Abstract Fermionic phase space representations are a promising method for studying correlated fermion systems. The fermionic Q-function and P-function have been defined using Gaussian operators of fermion annihilation and creation operators. The resulting phase-space of covariance matrices belongs to the symmetry class D, one of the non-standard symmetry classes. This was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is the type of structure that has led to recent experimental observations of Majorana fermions. Under a unitary transformation, it is possible to express these Gaussian operators using real anti-symmetric matrices and Majorana operators, which are much simpler mathematical objects. We derive differential identities involving Majorana fermion operators and an antisymmetric matrix which are relevant to the derivation of the corresponding Fokker–Planck equations on symmetric space. These enable stochastic simulations either in real or imaginary time. This formalism has direct relevance to the study of fermionic systems in which there are Majorana type excitations, and is an alternative to using expansions involving conventional Fermi operators. The approach is illustrated by showing how a linear coupled Hamiltonian as used to study topological excitations can be transformed to Fokker–Planck and stochastic equation form, including dissipation through particle losses.
We introduce and develop an approach to realizing a topological phase transition and non-Abelian statistics with dynamically induced Floquet Majorana Fermions (FMFs). When the periodic driving potential does not break fermion parity conservation, FMFs can encode quantum information. Quasi-energy analysis shows that a stable FMF zero mode and two other satellite modes exist in a wide parameter space with large quasi-energy gaps, which prevents transitions to other Floquet states under adiabatic driving. We also show that in the asymptotic limit FMFs preserve non-Abelian statistics and, thus, behave like their equilibrium counterparts.
In a one-dimensional (1D) disordered superconducting wire (Kitaev chain), the topological phase hosts Majorana zero modes (MZMs) at its boundaries, while the trivial phase hosts ordinary fermionic states. We identify the
Z2 Pfaffian invariant, defined as the sign of the Pfaffian of the real-space Majorana Hamiltonian matrix, as the precise topological invariant distinguishing these phases in the presence of strong disorder. We demonstrate that this invariant is exactly mirrored in the ground-state entanglement spectrum (ES). By bipartitioning the chain, the reduced density matrix of the ground state can be expressed via an entanglement Hamiltonian determined by the restricted single-particle correlation matrix. The
Z2 invariant is computed as the sign of the Pfaffian of the restricted correlation matrix, tracking the parity of its negative eigenvalues. In the trivial phase, the ES is non-degenerate; in the topological phase, the ES exhibits a universal two-fold degeneracy. The sign flip of this Pfaffian provides a sharp, falsifiable signature of the topological phase transition, robust to local disorder. ## 1. Introduction The Kitaev chain, a one-dimensional (1D) p-wave superconducting wire, provides a paradigmatic model for topological superconductivity. In its topological phase, the wire hosts Majorana zero modes (MZMs)—operators that are their own antiparticles—at its boundaries. In the presence of strong disorder, the distinction between the topological phase and the trivial phase hosting ordinary fermionic states becomes subtle, as momentum-space topological invariants are ill-defined. This raises the question of what precise topological invariant distinguishes these phases in real space, and whether this invariant can be extracted from the ground-state entanglement spectrum (ES) to provide a falsifiable, non-local signature of the topological phase transition. Here, we show that the
Z2 Pfaffian invariant of the real-space M