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arXiv:2103.16625 · Physical review. B./Physical review. B

Z2 topological order and first-order quantum phase transitions in systems with combinatorial gauge symmetry

Kai-Hsin Wu, Zhi-Cheng Yang, Dmitry Green, Anders W. Sandvik, Claudio Chamon

Published Aug 24, 2021• 4 citations
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Abstract

We study a generalization of the two-dimensional transverse-field Ising model, combining both ferromagnetic and antiferromagnetic two-body interactions, that hosts exact global and local ${\mathbb{Z}}_{2}$ gauge symmetries. Using exact diagonalization and stochastic series expansion quantum Monte Carlo methods, we confirm the existence of the topological phase in line with previous theoretical predictions. Our simulation results show that the transition between the confined topological phase and the deconfined paramagnetic phase is of first order, in contrast to the conventional ${\mathbb{Z}}_{2}$ lattice gauge model in which the transition maps onto that of the standard Ising model and is continuous. We further generalize the model by replacing the transverse field on the gauge spins with a ferromagnetic $XX$ interaction while keeping the local gauge symmetry intact. We find that the ${\mathbb{Z}}_{2}$ topological phase remains stable, while the paramagnetic phase is replaced by a ferromagnetic phase. The topological-ferromagnetic quantum phase transition is also of first order. For both models, we discuss the low-energy spinon and vison excitations of the topological phase and their avoided level crossings associated with the first-order quantum phase transitions.

DOI
10.1103/physrevb.104.085145

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We construct spin models with bond-dependent anisotropic interactions on the penta-coordinated maple-leaf and trellis lattices, which yield exact Z2\mathbb{Z}_2Z2​ quantum spin liquids akin to the Kitaev honeycomb model. We characterize the resulting ground states by their flux sectors, Chern numbers, and topological excitations. Using replica exchange quantum Monte Carlo simulations, we find that the gauge fluxes are ordered at sufficiently low temperatures such that every unit triangle has ±π/2\pm π/2±π/2-flux, where the sign is uniform across the system, and every unit hexagon (square) has 000-flux (πππ-flux) within the parameter space of interest. We map out the topological phase diagram for each model, which reveals parameter regimes hosting Z2\mathbb{Z}_2Z2​ and Ising topological orders, and we derive an analytical expression for the mass term of the Majorana fermions, the vanishing of which indicates a transition between these phases. We further establish the correspondence between individual vortices (i.e., flux excitations) and two species of anyons in the dimer limit, overcoming the obstacle faced by degenerate perturbation theory in treating odd-length elementary plaquettes. Interestingly, we find that two dimer limits of the maple-leaf model with distinct assignments of anyon species can be smoothly connected to each other in the vortex-free sector, but they are separated by fermion-gap-closing transitions in certain two-vortex sectors.
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