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arXiv:2605.20181

Band Structure and topology of a periodically deformed Kitaev honeycomb model

Abdullah Aljishi, Ali AlSwaid, Moayad Ekhwan, Hocine Bahlouli, Raditya Weda Bomantara, Michael Rudolf Vogl

Published May 19, 2026• 0 citations
arXivHTMLPDF

Abstract

Motivated by the growing interest in spin liquids and topological phases, as well as the rise of deformation engineering, we study the combined effects of deformation and magnetic fields on the honeycomb Kitaev model. The Kitaev model, as one of the prototypical and exactly solvable spin liquid-hosting models, serves as a simple platform that demonstrates the rich physics one can expect at the intersection of deformation physics and quantum spin liquids. Our work builds on a simplified solution to the undeformed base model that we present. This simplified solution allows for a straightforward extension of our analysis to the deformed case. After incorporating periodic deformations into the Kitaev model (chosen for its similarity to moiré physics), we investigate the effects of a hexagonally symmetric deformation on the band structure. We find that deformation leads to a smaller Brillouin zone with new band gaps at the edges, indicating the potential for topological transitions. Finally, we introduce a magnetic field to break time-reversal symmetry and thereby allow for non-trivial topology. We find that, under specific parameter conditions, the magnetic field leads to multiple band-gap closings and openings. An investigation into topological properties reveals nontrivial Chern numbers and a plethora of topological transitions. Our results suggest possible thermal Hall or Nernst-type responses. We also suggest a potential bulk measurement approach for he Chern numbers and possible path to physical realization. Most importantly, our results serve as a demonstration of the rich phenomenology that can arise due to the interplay between deformation and spin-liquid physics.

DOI
10.48550/arxiv.2605.20181

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Motivated by the growing interest in spin liquids and topological phases, as well as the rise of deformation engineering, we study the combined effects of deformation and magnetic fields on the honeycomb Kitaev model. The Kitaev model, as one of the prototypical and exactly solvable spin liquid-hosting models, serves as a simple platform that demonstrates the rich physics one can expect at the intersection of deformation physics and quantum spin liquids. Our work builds on a simplified solution to the undeformed base model that we present. This simplified solution allows for a straightforward extension of our analysis to the deformed case. After incorporating periodic deformations into the Kitaev model (chosen for its similarity to moiré physics), we investigate the effects of a hexagonally symmetric deformation on the band structure. We find that deformation leads to a smaller Brillouin zone with new band gaps at the edges, indicating the potential for topological transitions. Finally, we introduce a magnetic field to break time-reversal symmetry and thereby allow for non-trivial topology. We find that, under specific parameter conditions, the magnetic field leads to multiple band-gap closings and openings. An investigation into topological properties reveals nontrivial Chern numbers and a plethora of topological transitions. Our results suggest possible thermal Hall or Nernst-type responses. We also suggest a potential bulk measurement approach for he Chern numbers and possible path to physical realization. Most importantly, our results serve as a demonstration of the rich phenomenology that can arise due to the interplay between deformation and spin-liquid physics.
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.
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 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.
The Kitaev spin liquid, stabilized as the ground state of the Kitaev honeycomb model, is a paradigmatic example of a topological Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid. The fate of the Kitaev spin liquid in presence of an external magnetic field is at present, a subject of intense scrutiny due to recent experimental signatures pointing to a Z2{\mathbb{Z}}_{2}Z2​ topological phase in certain materials, as well as theoretical studies predicting the emergence of a quantum spin liquid phase of debated nature. In this work, we employ hierarchical mean-field theory, a method based on the use of clusters preserving relevant symmetries and short-range quantum correlations, to investigate the quantum phase diagram of the antiferromagnetic Kitaev's model in a [111] magnetic field. By using clusters of 24 sites, we predict that the Kitaev spin liquid transits through two intermediate phases, characterized by emerging stripe and chiral orders, respectively, before entering the trivial partially polarized phase, differing from previous studies. We assess our results by performing exact diagonalization and computing the scaling of different observables, including the many-body Chern number and other topological quantities, thus establishing hierarchical mean-field theory as a method to study models of frustrated quantum magnetism potentially hosting topological quantum spin liquids.
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.
A large part of modern condensed matter concerns non-local or global properties of systems under the broad term of topology. Closely intertwined with certain senses of topology are symmetries of the system, which physically constrain our description and offer non-perturbative insights into material properties. The range of possible phases of matter is greatly enriched by quantum mechanics, including certain types of non-local entanglement that cannot be understood in the usual symmetry-breaking paradigm of local order parameters. Beyond weakly interacting topology we find the stronger notion of \textit{topological order} in correlated systems, a prominent example being the S=12S = \tfrac{1}{2}S=21​ Kitaev honeycomb model in a weak magnetic field.Through a remarkable combination of lattice geometry, strong spin-orbit coupling, and electron interactions, several material candidates have been put forward to realize Kitaev physics, including the 4d4d4d Mott insulator α\alphaα-RuCl3_{3}3​ which offers tantalizing experimental hints of topological order. In this thesis we explore several aspects of Kitaev candidates and related systems under two broad themes. One involves symmetry-informed and constrained observables, including a proposed singularity in a thermodynamic quantity accompanying phase transitions between two Ising topological orders (ITOs). The ITO physics realized by the Kitaev model requires time-reversal symmetry to be broken, and much of the non-trivial physics is found at finite magnetic field. To this end, we study a microscopic spin model appropriate for α\alphaα-RuCl3_{3}3​ and show that it exhibits an interesting field-induced paramagnetic state between the low- and high-field magnetic orders, consistent with recent experimental findings. We determine several properties of a lower-dimensional spin-SSS analogue of the Kitaev model, including a multi-critical point at finite field separating different topological phases. Finally, we derive quantum state purities of entangled photons emitted from a spin-triplet superconducting heterostructure and discuss its use in a new paradigm of entangled spectroscopy.