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...

We investigate dynamical evolution of a topological memory that consists of two p-wave superconducting wires separated by a non-topological junction, focusing on the primary errors (i.e., qubit-loss) and secondary errors (bit and phase-flip) that arise due to non-adiabaticity. On the question of qubit-loss we examine the system's response to both periodic boundary driving and deliberate shuttling of the Majorana bound states. In the former scenario we show how the frequency dependent rate of qubit-loss is strongly correlated with the local density of states at the edge of wire, a fact that can make systems with a larger gap more susceptible to high frequency noise. In the second scenario we confirm previous predictions concerning super-adiabaticity and critical velocity, but see no evidence that the coordinated movement of edge boundaries reduces qubit-loss. Our analysis on secondary bit flip errors shows that it is necessary that non-adiabaticity occurs in both wires and that inter-wire tunnelling be present for this error channel to be open. We also demonstrate how such processes can be minimised by disordering central regions of both wires. Finally we identify an error channel for phase flip errors, which can occur due to mismatches in the energies of states with bulk excitations. In the non-interacting system considered here this error systematically opposes the expected phase rotation due to finite size splitting in the qubit subspace. PACS numbers: 74.78.Na 74.20.Rp 03.67.Lx 73.63.Nm arXiv:1905.06923v1 [cond-mat.mes-hall] 16 May 2019 * For further information contact Aaron.Conlon@mu.ie 1 A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001). 2 A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303 2 (2003) 3 A. Y. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321 2 (2006).

Endowing mesh routers with multiple radios is a recent solution to improve the performance of wireless mesh networks. The consequent problem to assign channels to radios has been recently investigated and its relation to the routing problem has been revealed. The joint channel assignment and routing problem has been shown to be NP-complete and hence mainly heuristics have been proposed. However, such heuristics consider wireless links just like wired links, whereas disregarding their peculiar features. In this paper, we consider the impact of tuning the transmission power and rate of the wireless links on the efficiency of the channel assignment. Then, we present a channel, power and rate assignment heuristic and compare its performance to previously proposed algorithms

We investigate the existence, normalization and explicit construction of edge zero modes in topologically ordered spin chains. In particular we give a detailed treatment of zero modes in a Z 3 generalization of the Ising/Kitaev chain, which can also be described in terms of parafermions. We analyze when it is possible to iteratively construct strong zero modes, working completely in the spin picture. An important role is played by the so called total domain wall angle, a symmetry which appears in all models with strong zero modes that we are aware of. We show that preservation of this symmetry guarantees locality of the iterative construction, that is, it imposes locality conditions on the successive terms appearing in the zero mode's perturbative expansion. The method outlined here summarizes and generalizes some of the existing techniques used to construct zero modes in spin chains and sheds light on some surprising common features of all these types of methods. We conjecture a general algorithm for the perturbative construction of zero mode operators and test this on a variety of models, to the highest order we can manage. We also present analytical formulas for the zero modes which apply to all models investigated, but which feature a number of model dependent coefficients.