Exact quantum spin liquids with topological order on maple-leaf and trellis lattices

Abstract

We construct spin models with bond-dependent anisotropic interactions on the penta-coordinated maple-leaf and trellis lattices, which yield exact $\mathbb{Z}_2$ 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 $\pm π/2$-flux, where the sign is uniform across the system, and every unit hexagon (square) has $0$-flux ($π$-flux) within the parameter space of interest. We map out the topological phase diagram for each model, which reveals parameter regimes hosting $\mathbb{Z}_2$ 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.