Signatures of SPT Physics in 1D Bulk-Translationally Invariant QBLs

This chapter contains the main result of Part II, namely, the discovery of SPT-like bosonic edge zero modes and symmetries that manifest, and persist, in a transient dynamical regime whose duration increases with system size. We begin by spelling out the necessary features any system supporting such entities must possess: Non-trivial spectral topology, bulk-instabilities, and highly non-normal dynamical matrices. We naturally arrive at the mathematical theory of pseudospectra. Following a brief mathematical digression, we proceed to uncover conditions for anomalous transient dynamics in QBLs. Two relevant dynamical phases emerge: an anomalously relaxing phase, and a dynamically metastable phase. The anomalously relaxing phase is characterized by exponential relaxation of generic observables in a two-step manner. The first step is characterized by a decay rate set by the bulk (infinite-size) Lindblad gap, while the second rate is set by the finite-size Lindblad gap which, remarkably, differs from its infinite-size counterpart by a non-zero amount, independent of system-size. A dynamically metastable system is one for which all finite open boundary truncations are dynamically stable, despite possessing an unstable infinite-size limit. Such systems possess bulk instabilities that are suppressed by imposing hard-wall boundaries. The evolution of dynamically metastable systems is characterized by a transient regime whereby generic observables are amplified, followed by an asymptotic exponential relaxation at a rate set by the finite-size Lindblad gap. Both the length of the transient, and the degree of amplification increase with system size. We zoom in on those dynamically stable QBLs that possess non-trivial bulk topological invariants which, we deem topologically metastable. In the simplest cases, these invariants are the winding numbers of the bulk bands. The combined assumptions of dynamical metastability and non-trivial topology facilitate the emergence of pairs of operators consisting of one approximate zero mode and one approximate SG. These modes are Hermitian,macroscopically separated, andcanonically conjugate. We call the members of these pairs Majorana bosons (MBs) due to their similarities with Majorana fermions. Consequentially, we identify the emergence of a manifold of long-lived quasi-steady states alongside MBs. We then explore the implications of a global number symmetry on topologically metastable systems, resulting in the discovery of edge modes, which we call Dirac bosons (DBs), reminiscent of the Dirac fermion edge modes in topological insulators.