Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new user$mathbb{Z}_3$ quantum double in a superconducting wire array
122
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that a $mathbb{Z}_3$ quantum double can be realized in an array of superconducting wires coupled via Josephson junctions. With a suitably chosen magnetic flux threading the system, the inter-wire Josephson couplings take the form of a complex Hadamard matrix, which possesses combinatorial gauge symmetry -- a local $mathbb{Z}_3$ symmetry involving permutations and shifts by $pm 2pi/3$ of the superconducting phases. The sign of the star potential resulting from the Josephson energy is inverted in this physical realization, leading to a massive degeneracy in the non-zero flux sectors. A dimerization pattern encoded in the capacitances of the array lifts up these degeneracies, resulting in a $mathbb{Z}_3$ topologically ordered state. Moreover, this dimerization pattern leads to a larger effective vison gap as compared to the canonical case with the usual (uninverted) star term. We further show that our model maps to a quantum three-state Potts model under a duality transformation. We argue, using a combination of bosonization and mean field theory, that altering the dimerization pattern of the capacitances leads to a transition from the $mathbb{Z}_3$ topological phase into a quantum XY-ordered phase. Our work highlights that combinatorial gauge symmetry can serve as a design principle to build quantum double models using systems with realistic interactions.
rate research
Read More
Studies of free particles in low-dimensional quantum systems such as two-leg ladders provide insight into the influence of statistics on collective behaviour. The behaviours of bosons and fermions are well understood, but two-dimensional systems also admit excitations with alternative statistics known as anyons. Numerical analysis of hard-core $mathbb{Z}_3$ anyons on the ladder reveals qualitatively distinct behaviour, including a novel phase transition associated with crystallisation of hole degrees of freedom into a periodic foam. Qualitative predictions are extrapolated for all Abelian $mathbb{Z}_q$ anyon models.
We continue recent efforts to discover examples of deconfined quantum criticality in one-dimensional models. In this work we investigate the transition between a $mathbb{Z}_3$ ferromagnet and a phase with valence bond solid (VBS) order in a spin chain with $mathbb{Z}_3timesmathbb{Z}_3$ global symmetry. We study a model with alternating projective representations on the sites of the two sublattices, allowing the Hamiltonian to connect to an exactly solvable point having VBS order with the character of SU(3)-invariant singlets. Such a model does not admit a Lieb-Schultz-Mattis theorem typical of systems realizing deconfined critical points. Nevertheless, we find evidence for a direct transition from the VBS phase to a $mathbb{Z}_3$ ferromagnet. Finite-entanglement scaling data are consistent with a second-order or weakly first-order transition. We find in our parameter space an integrable lattice model apparently describing the phase transition, with a very long, finite, correlation length of 190878 lattice spacings. Based on exact results for this model, we propose that the transition is extremely weakly first order, and is part of a family of DQCP described by walking of renormalization group flows.
We propose and investigate a simple one-dimensional model for a single-channel quantum wire hosting electrons that interact repulsively and are subject to a significant spin-orbit interaction. We show that an external Zeeman magnetic field, applied at the right angle to the Rashba spin-orbit axis, drives the wire into a correlated spin-density wave state with gapped spin and gapless charge excitations. By computing the ground-state degeneracies of the model with either (anti-)periodic or open boundary conditions, we conclude that the correlated spin-density state realizes a gapless symmetry-protected topological phase, as the ground state is unique in the ring geometry while it is two-fold degenerate in the wire with open boundaries. Microscopically the two-fold degeneracy is found to be protected by the conservation of the magnetization parity. Open boundaries induce localized zero-energy (midgap) states which are described, at the special Luther-Emery point of the model, by Majorana fermions. We find that spin densities at the open ends of the wire exhibit unusual long-ranged correlations despite the fact that all correlations in the bulk of the wire decay in a power-law or exponential fashion. Our study exposes the crucial importance of the long-ranged string operator needed to implement the correct commutation relations between spin densities at different points in the wire. Along the way we rederive the low-energy theory of Galilean-invariant electron systems in terms of current operators.
Recent experiments on a one-dimensional chain of trapped alkali atoms [arXiv:1707.04344] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous $mathbb{Z}_3$ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley $et, ,al.$ [arXiv:cond-mat/0309438]. By symmetry arguments, the transition is expected to be in the universality class of the $mathbb{Z}_3$ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order-disorder transition in these models, and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization group techniques. We use finite-size scaling to determine the dynamical critical exponent $z$ and the correlation length exponent $ u$. Our analysis presents the only known instance of a strongly-coupled transition between gapped states with $z e 1$, implying an underlying nonconformal critical field theory.
We present a study of a simple model antiferromagnet consisting of a sum of nearest neighbor SO($N$) singlet projectors on the Kagome lattice. Our model shares some features with the popular $S=1/2$ Kagome antiferromagnet but is specifically designed to be free of the sign-problem of quantum Monte Carlo. In our numerical analysis, we find as a function of $N$ a quadrupolar magnetic state and a wide range of a quantum spin liquid. A solvable large-$N$ generalization suggests that the quantum spin liquid in our original model is a gapped ${mathbb Z}_2$ topological phase. Supporting this assertion, a numerical study of the entanglement entropy in the sign free model shows a quantized topological contribution.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
