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arXiv:2312.14729 · Physical review. B./Physical review. B

Effective models for dense vortex lattices in the Kitaev honeycomb model

David J. Alspaugh, Jean-Noël Fuchs, Anna Ritz-Zwilling, Julien Vidal

Published Mar 4, 2024• 2 citations
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Abstract

We introduce low-energy effective models for dense configurations of vortices in the Kitaev honeycomb model. Specifically, we consider configurations of vortices in which vortex-free plaquettes form triangular lattices against a vortex-full background. Depending on the vortex density, these ``dual'' configurations belong to either one of two families classified by translation and inversion symmetry. As a function of a time-reversal symmetry breaking term, one family exhibits gapped phases with even Chern numbers separated by extended gapless phases, while the other exhibits gapped phases with even or odd Chern numbers, separated by critical points. We construct an effective model for each family, determine the parameters of these models by fitting the integrated density of states, and reproduce energy spectra and Chern numbers of the Kitaev honeycomb model. We also derive phase diagrams and determine these models' validity.

DOI
10.1103/physrevb.109.115107

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We introduce low-energy effective models for dense configurations of vortices in the Kitaev honeycomb model. Specifically, we consider configurations of vortices in which vortex-free plaquettes form triangular lattices against a vortex-full background. Depending on the vortex density, these ``dual'' configurations belong to either one of two families classified by translation and inversion symmetry. As a function of a time-reversal symmetry breaking term, one family exhibits gapped phases with even Chern numbers separated by extended gapless phases, while the other exhibits gapped phases with even or odd Chern numbers, separated by critical points. We construct an effective model for each family, determine the parameters of these models by fitting the integrated density of states, and reproduce energy spectra and Chern numbers of the Kitaev honeycomb model. We also derive phase diagrams and determine these models' validity.
In two dimensions, topological phases of free Majorana fermions coupled to a Z2{\mathbb{Z}}_{2}Z2​ gauge field are known to be classified according to the Chern number $\ensuremath{\nu}\ensuremath{\in}\mathbb{Z}$. Its value mod 16 specifies the type of anyonic excitations. In this paper, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}7$. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.
The Kitaev model is exactly solvable in terms of Majorana fermions hopping on a honeycomb lattice and coupled to a static Z2\mathbb{Z}_2Z2​ gauge field, giving the possibility of πππ-vortices in hexagonal plaquettes. In the vortex-full sector and in the presence of a time-reversal-breaking three-spin term of strength κκκ, the energy spectrum is gapped and the ground state possesses an even Chern number. An isolated vortex-free plaquette acts as a ``dual vortex'' and binds a fermionic mode at finite energy εεε in the bulk gap. This mode is equivalent to two coupled Majorana zero modes located on the same dual vortex. In a continuum approximation, we analytically compute the Majorana wavefunctions and their coupling εεε in the two limits of small or large κκκ. The analytical approach is confirmed by numerical perturbation theory directly on the lattice. The latter is in excellent agreement with the full numerics on a finite-size system. We contrast our results with states bound to an isolated vortex in a topological superconductor with even Chern number.
The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge symmetry. We mapped out the phase diagram of the model and found gapped and gapless spin-liquid phases. While the low-energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in small magnetic fields, where we showed that the ground state has a finite topological entanglement entropy. We argue that the gapped phases could be possibly described by exotic excitations, and their corresponding spectrum is richer than the Ising phase of the Kitaev model.
We present a comprehensive theory of mapping flux excitations, or vortices, to electric and magnetic particles in the toric code phases of generalizations of the Kitaev honeycomb model in two spatial dimensions. Our method, which is formulated with the Majorana fermion representation, utilizes the fusion rule of the Abelian anyons and the physical constraint on the fermion parity, and applies to generic model parameters including any perturbative limit. Not only are we able to reproduce the known mapping scheme in the dimer limit, we also derive the conditions for the invariance of anyon species of individual vortices. We prove that the mapping of anyons is left unchanged by any continuous evolution of model parameters that does not close the fermion gap in both the vortex-free and two-vortex sectors, which enables precise demarcations between multiple regimes associated with different maps within a single phase characterized by a trivial Chern number. We illustrate our theory via extensive computations for a number of selected models, in particular those defined on the square-octagon lattice and the honeycomb lattice with a Kekulé structure. We also demonstrate that distinct mappings of anyons can nevertheless exhibit the same weak symmetry breaking, and further argue that they belong to the same symmetry-enriched topological order.
We study the Kitaev--Heisenberg model on a triangular lattice by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy and spin structure factors, we obtain a ground-state phase diagram of the Kitaev--Heisenberg model. As suggested by previous studies, we find a 120∘^\circ∘ antiferromagnetic (AFM) phase, a Z2\mathbb{Z}_2Z2​-vortex crystal phase, a nematic phase, a dual Z2\mathbb{Z}_2Z2​-vortex crystal phase (the dual counterpart of the Z2\mathbb{Z}_2Z2​-vortex crystal phase), a Z6\mathbb{Z}_6Z6​ ferromagnetic phase, and a dual ferromagnetic phase (the dual counterpart of the $\mathbb{Z}_6 $ ferromagnetic phase). Spin correlations discontinuously change at phase boundaries because of first-order phase transitions. We also study the relation among the von Neumann entanglement entropy, entanglement spectrum, and phase transitions of the model. We find that the Schmidt gap closes at phase boundaries and thus the entanglement entropy clearly changes as well. This is different from the Kitaev--Heisenberg model on a honeycomb lattice, where the Schmidt gap and entanglement entropy are not necessarily a good measure of phase transitions.
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.