Parafermionic clock models and quantum resonance

Abstract

We explore the ZN parafermionic clock-model generalisations of the p-wave Majorana wire model. In particular we examine whether zero-mode operators analogous to Majorana zero-modes can be found in these models when one introduces chiral parameters to break time reversal symmetry. The existence of such zero-modes implies N -fold degeneracies throughout the energy spectrum. We address the question directly through these degeneracies by characterising the entire energy spectrum using perturbation theory and exact diagonalisation. We find that when N is prime, and the length L of the wire is finite, the spectrum exhibits degeneracies up to a splitting that decays exponentially with system size, for generic values of the chiral parameters. However, at particular parameter values (resonance points), band crossings appear in the unperturbed spectrum that typically result in a splitting of the degeneracy at finite order. We find strong evidence that these preclude the existence of strong zero-modes for generic values of the chiral parameters. In particular we show that in the thermodynamic limit, the resonance points become dense in the chiral parameter space. When N is not prime, the situation is qualitatively different, and degeneracies in the energy spectrum are split at finite order in perturbation theory for generic parameter values, even when the length of the wire L is finite. Exceptions to these general findings can occur at special "anti-resonant" points. 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 1-6 . These models generalise the Kitaev wire model 7 which exhibits localised unpaired Majorana zero-modes at each end. The recent surge of interest is inspired in part by proposals for their physical realisation and their potential application to universal topological quantum computation 8-10 , something which is not possible with Majorana zero-modes.Clock-like systems of this type have been studied earlier 11,12 , and much is known about the exactly solvable chiral Potts models that occur at special values of the coupling constants (see e.g. Refs 13 and 14). From the perspective of topological quantum computation however, there are a number of interesting open problems surrounding the topological degeneracies and potential zero-modes of the models. In particular, one may ask whether zero-modes exist which result in topological degeneracies throughout the energy spectrum 3,5 . In this context one speaks of strong vs. weak zero-modes (see e.g. Ref. 15) and the question is important because degeneracies that exist at energies above the ground-state potentially allow for topologically fault-tolerant quantum devices at higher temperatures 16 . This area is of course also interesting on a fundamental level, as it addresses if and when decoupled/fr...

Localization enhanced and degraded topological order in interacting p -wave wires

“…For the Z n models, apart from some special cases (e.g. [50]) there are no obvious non-interacting limits. There are however a number of works that point to exactlysolvable/free-fermion ground-states [115,116] Moreover, there are clear indications that special points of clean Z n models, where n is prime, can contain strong-zero-modes.…”

“…The experimental observations in proximity coupled systems are typically well described within a quasi-particle framework (see e.g [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]), suggesting that at temperatures well below the gap, the properties of these systems are stable to imperfect conditions such as electron-electron interactions. Recently there have been efforts to understand the stability of these non-abelian excitations at energies and temperatures well above the topological gap [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. These studies directly relate to the effectiveness of symmetry-protected-topological (SPT) systems as platforms for quantum memories.…”

“…In this manuscript we address these questions from the perspective of many-body-zero-modes [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57] . We focus on the simplest topological superconductor, the Kitaev chain, in the presence of short range interactions and potential disorder.…”

Quantifying the effect of interactions in quantum many-body systems

“…This relation allows to determine the properties of the Z 2 n parafermions in terms of the known physics of Majorana fermions, such us their stability in terms of perturbations. Recently, several studies have focused on investigating the description of the Z 4 model in terms of free fermions [62,63].…”

General phase spaces: from discrete variables to rotor and continuum limits

“…VI). This model is equivalent to a parafermion chain via a nonlocal extension of the Jordan-Wigner transformation [85]. The operator for the former symmetry is simply Z Z Z ⊗K ; we will keep track of this symmetry as the systems travels to the cv and rot phase spaces.…”