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Entanglement and dynamics of diffusion-annihilation processes with Majorana defects

Coupling a many-body system to a thermal environment typically destroys the quantum coherence of its state, leading to an effective classical dynamics at the longest time scales. We show that systems with anyon-like defects can exhibit universal late-time dynamics that is stochastic, but fundamentally non-classical, because some of the quantum information about the state is topologically protected from the environment. Our coarse-grained model describes one-dimensional systems with domain-wall defects carrying Majorana modes. These defects undergo Brownian motion due to coupling with a bath. Since the fermion parity of a given pair of defects is nonlocal, it cannot be measured by the bath until the two defects happen to come into contact. We examine how such a system anneals to zero temperature via the diffusion and pairwise annihilation of Majorana defects, and we characterize the nontrivial entanglement structure that arises in such stochastic processes. Separately, we also investigate simplified "quantum measurement circuits" in one or more dimensions, involving repeated pairwise measurement of fermion parities for a lattice of Majoranas. The dynamics of these circuits can be solved by exact mappings to classical loop models. They yield analytically tractable examples of measurement-induced phase transitions, with critical entanglement structures that are governed by nonunitary conformal fixed points. In the system of diffusing and annihilating Majorana defects, the relaxation to the ground state is analogous to coarsening in a classical 1D Ising model via domain wall annihilation. Here, however, configurations are labeled not only by the defect positions but by a nonlocal entanglement structure. The resulting dynamical process is a new universality class for the coarsening of topological domain walls, whose universal properties can be obtained from an exact mapping.