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arXiv:2411.08105 · Physical review. B./Physical review. B

Anyon polarons as a window into competing phases of the Kitaev honeycomb model under a Zeeman field

Chuan Chen, Inti Sodemann

Published Jun 3, 2025• 4 citations
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Abstract

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.

DOI
10.1103/xxym-pc1t

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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.
Abstract We investigate the dispersions of anyon quasiparticles in the Kitaev honeycomb spin liquid perturbed by Γ and Γ′{\Gamma }^{{\prime} }Γ′ Γ ′ couplings in order to understand phase transitions into competing states through anyon gap-closing instabilities. We demonstrate how anyon gap closings provide a way to understand phase transitions into a plethora of previously identified competing phases—including zigzag, stripy, 120°, and incommensurate spiral phases—and are in agreement with numerical studies not only on the nature of the phases, but also on the specific critical values of Γ and Γ′{\Gamma }^{{\prime} }Γ′ Γ ′ couplings. Remarkably, when the antiferromagnetic Kitaev model is perturbed by a negative Γ interaction, we find that the single-vison and fermion gaps remain open while the gap of a magnon-like local boson vanishes. This points to a plausible candidate coexistence regime where long-range magnetic order emerges while fractionalized excitations inherited from the Kitaev spin liquid may persist. The magnetic order can be either stripy antiferromagnetic or incommensurate spiral, depending on the sign of Γ′{\Gamma }^{{\prime} }Γ′ Γ ′ .
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.
We investigate the non-Abelian topological chiral spin-liquid phase in the two-dimensional Kitaev honeycomb model subject to a magnetic field. By combining density matrix renormalization group and exact diagonalization we study the energy spectra, entanglement, topological degeneracy, and expectation values of Wilson loop operators, allowing for a robust characterization. While the ferromagnetic Kitaev spin liquid is already destroyed by a weak magnetic field with Zeeman energy ${H}_{*}^{\text{FM}}\ensuremath{\approx}0.02$, the antiferromagnetic (AFM) spin liquid remains robust up to a magnetic field that is an order of magnitude larger, ${H}_{*}^{\text{AFM}}\ensuremath{\approx}0.2$. Interestingly, for larger fields H∗AFM<H<H∗∗AFM{H}_{*}^{\text{AFM}}<H<{H}_{**}^{\text{AFM}}H∗AFM​<H<H∗∗AFM​, an intermediate gapless phase is observed, before a second transition to the high-field partially polarized paramagnet. We attribute this rich phase diagram, and the remarkable stability of the chiral topological phase in the AFM Kitaev model, to the interplay of strong spin-orbit coupling and frustration enhanced by the magnetic field. Our findings suggest relevance to recent experiments on RuCl3{\mathrm{RuCl}}_{3}RuCl3​ under magnetic fields.
Recent numerical studies indicate that the antiferromagnetic Kitaev honeycomb lattice model undergoes a magnetic-field-induced quantum phase transition into a new spin-liquid phase. This intermediate-field phase has been previously characterized as a gapless spin liquid. By implementing a recently developed variational approach based on the exact fractionalized excitations of the zero-field model, we demonstrate that the field-induced spin liquid is gapped and belongs to Kitaev's 16-fold way. Specifically, the low-field non-Abelian liquid with Chern number C = ±1 transitions into an Abelian liquid with C = ±4. The critical field and the field-dependent behaviors of key physical quantities are in good quantitative agreement with published numerical results. Furthermore, we derive an effective field theory for the field-induced critical point which readily explains the ostensibly gapless nature of the intermediate-field spin liquid.
Non-Abelian anyons in quantum spin liquids (QSLs) provide a promising route to fault-tolerant topological quantum computation. In the exactly solvable Kitaev honeycomb model, such anyons of the QSL state can be bound to nonmagnetic spin vacancies and endowed with non-Abelian statistics by an infinitesimal magnetic field. Here, we investigate how this approach for stabilizing non-Abelian anyons extends to a finite magnetic field represented by a proper Zeeman term. Through large-scale density-matrix renormalization group simulations, we compute the vacancy-anyon binding energy as a function of magnetic field for both the ferromagnetic and antiferromagnetic Kitaev models. We find that anyon binding remains robust within the entire QSL phase for the ferromagnetic Kitaev model but breaks down already inside this phase for the antiferromagnetic Kitaev model. To compute a binding energy several orders of magnitude below the magnetic energy scale, we introduce both a refined definition and an extrapolation scheme based on carefully tailored perturbations.
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.