We consider a semiconducting nanowire with Rashba spin-orbit interaction subjected to a magnetic field and in the presence of strong electron-electron interactions. When the ratio between Fermi and Rashba momenta is tuned to $1/2$, two competing resonant multiparticle scattering processes are present simultaneously and the interplay between them brings the system into a gapless critical parafermion phase. This critical phase is described by a self-dual sine-Gordon model, which we are able to map explicitly onto the low-energy sector of the ${\mathbb{Z}}_{4}$ parafermion clock chain model. Finally, we show that by alternating regions in which only one of these two processes is present one can generate localized zero-energy parafermion bound states.
We consider a semiconducting nanowire with Rashba spin-orbit interaction subjected to a magnetic field and in the presence of strong electron-electron interactions. When the ratio between Fermi and Rashba momenta is tuned to 1/2, two competing resonant multiparticle scattering processes are present simultaneously and the interplay between them brings the system into a gapless critical parafermion phase. This critical phase is described by a self-dual sine-Gordon model, which we are able to map explicitly onto the low-energy sector of the Z4 parafermion clock chain model. Finally, we show that by alternating regions in which only one of these two processes is present one can generate localized zero-energy parafermion bound states.
We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read-Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors $\ensuremath{\nu}=2/(2m\ensuremath{-}p)$ for odd integers m and p, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge c=1 at radius R=p/2. When p=1 the state is Abelian. The state with p=3 is the Z4 Read-Rezayi state, and the series of $p\ensuremath{\ge}3$ defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model is based on clustering of electrons in groups of four, and is formulated as a two fluid model in which each wire exhibits two phases: a weak clustered phase, where charge e electrons coexist with charge 4e bosons and a strong clustered phase where the electrons are strongly bound in groups of four. The transition between these two phases on a wire is mapped to the critical point of the four-state clock model, which in turn is described by the orbifold conformal field theory. For an array of wires coupled in the presence of a perpendicular magnetic field, strongly clustered wires form a charge 4e bosonic Laughlin state with a chiral charge mode at the edge, but no neutral mode and a gap for single electrons. Coupled wires near the critical state form quantum Hall states with a gapless neutral mode described by the orbifold theory. The coupled wire approach allows us to employ the Abelian bosonization technique to fully analyze the physics of single wire, and then to extract most topological properties of the resulting non-Abelian quantum Hall states. These include the list of quasiparticles, their fusion rules, the correspondence between bulk quasiparticles and edge topological sectors, and most of the phases associated with quasiparticles winding one another.
In this work, we numerically study critical phases in translation-invariant ZN parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a ZN spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translational invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual ZN clock models. For $3\ensuremath{\le}N\ensuremath{\le}6$, we match the numerical results to the known conformal field theory(CFT) identification. We then analyze in detail the phase diagram of a N=3 chain with both nearest and next-nearest-neighbor hopping and six critical phases with central charges being 4/5, 1, or 2 are found. We find continuous phase transitions between c=1 and 2 phases, while the phase transition between c=4/5 and 1 is conjectured to be of Kosterlitz-Thouless type.
Clock-spin models are attracting great interest, due to both their rich phase diagram and their connection to parafermions. In this context, we derive an exact local mapping from clock-spin to fermionic partition functions. Such mapping, akin to techniques introduced by Fedotov and Popov for spin \frac{1}{2} 1 2 chains, grants access to well established numerical tools for the perturbative treatment of fermionic systems in the clock-spin framework. Moreover, in one dimension, it allows to use bosonization to access the low energy properties of clock-spin models. Finally, aside from the direct application in clock-spin models, this new mapping enables the conception of interesting fermionic models, based on the clock-spin counterparts.
We construct a model of a three-dimensional helical second-order topological superconductor formed by an array of weakly coupled Rashba nanowires. We identify the parameter regime in which there are energy gaps in both the bulk and surface energy spectra, while a pair of helical Z 2 q parafermionic modes (with q being an odd integer) remains gapless along a closed path of one-dimensional hinges. The precise trajectory of these hinge modes is dictated by the hierarchy of interwire couplings and the boundary termination of the sample. In the noninteracting limit q = 1 , the system hosts gapless Majorana hinge modes.