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arXiv:2509.13057 · Physical review. B./Physical review. B

Nature of the topological transition of the Kitaev model in [111] magnetic field

S. Thiagarajan, C. Watson, Tarkan Yzeiri, Huanzhi Hu, Bruno Uchoa, Frank Krüger

Published Jan 20, 2026• 1 citation
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Abstract

We investigate the nature of the topological phase transition of the antiferromagnetic Kitaev model on a honeycomb lattice in the presence of a magnetic field along the [111] direction. The field opens a topological gap in the Majorana fermion spectrum and leads to a sequence of topological phase transitions before the field-polarized state is reached. At mean-field level, the gap first closes at the three M points in the Brillouin zone, where the Majorana fermions form Dirac cones, resulting in a change of Chern number by three. An odd number of Dirac fermions in infrared is unusual and requires Berry curvature compensation in ultraviolet, which occurs via topological, ringlike hybridization gaps with higher-energy bands. We perform a renormalization-group analysis of the topological phase transition at the three M points within the Yukawa theory, allowing for intravalley and intervalley fluctuations of the spin-liquid bond operators. We find that the latter lead to a breaking of Lorentz invariance and hence a different universality compared with the standard Ising Gross-Neveu-Yukawa class.

DOI
10.1103/jjj6-cx8l

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We investigate the nature of the topological phase transition of the antiferromagnetic Kitaev model on a honeycomb lattice in the presence of a magnetic field along the [111] direction. The field opens a topological gap in the Majorana fermion spectrum and leads to a sequence of topological phase transitions before the field-polarized state is reached. At mean-field level, the gap first closes at the three M points in the Brillouin zone, where the Majorana fermions form Dirac cones, resulting in a change of Chern number by three. An odd number of Dirac fermions in infrared is unusual and requires Berry curvature compensation in ultraviolet, which occurs via topological, ringlike hybridization gaps with higher-energy bands. We perform a renormalization-group analysis of the topological phase transition at the three M points within the Yukawa theory, allowing for intravalley and intervalley fluctuations of the spin-liquid bond operators. We find that the latter lead to a breaking of Lorentz invariance and hence a different universality compared with the standard Ising Gross-Neveu-Yukawa class.
We investigate the nature of the topological quantum phase transition between the gapless and gapped Kitaev quantum spin liquid phases away from the exactly solvable point. The transition is driven by anisotropy of the Kitaev couplings. At the critical point, the two Dirac points of the gapless Majorana modes merge, resulting in the formation of a semi-Dirac point with quadratic and linear band touching directions. We derive an effective Gross-Neveu-Yukawa-type field theory that describes the topological phase transition in the presence of additional magnetic interactions. We obtain the infrared scaling form of the propagator of the dynamical Ising order parameter field and perform a renormalization-group analysis. The universality of the transition is found to be different from that of symmetry-breaking phase transitions of semi-Dirac electrons. However, as in the electronic case, the Majorana fermions acquire an anomalous dimension, indicative of the breakdown of the fractionalized quasiparticle description.
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.
Magnetic fields can give rise to a plethora of phenomena in Kitaev spin systems, such as the formation of nontrivial spin liquids in two and three spatial dimensions. For the original honeycomb Kitaev model, it has recently been observed that the sign of the bond-directional exchange is of crucial relevance for the field-induced physics, with antiferromagnetic couplings giving rise to an intermediate spin liquid regime between the low-field gapped Kitaev spin liquid and the high-field polarized state, which is not present in the ferromagnetically coupled model. Here, by employing a Majorana mean-field approach for a magnetic field pointing along the [001] direction, we present a systematic study of field-induced spin liquid phases for a variety of two and three-dimensional lattice geometries. We find that antiferromagnetic couplings generically lead to (i) spin liquid phases that are considerably more stable in field than those for ferromagnetic couplings, and (ii) an intermediate spin liquid phase which arises from a change in the topology of the Majorana band structure. Close inspection of the mean-field parameters reveal that the intermediate phase occurs due to a field-driven sign change in an `effective' zzz-bond energy parameter. Our results clearly demonstrate the richness of the Majorana physics of the antiferromagnetic Kitaev models, in comparison to their ferromagnetic counterparts.
In this work we study the phase diagram of the Kekul\'e-Kitaev model. The model is defined on a honeycomb lattice with bond-dependent anisotropic exchange interactions, making it exactly solvable in terms of Majorana representation of spins, in close analogy to the Kitaev model. However, the energy spectrum of Majorana fermions has a multiband structure characterized by Chern numbers 0, $\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$. We obtained the phase diagram of the model in the plane of exchange couplings and in the presence of a magnetic field and found chiral topological and trivial spin-liquid ground states. In the absence of magnetic field most of the phase diagram is a trivial gapped phase continuously connected to an Abelian phase, while in the presence of the magnetic field a topological phase arises. Furthermore, motivated by recent thermal measurements on the spin-liquid candidate $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$, we calculated the thermal Hall conductivity at different regimes of parameters and temperatures and found the latter is quantized over a wide range of temperatures.
We unveil an interesting example of topological flat bands of Majorana fermions in quantum spin liquids. We study the Kitaev model on a periodically depleted honeycomb lattice, under a magnetic field within the perturbation theory. The model can be straightforwardly extended while maintaining the exact solvability, and its ground state is a quantum spin liquid as on the honeycomb lattice. As fractionalized excitations, there are unpaired localized Majorana fermions in addition to the itinerant Majorana fermions and Z2{\mathbb{Z}}_{2}Z2​ fluxes. We show that, in the absence of the magnetic field, the Majorana fermions exhibit completely flat bands at zero energy, and by applying the magnetic field, these bands turn into topological flat bands with nonzero Chern number. By varying the anisotropy of the interactions and the magnitude of the magnetic field, we clarify that the system hosts a variety of topological phases that do not appear in the original honeycomb model. We emphasize that the topological flat bands underlying this rich topology come from the hybridization of the Majorana flat bands and unpaired Majorana fermions, which is unique to the flat bands of fractionalized excitations in quantum spin liquids. Our findings would stimulate the exploration of a new class of Kitaev materials exhibiting rich topology arising from topological Majorana flat bands.
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.