We investigate the dynamical interplay between the topological state of matter and a non-Markovian dissipation, which gives rise to a crucial timescale into the system dynamics due to its quantum memory. We specifically study a one-dimensional polaronic topological superconductor with phonon-dressed
p-wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above a critical memory depth, the system can approach a new dressed state of the topological superconductor in dynamical equilibrium with phonons, with nearly full buildup of the Majorana correlation.
We introduce a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory, which is based on a SO(7)_{1}/(G_{2})_{1} coset factorization, leads to a solvable one-dimensional model that is extended to two dimensions using a network construction. In addition to providing a description of the Fibonacci phase without parafermions, our theory predicts a closely related "anti-Fibonacci" phase, whose topological order is characterized by the tricritical Ising model. We show that Majorana fermions can split into a pair of Fibonacci anyons, and propose an interferometer that generalizes the Z_{2} Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.
Majorana Zero Modes, Super-Modular Categories, and the Indistinguishability of
π/2 Berry Phase in Vortex Exchange
The exchange statistics of Majorana zero modes (MZMs) in a two-dimensional topological superconductor are commonly modeled using the Ising anyon theory. However, the intrinsic topological order of a fermionic system is described by a super-modular tensor category (SMC), specifically sIsing, which contains a transparent fermion. We rigorously demonstrate that the modular data of the underlying SMC is insufficient to uniquely determine the braiding statistics of vortices binding MZMs. Fixing the exchange statistics requires specifying a minimal modular extension (MME) of the SMC. When the Ising unitary modular tensor category (UMTC) is chosen as the MME, the vortex exchange statistics exactly match those of Ising anyons, yielding a non-Abelian Berry phase difference of
π/2 between fusion channels. Consequently, this
π/2 phase is not distinguishable from standard Ising anyon braiding in interferometric measurements; rather, it is identical to it. We conclude that distinguishing fermionic topological orders requires probing the specific MME via observables beyond standard anyon interferometry. ## 1. Introduction Majorana zero modes (MZMs) localized at vortices in two-dimensional (2D) topological superconductors are prime candidates for topological quantum computation. The non-Abelian exchange statistics of these vortices are widely described using the Ising anyon theory. However, a 2D topological superconductor is an intrinsically fermionic system, meaning its topological order is not strictly captured by a standard unitary modular tensor category (UMTC). Instead, it is described by a super-modular tensor category (SMC). This raises a fundamental question: can the exchange statistics of MZMs be uniquely derived from the modular data of the underlying SMC? Furthermore, does this derivation predict a non-Abelian Berry phase of
π/2 for vortex exchange that is distinguishable from standard Ising anyon braiding in interferometry? In this preprint, we address t