Symmetry-Protected Topological Edge Modes and Emergent Partial Time-Reversal Symmetry Breaking in Open Quantum Many-Body Systems

Symmetry-protected topological edge modes are one of the most remarkable phenomena in topological physics. Here, we formulate and quantitatively examine the effect of a quantum bath on these topological edge modes. Using the density matrix renormalization group method, we study the ground state of a composite system of spin-1 quantum chain, where the system and the bath degrees of freedom are treated on the same footing. We focus on the dependence of these edge modes on the global and partial symmetries of system-bath coupling and on the features of the quantum bath. It is shown that the time-reversal symmetry (TRS) plays a special role for an open quantum system, where an emergent partial TRS breaking will result in a TRS-protected topological mode diffusing from the system edge into the bath, thus make it useless for quantum computation.

Figure 1

Schematic of the symmetry-protected topological systems (AKLT state) coupled to different quantum baths via various SB couplings.

Figure 2

Finite size scaling of the magnetization on the edge site of the system (upper panels) and within the bath chain (lower panels) for the AKLT model coupled to the $XX$ model via SB coupling: (a) $H_{sb}^{(a)}$ and (b) $H_{sb}^{(b)}$ in the presence of various small local magnetic fields $h_z$ (the results of $h_z=0$ are obtained via $h_z\to 0$ extrapolations). (c) Spatial distribution of magnetization in the ground state of the composite system with $L_s=L_b=32$ and different SB couplings (square for $H_{sb}^{(a)}$, circle for $H_{sb}^{(b)}$, and triangle for $H_{sb}^{(c)}$). $J_b=0.1J$ for (a)–(c) and $J^{\prime }=0.02J$ for (a) and $J^{\prime }=0.2J$ for (b) and (c).

Figure 3

The spatial distribution of the magnetization in a SPT system coupled to a (a) trivial gapped bath and (b) topologically gapped bath via different SB couplings (square for $H_{sb}^{(a)}$, circle for $H_{sb}^{(b)}$, and triangle for $H_{sb}^{(c)}$). (c) Entanglement spectrum in the SB coupling bond for the SB coupling $H_{sb}^{(a)}$ (left panel) and $H_{sb}^{(b)}$ (right panel). (d) Magnetization on the edge of the system $m_{L_s}$ as a function of the small local magnetic field $h_z$. We choose $D=0.5J$ for (a) and $L_s=L_b=32$, $J_b=0.1J$, and $J^{\prime }=0.2J$ for (a)–(d).