Classification of symmetry enriched topological phases with exactly solvable models

Abstract

Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this paper we present a non-trivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave-functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wavefunc-tions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the non-approximate study of topological one-dimensional parafermionic chains with spatial-inversion and time-reversal symmetry in the absence of strong edge modes. Introduction. The study of topological order (TO) is currently one of the most active research fields in condensed-matter physics. From the AKLT model [1] to the Laughlin wavefunction [2], from the Kitaev chain [3] to the Toric code [4], this study has always benefited from the development of exactly-solvable models and of paradigmatic wavefunctions, whose detailed analysis permits the formation of a clear physical intuition, to be used in the understanding of complex experimental setups. In this letter we focus on parafermions, a generalization of Majorana fermions [5]. After the experimental clarification that two zero-energy Majorana modes can be localized at the edges of a one-dimensional fermionic wire [6, 7], the possibility of localizing parafermionic modes, and letting them interact, is currently under deep investigation. These excitations cannot appear in strictly one-dimensional spinless fermionic systems [8, 9], but may emerge at the edge of a two-dimensional fractional topological insulator coupled to alternating ferro-magnetic and superconducting materials [5, 10-15], as well as in other nanostructures or models [16-24]. In these setups, one-dimensional chains of interacting parafermions arise, which, in certain circumstances, display TO and edge Z N parafermionic modes [5, 25-34]. Such edge modes are called strong when they commute with the Hamiltonian [35] and thereby generate a N-fold degeneracy in the entire spectrum, and weak when the commutation property and associated degeneracy are restricted to the ground state manifold. TO survives weak perturbations and hosts indistinguishably weak or strong modes [36]. The importance of parafermionic zero-modes for topological quantum computation [37] motivates further investigations of these fractionalized systems. In this letter we provide a non-trivial family of parafermionic models for which the properties of the ground states can be exactly characterized. These models are gapped, display TO, have spatial-inversion and time-reversal symmetries, and feature weak edge modes; they thus belong to the same symmetry class for which weak edge modes have been discussed so far with numerical and perturbative analytical methods [28, 31, 36], with the advantage of being easy to handle. We analytically establish ...

Symmetry-protected topological phases from decorated domain walls

“…One can now gauge the Z 2 symmetry to obtain a topologically ordered state that is distinct from the regular Z 2 gauge theory 38,39,[43][44][45] . In fact, the particle excitations in this theory are semions and anti-semions, and this is termed the double-semion theory.…”

Ground-State Degeneracy of Topological Phases on Open Surfaces

“…We now present here a fascinating example-the Z 3 2 twisted quantum double (TQD) [28][29][30] -that bears more than one set of nontrivial gapped boundary conditions. As a twisted version of the G = Z 3 2 Kitaev model, this model contains 22 distinct anyons.…”

Symmetry enrichment in three-dimensional topological phases

“…SPT phases are short-range entangled [1] with a global symmetry and have been studied intensively in stronglycorrelated bosonic systems . Much progress has also been made in two-dimensional (2D) SETs [36][37][38][39][40][41][42][43][44][45][46][47], which are partially driven by tremendous efforts in quantum spin liquids (QSL) [36,48] that respect a certain global symmetry (e.g., spatial reflection, time-reversal, Ising Z 2 , U(1) and SU(2) spin rotations, etc.). In contrast to SPTs, SETs are long-range entangled [1] and support emergent excitations, such as anyons in 2D systems.…”