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 study a supersymmetric model for strongly interacting lattice fermions in the presence of a staggering parameter. The staggering is introduced as a tunable parameter in the manifestly supersymmetric Hamiltonian. We obtain analytic expressions for the ground states in the limit of small and large staggering for the model on the class of doubly decorated lattices. On this type of lattice there are two ground states, each with a different density. In one limit we find these ground states to be a simple Wigner crystal and a valence bond solid state. In the other limit we find two types of quantum liquids. As a special case, we investigate the quantum liquid state on the one dimensional chain in detail. It is characterized by a massless kink that separates two types of order.

Despite the increasing availability of Open Science (OS) infrastructure and the rise in policies to change behaviour, OS practices are not yet the norm. While pioneering researchers are developing OS practices, the majority sticks to status quo. To transition to common practice, we must engage a critical proportion of the academic community. In this transition, OS Communities (OSCs) play a key role. OSCs are bottom-up learning groups of scholars that discuss OS within and across disciplines. They make OS knowledge more accessible and facilitate communication among scholars and policymakers. Over the past two years, eleven OSCs were founded at several Dutch university cities. In other countries, similar OSCs are starting up. In this article, we discuss the pivotal role OSCs play in the large-scale transition to OS. We emphasize that, despite the grassroot character of OSCs, support from universities is critical for OSCs to be viable, effective, and sustainable.