infinite chain by identifying every $N$ sites.