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arXiv:1702.01359 · Physical review. B./Physical review. B

Quantum spin liquid with seven elementary particles

Haoyu Wang, Hitesh J. Changlani, Yuan Wan, Oleg Tchernyshyov

Published Apr 18, 2017• 2 citations
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Abstract

We present an exactly solvable model of a quantum spin liquid with Abelian anyons in $d=2$ spatial dimensions. With spins 1/2 on a triangular lattice and six-body interactions, our model has zero spin correlation length and localized elementary excitations like the toric codes of Kitaev and Wen. In contrast to those earlier models, it has more elementary particles---four bosons and three fermions---and higher topological degeneracy of 64 on a torus. Elementary excitations are boson-fermion pairs that come in 12 distinct flavors. We use string operators to expose the topological nature of the model.

DOI
10.1103/physrevb.95.144425

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The spin-1/2 Kitaev model offers the exactly solvable example of quantum spin liquids.Possible material realizations of the spin-1/2 Kitaev systems and the prospect of using the Majorana fermion excitations for quantum computations have revolutionized quantum spin liquids research.Recently it has been suggested that higher-spin, especially spin-1, Kitaev exchange interactions can be realized in a variety of materials.Numerical computations on small clusters indicate that the ground state of the spin-1 Kitaev model may also be a quantum spin liquid.On the other hand, the nature of the ground state remains elusive since the spin-1 model is not exactly solvable in contrast to the spin-1/2 model.In this work, using the quantum-entanglement based tensor network approach, we construct an explicit ground-state wave function for the spin-1 Kitaev model, which is written only in terms of physical spin operators.We establish the existence of distinct topological sectors on a torus by constructing the minimally entangled states in the degenerate ground-state manifold and evaluating topological entanglement entropy.Our results suggest that the ground state of the spin-1 Kitaev model is a gapped quantum spin liquid with Z 2 gauge structure and Abelian quasiparticles.We explain the subtle differences between the spin-1/2 and spin-1 Kitaev quantum spin liquids.
The Z2{Z}_{2}Z2​ topological order in Z2{Z}_{2}Z2​ spin liquid and in the exactly solvable Kitaev toric code model is the simplest topological order for 2 + 1D bosonic systems. More general 2 + 1D bosonic topologically ordered states can be constructed via exactly solvable string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly solvable models and fermionic topologically ordered states.
Abstract Elementary excitations in entangled states such as quantum spin liquids may exhibit exotic statistics different from those obeyed by fundamental bosons and fermions. Non-Abelian anyons exist in a Kitaev spin liquid—the ground state of an exactly solvable model. A smoking-gun signature of these excitations, namely a half-integer quantized thermal Hall conductivity, was recently reported in α-RuCl3. While fascinating, a microscopic theory for this phenomenon remains elusive because the pure Kitaev model cannot display this effect in an intermediate magnetic field. Here we present a microscopic theory of the Kitaev spin liquid emerging between the low- and high-field states. Essential to this result is an antiferromagnetic off-diagonal symmetric interaction which allows the Kitaev spin liquid to protrude from the ferromagnetic Kitaev limit under a magnetic field. This generic model displays a strong field anisotropy, and we predict a wide spin liquid regime when the field is perpendicular to the honeycomb plane.
Abstract Elementary excitations in entangled states such as quantum spin liquids may exhibit exotic statistics different from those obeyed by fundamental bosons and fermions. Non-Abelian anyons exist in a Kitaev spin liquid—the ground state of an exactly solvable model. A smoking-gun signature of these excitations, namely a half-integer quantized thermal Hall conductivity, was recently reported in α-RuCl3. While fascinating, a microscopic theory for this phenomenon remains elusive because the pure Kitaev model cannot display this effect in an intermediate magnetic field. Here we present a microscopic theory of the Kitaev spin liquid emerging between the low- and high-field states. Essential to this result is an antiferromagnetic off-diagonal symmetric interaction which allows the Kitaev spin liquid to protrude from the ferromagnetic Kitaev limit under a magnetic field. This generic model displays a strong field anisotropy, and we predict a wide spin liquid regime when the field is perpendicular to the honeycomb plane.
We find an exact spin liquid state without time reversal symmetry on the kagome lattice with odd number of electrons per unit cell and explicit wave functions for all eigenstates. We also obtain that all spin-spin correlations are zero except trivial cases. We then show that there are anyonic excitations in our model. Finally, we demonstrate the existence of ground state degeneracy and gapless edge states indicating nontrivial topological order in our model. We also label all eiginstates on torus by local and global string operators.
We present a two-dimensional lattice model for quantum spin-1/2 for which the low-energy limit is governed by four flavors of strongly interacting Majorana fermions. We study this low-energy effective theory using two alternative approaches. The first consists of a mean-field approximation. The second consists of a random phase approximation (RPA) for the single-particle Green's functions of the Majorana fermions built from their exact forms in a certain one-dimensional limit. The resulting phase diagram consists of two competing chiral phases, one with Abelian and the other with non-Abelian topological order, separated by a continuous phase transition. Remarkably, the Majorana fermions propagate in the two-dimensional bulk, as in the Kitaev model for a spin liquid on the honeycomb lattice. We identify the vison fields, which are mobile (they are static in the Kitaev model) domain walls propagating along only one of the two space directions.