We study the mechanism of decay of a topological (winding-number) excitation due to finite-size effects in a two-dimensional valence-bond solid state, realized in an $S=1/2$ spin model ($J$-$Q$ model) and studied using projector Monte Carlo simulations in the valence bond basis. A topological excitation with winding number $|W|>0$ contains domain walls, which are unstable due to the emergence of long valence bonds in the wave function, unlike in effective descriptions with the quantum dimer model. We find that the life time of the winding number in imaginary time diverges as a power of the system length $L$. The energy can be computed within this time (i.e., it converges toward a "quasi-eigenvalue" before the winding number decays) and agrees for large $L$ with the domain-wall energy computed in an open lattice with boundary modifications enforcing a domain wall. Constructing a simplified two-state model and using the imaginary-time behavior from the simulations as input, we find that the real-time decay rate out of the initial winding sector is exponentially small in $L$. Thus, the winding number rapidly becomes a well-defined conserved quantum number for large systems, supporting the conclusions reached by computing the energy quasi-eigenvalues. Including Heisenberg exchange interactions which brings the system to a quantum-critical point separating the valence-bond solid from an antiferromagnetic ground state (the putative "deconfined" quantum-critical point), we can also converge the domain wall energy here and find that it decays as a power-law of the system size. Thus, the winding number is an emergent quantum number also at the critical point, with all winding number sectors becoming degenerate in the thermodynamic limit. This supports the description of the critical point in terms of a U(1) gauge-field theory.
Emergent Symmetry and Phase Transitions on the Domain Wall of $\mathbb{Z}_{2}$ Topological Orders
Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences（中国科学院理论物理研究所与拓扑量子计算卓越中心，中国科学院大学）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）
Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Department of Physics, University of California at San Diego（加州大学圣地亚哥分校物理系）；Department of Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校物理系）
Emergent Topological Complexity in the Barabasi-Albert Model with Higher-Order Interactions
University of Mohaghegh Ardabili（莫哈格赫阿尔达比勒大学）；Shahid Beheshti University（沙希德贝赫什蒂大学）；Swiss Federal Institute for Forest, Snow and Landscape Research, WSL（瑞士联邦森林、雪地和景观研究所）
