Topological Phases of Parafermions: A Model with Exactly Solvable Ground States

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 wavefunctions, 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 wavefunctions, 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 ferromagnetic and superconducting materials [5,[10][11][12][13][14][15], as well as in other nanostructures or models [16][17][18][19][20][21][22][23][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][26][27][28][29][30][31][32][33][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], w...

Exact zero modes in twisted Kitaev chains

“…From an experimental point of view, it is essential to investigate the effects of disorder [16][17][18][19][20] and interactions [21][22][23][24][25][26][27][28][29][30][31][32] . Furthermore, various theoretical aspects have been revealed, including the connection with supersymmetry 30,31,[33][34][35] , the generalization to parafermion modes [36][37][38][39][40][41][42][43] , and the construction of topologically invariant defects 44 . The emergence of Majorana zero modes in condensed matter systems was first proposed by Kitaev 45 .…”

Parafermionic clock models and quantum resonance

“…where P d = i σ x i,d and P u = i σ x i,u are the Z 2 symmetry operators (or fermionic parity operators) of the upper and lower chains, one may check directly that the α L and α R given above satisfy the correct parafermionic relations (6)…”

“…Here the evidence points to the existence of strong zero modes and, in the case of the achiral point of the the N = 4 model, we are able to construct these modes exactly. There has recently been growing interest in a class of Z N symmetric one dimensional lattice models known as parafermion chain or quantum clock models [1][2][3][4][5][6] . These models generalise the Kitaev wire model 7 which exhibits localised unpaired Majorana zero-modes at each end.…”

Exact results for a Z3 -clock-type model and some close relatives

“…For instance, it is interesting to note 49 that the construction of the local operators that permute the ground states can be extended to the model of Ref. 35.…”