In this paper, we explore the decoherence dynamics of a probing spin coupled to a spin bath, where the spin bath is given by a controllable 1D transverse-field Ising chain. The 1D transverse-field Ising chain with free-end boundary condition is equivalent to a modified Kitaev model with nonlocal Majorana bound states in its topological phase. We find that the non-Markovian decoherence dynamics of the probing spin can manifest the topological structure of the spin chain. By controlling the external magnetic field on the Ising chain, we find a close relationship between the topological phase transition and the non-Markovian dynamics in the real-time domain. We also investigate the corresponding quantum entanglement dynamics in this topological system.
We gauge the fermion parity symmetry of the Kitaev chain. While the bulk of the model becomes an Ising chain of gauge-invariant spins in a tilted field, near the boundaries the global fermion parity symmetry survives gauging, leading to local gauge-invariant Majorana operators. In the absence of vortices, the Higgs phase exhibits fermionic symmetry-protected topological (SPT) order distinct from the Kitaev chain. Moreover, the deconfined phase can be stable even in the presence of vortices. We also undertake a comprehensive study of a gently gauged model which interpolates between the ordinary and gauged Kitaev chains. This showcases rich quantum criticality and illuminates the topological nature of the Higgs phase. Even in the absence of superconducting terms, gauging leads to an SPT phase which is intrinsically gapless due to an emergent anomaly.
Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi-one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to
ln(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such a topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi-1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.