Free-fermion descriptions of parafermion chains and string-net models

Abstract

Topological phases of matter remain a focus of interest due to their unique properties -fractionalisation, ground state degeneracy, and exotic excitations. While some of these properties can occur in systems of free fermions, their emergence is generally associated with interactions between particles. Here we quantify the role of interactions in general classes of topological states of matter in all spatial dimensions, including parafermion chains and string-net models. Using the interaction distance [Nat. Commun. 8, 14926 (2017)], we measure the distinguishability of states of these models from those of free fermions. We find that certain topological states can be exactly described by free fermions, while others saturate the maximum possible interaction distance. Our work opens the door to understanding the complexity of topological models and to applying new types of fermionisation procedures to describe their low-energy physics.

Fermionized parafermions and symmetry-enriched Majorana modes

“…(More precisely, for any fixed configuration of m a 's the Hamiltonian is quadratic.) This observation connects with the recent work of Meichanetzidis et al [26] that inferred free-fermion eigenstates from an analytic solution of the f = 0 fixed point combined with an interesting numerical diagnostic for the general case [69]. In terms of the clock-model states in Eq.…”

“…Second, Ref. [26] used complementary analytical and numerical methods to infer that the eigenstates of certain Z 4 parafermion chains can be described in terms of free fermions. Third, Zhang and Kane [27] and Orth et al [28] showed that proximitized edge states of a two-dimensional quantum-spin-Hall insulator can support zero modes reminiscent of Z 4 parafermions (see also Refs.…”

Quantifying the effect of interactions in quantum many-body systems

“…The condition D F = 0 corresponds to a system that can be exactly described by free fermions, while D F = 3 − 2 √ 2 is the maximum distance a density matrix can have from a free description. Note that even this value can be attained only for asymptotically large systems [50].…”

Interaction distance in the extended XXZ model

“…Importantly, the optimal free state σ, i.e., the one with the smallest distance from ρ, does not need to be expressed in terms of the original degrees of freedom, c j , that define the Hamiltonian. Moreover, in general the optimal free state may not be unique, although in many cases it was indeed found to be 3,32 . When ρ is chosen to be the reduced density matrix, D F measures the distance of the entanglement spectrum, corresponding to the given state |ψ and the given partition, from the closest possible freefermion entanglement spectrum, {E f k }, given by Eq.…”