In this paper, we generalized the Peschel-Emery line of the interacting transverse field Ising model to a model based on three-state clock variables. Along this line, the model has exactly degenerate ground states, which can be written as product states. In addition, we present operators that transform these ground states into each other. Such operators are also presented for the Peschel-Emery case. We numerically show that the generalized model is gapped. Furthermore, we study the spin-S generalization of interacting Ising model and show that along a Peschel-Emery line they also have degenerate ground states. We discuss some examples of excited states that can be obtained exactly for all these models.

We present a detailed study of the phase diagram of the Kitaev-Hubbard chain, that is the Kitaev chain in the presence of a nearest-neighbour density-density interaction, using both analytical techniques as well as DMRG. In the case of a moderate attractive interaction, the model has the same phases as the non-interacting chain, a trivial and a topological phase. For repulsive interactions, the phase diagram is more interesting. Apart from the previously observed topological, incommensurate and charge density wave phases, we identify the 'excited state charge density wave' phase. In this phase, the ground state resembles an excited state of an ordinary charge density phase, but is lower in energy due to the frustrated nature of the model. We find that the dynamical critical exponent takes the value z 1.8. Interestingly, this phase only appears for even system sizes, and is sensitive to the chemical potential on the edges of the chain. For the topological phase, we present an argument that excludes the presence of a strong zero mode for a large part of the topological phase. For the remaining region, we study the time dependence of the edge magnetization (using the bosonic incarnation of the model). These results further expand the region where a strong zero mode does not occur. arXiv:1911.03156v2 [cond-mat.str-el]

We present an analytical solution for the full spectrum of Kitaevʼs one-dimensional p-wave superconductor with arbitrary hopping, pairing amplitude and chemical potential in the case of an open chain. We also discuss the structure of the zero-modes in the presence of both phase gradients and next nearest neighbor hopping and pairing terms. As observed by Sticlet et al, one feature of such models is that in a part of the phase diagram, zero-modes are present at one end of the system, while there are none on the other side. We explain the presence of this feature analytically, and show that it requires some fine-tuning of the parameters in the model. Thus as expected, these 'one-sided' zeromodes are neither protected by topology, nor by symmetry. IntroductionOne of the characteristic features of many topological phases is the presence of gapless boundary modes. The (fractional) quantum Hall states are a prime example [1-3], and their boundary modes provide strong evidence of the topological nature of these states. Another prime example is the Kitaev chain, whose topological p-wave superconducting phase features so-called 'Majorana zero modes' at its edges [4]. Trying to establish the existence of the topological phase is often done by trying to establish the presence of the boundary modes. This has led to strong evidence for the topological phase in for instance strongly spin-orbit coupled nano-wires that are proximity coupled to an s-wave superconductor in the presence of a magnetic field [5-9], or in chains of magnetic ad-atoms [10][11][12][13]. It has been proposed that the zero energy Majorana bound states can be used as topologically protected q-bits, for quantum information processing purposes [14,15]. By now, there exist various proposals to manipulate these q-bits, either in T-junction systems, in which the Majorana bound states can be braided explicitly [16], or in Josephson coupled Kitaev chains, in which the coupling of the various chains allows operation on the q-bits [17].Despite the intense research on the Kitaev chain models, there are still interesting features that deserve attention. In this paper, we look into one of them. It was observed by Sticlet et al [18], that the zero-modes of Kitaev chains carrying a current, i.e., in the presence of a gradient in the phase of the order parameter, have interesting properties. The most striking feature is that is it possible that at one edge of the chain, there is pair of Majorana bound states (or better, one 'ordinary' Dirac zero mode), while there is no zero mode at the other end of the chain. Clearly, from a topological point of view, this means that the chain is in a trivial phase, but it is nevertheless worthwhile to investigate these zero-modes further. In this paper, we explain the presence of these zero-modes, via an exact solution of the zero modes of an extended Kitaev chain, i.e., in the presence of both complex and next nearest-neighbor hopping an pairing terms. We show that it is necessary to fine tune the couplings in order that these 'one-sid...

We study a tight binding model of \mathbb{Z}_3ℤ3-Fock parafermions with single-particle and pair-hopping terms. The phase diagram has four different phases: a gapped phase, a gapless phase with central charge \boldsymbol{c=2}𝐜=2, and two gapless phases with central charge \boldsymbol{c=1}𝐜=1. We characterise each phase by analysing the energy gap, entanglement entropy and different correlation functions. The numerical simulations are complemented by analytical arguments.

Formation of localized magnetic states in a metallic host is a classic problem of condensed matter physics formalized by P. W. Anderson within the so called single impurity Anderson model (SIAM). The general picture in a host of a simple one-band metal is that a large Hubbard U in the impurity orbital is pre-requisite for the formation of localized magnetic states. In recent years three dimensional (3D) Dirac solids have emerged the hallmark of which is strong spin-orbit interaction. In this work we show that such a strong spin-orbit interaction allows to form localized magnetic states even with small values of Hubbard U . This opens up the fascinating possibility of forming magnetic states with s or p orbital impurities -a different from traditional paradigms of d or f orbital based magnetic moments.