Edge bits in average symmetry protected topological mixed state

Abstract

Edge bit in an average symmetry protected topological (ASPT) mixed state is studied. The state is protected by one strong $Z_2$ and one weak (average) $Z_2$ symmetries. As analogous objects of pure symmetry protected topological (SPT) states, the ASPT possesses edge bits. In particular, the analogous operator response exists, that is, symmetry fractionalization. The fractionalization preserves the presence of the ASPT in the bulk, and the fractionalized edge operators acting on the edge bits of the ASPT. %analogous to the ones in the pure SPTs. In this work, based on the cluster model and by employing Choi mapping, we discuss generic features of the edge bits and numerically clarify the behavior of the edge bits and their robustness for varying decoherence and perturbative interactions. By using an operator-space mutual information (OSMI), we track the flow of quantum correlations between the two edges. Remarkably, even in the ASPT regime, a finite portion of the initial edge-to-edge correlation survives.

Figures (5)

• Figure 1: (a) Schematic phase structure. There are three states in the system under the decoherence $\mathcal{E}^{Z}_{e}$ and the nearest-neighbor $XX$ interactions. (b) Table of behavior of the order parameters of the symmetry fractionalization. The patterns of values of the order parameters can characterize each phase from the viewpoint of the presence of edge operators.
• Figure 2: Behaviors of the order parameters of the symmetry fractionalization. (a) $J_{xx}$-dependence of $M_{\rm FEO}$. (b) $J_{xx}$-dependence of $M_{\rm WFO}$. In (a) and (b), the data for different $p_z$'s are displayed. (c) $J_{xx}$-dependence of $M_{\rm SFO}$ for different $J_{xx}$'s. $L=23$ ($23\times 2$-sites ladder). The system size dependence for (c) is shown in Appendix F.
• Figure 3: $p_z$-dependence of the OSMI. In the upper panel, the partition of the entanglement entropy for the doubled system (ladder geometry). The subsystem $A$ and $B$ includes the edge-sites of the upper and lower Hilbert space. We set $L=23$ ($23\times 2$-sites ladder). The system size dependence for $J_{xx}=0$ is shown in Appendix F.
• Figure A1: (a) System-size dependence of the symmetry fractionalization order parameter $M_{\rm SFO}$ with $J_{xx}=0$. (b) System-size dependence of the OSMI $I^{\rm OS}(A,B)$ with $J_{xx}=0$. The blue and gley lines represent $I^{\rm OS}(A,B)=2\ln 2$ and $\ln 2$, respectively. The simulation is carried out on the ladder geometry where the total number of sites is $2L$.
• Figure A2: Spacial distribution of the expectation value of $Z_j$. (a) $p_z=1/2$ and $J_{xx}=0$. (b) $p_z=1/2$ and $J_{xx}=0.4$. (c) $p_z=1/2$ and $J_{xx}=0.8$. For all case, we set $L=23$.