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arXiv:2207.09492 · Physical review. B./Physical review. B

Nature of visons in the perturbed ferromagnetic and antiferromagnetic Kitaev honeycomb models

Chuan Chen, Inti Sodemann

Published Jan 10, 2023• 18 citations
arXivHTMLPDF

Abstract

The Kitaev honeycomb model features emergent anyons in the form of a Majorana fermion and a non-Abelian excitation known as the vison. While visons are static in the ideal model, certain perturbations can induce their motion. Here, it is demonstrated that the vison dynamics induced by Zeeman perturbations are sharply distinct in the ferromagnetic versus the antiferromagnetic model. Namely, in the ferromagnetic model the vison has a trivial translational symmetry and zero Berry curvature, whereas in the antiferromagnetic model it experiences a $\ensuremath{\pi}$ flux per unit cell and has two bands with opposite Chern numbers, leading to an intrinsic contribution to the thermal Hall effect.

DOI
10.1103/physrevb.107.045114

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The Kitaev honeycomb model features emergent anyons in the form of a Majorana fermion and a non-Abelian excitation known as the vison. While visons are static in the ideal model, certain perturbations can induce their motion. Here, it is demonstrated that the vison dynamics induced by Zeeman perturbations are sharply distinct in the ferromagnetic versus the antiferromagnetic model. Namely, in the ferromagnetic model the vison has a trivial translational symmetry and zero Berry curvature, whereas in the antiferromagnetic model it experiences a \ensuremathπ\ensuremath{\pi}\ensuremathπ flux per unit cell and has two bands with opposite Chern numbers, leading to an intrinsic contribution to the thermal Hall effect.
A vison is an excitation of the Kitaev spin liquid which carries a Z 2 gauge flux.While immobile in the pure Kitaev model, it becomes a dynamical degree of freedom in the presence of perturbations.We study an isolated vison in the isotropic Kitaev model perturbed by a small external magnetic field h, an off-diagonal exchange interaction Γ, and a Heisenberg coupling J.In the ferromagnetic Kitaev model, the dressed vison obtains a dispersion linear in Γ and h and a fully universal low-T mobility, μ ¼ 6ℏv 2 m =ðk B TÞ 2 , where v m is the velocity of Majorana fermions.In contrast, in the antiferromagnetic (AFM) Kitaev model interference effects suppress coherent propagation and an incoherent Majorana-assisted hopping leads to a T-independent mobility.The motion of a single vison due to Heisenberg interactions is strongly suppressed for both signs of the Kitaev coupling.Vison bands in AFM Kitaev models can be topological and may lead to characteristic features in the thermal Hall effects in Kitaev materials.Furthermore, we argue that vison diffusion leads to universal signatures in quench experiments.
We compute the spectra of anyon quasiparticles in all three super-selection sectors of the Kitaev model (i.e., visons, fermions and bosons), perturbed by a Zeeman field away from its exactly solvable limit, to gain insights on the competition of its non-abelian spin-liquid with other nearby phases, such as the mysterious intermediate state observed in the antiferromagnetic model. Both for the ferro- and antiferro-magnetic models we find that the fermions and visons become gapless at nearly identical critical Zeeman couplings. In the ferromagnetic model this is consistent with a direct transition into a polarized state. In the anti-ferromagnetic model this implies that previous theories of the intermediate phase viewed as a spin liquid with a different fermion Chern number are inadequate, as they presume that the vison gap does not close. In the antiferromagnetic model we also find that a bosonic quasiparticle becomes gapless at nearly the same critical field as the fermions and visons. This boson carries the quantum numbers of an anti-ferromagnetic order parameter, suggesting that the intermediate phase has spontaneously broken symmetry with this order.
We introduce an extension of the Kitaev honeycomb model by including four-spin interactions that preserve the local gauge structure and, hence, the integrability of the original model. The extended model has a rich phase diagram containing five distinct vison crystals, as well as a symmetric π-flux spin liquid with a Fermi surface of Majorana fermions and a sequence of Lifshitz transitions. We discuss possible experimental signatures and, in particular, present finite-temperature Monte Carlo calculations of the specific heat and the static vison structure factor. We argue that our extended model emerges naturally from generic perturbations to the Kitaev honeycomb model.
We microscopically construct an Abelian mutual Chern-Simons lattice gauge theory for magnetic field-tuned topological transitions in the Kitaev model and obtain a complete characterization of the phases, including their quasiparticles. At low fields for both ferro (FM) and antiferromagnetic (AFM) Kitaev interactions, we demonstrate non-Abelian Ising topological order (ITO), and explicitly construct the Majorana anyon as an intrinsic excitation---a twist defect in our Abelian gauge theory. For the AFM case, an Abelian chiral phase appears at intermediate fields with trivial topological order and fermionic bulk excitations. Remarkably, both the ITO phase and the intermediate phase have the same chiral central charge c=1/2,c=1/2,c=1/2, implying no change in the quantized thermal Hall response across the transition. For the FM case, there is a direct transition from ITO to a partially polarized nontopological phase. Our paper completes the proof of Kitaev's original proposal of the low-field ITO, with c=1/2c=1/2c=1/2 going beyond his mean-field arguments by including the crucial effect of the gauge fluctuations, and provides a resolution of the debate surrounding the intermediate field phase in the AFM case.
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 quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum---line node formation interconnecting two Majorana cones. The peculiar change of the Majorana band topology is rationalized by a sign change of the effective Kitaev coupling by the magnetic field, which does not occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away from [001], the two gapless phases become gapped and topologically nontrivial, characterized by nonzero Chern numbers with different signs. The sign change of the Chern number leads to a reversal of the thermal edge current in the half-quantized thermal Hall effect.