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arXiv:2506.17981 · Physical review. B./Physical review. B

Kitaev model in regular hyperbolic tilings

Julien Vidal, Rémy Mosseri

Published Nov 7, 2025• 7 citations
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Abstract

We study the Kitaev model on regular hyperbolic trivalent tilings. Depending on the length $p$ of the elementary polygons, we examine two distinct tri-colorings of the tiling. Using a recent conjecture on the ground-state flux sector, we compute the phase diagram via exact diagonalizations and derive analytical expressions for the effective Hamiltonians in the isolated-dimer limit which are valid for all values of $p$. Our results interpolate between the Euclidean honeycomb lattice and the trivalent Bethe lattice ($p=\ensuremath{\infty}$) for which we derive the exact solution of the phase boundaries.

DOI
10.1103/mx1t-74dm

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We analyze the Kitaev model on the {9,3} hyperbolic lattice. The {9,3} is formed by a regular tricoordinated tiling of nonagons, where the three-color coding of bonds according to the inequivalent Kitaev Ising spin couplings yields the natural generalization of the original Kitaev model for Euclidean regular honeycomb tiling. Upon investigation of the bulk spectrum for large finite size droplets, we identify a gapless chiral Z_{2} spin liquid state featuring spontaneous time-reversal symmetry breaking. Because of its noncommutative translation group structure, such type of hyperbolic spin liquid is conjectured to feature chiral quasiparticles with a potentially non-Abelian Bloch profile.
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We consider an extension of the Kitaev honeycomb model based on arbitrary dimer coverings satisfying matching rules. We focus on three different dimer coverings having the smallest unit cells for which we calculate the ground-state phase diagram. We also study one- and two-vortex properties for these coverings in the Abelian phases and show that vortex–vortex interactions can be attractive or repulsive. These qualitative differences are confirmed analytically by high-order perturbative expansions around the isolated-dimer limit. Similarities and differences with the original Kitaev honeycomb model are discussed.
These lecture notes focus on the Kitaev honeycomb model, a two-dimensional quantumsystem with three distinct Ising-type interactions, depending on the direction of eachbond. This model has proven particularly important for describing quantum spin liquids,as well as for applications in topological quantum computing, due to its exotictopological properties. It constitutes the first exactly solvable theoretical model thatfully captures the behavior of a quantum spin liquid. We begin by presenting the geometryof the model, the Hamiltonian that describes the system, and Alexei Kitaev's effort toobtain an exact analytical solution, based on the decomposition of the Hilbert spaceinto flux sectors. We then present Ettore Majorana's observation that the Dirac equationalso admits real solutions, leading to the theoretical prediction of Majorana fermions-- particles that are identical to their antiparticles. We examine how the spatialseparation of Majorana operators leads to the rise of unbound Majorana fermions, andshowcase the exotic topological properties that manifest in physical systems such assuperconducting wires. Next, we represent the Hamiltonian in the basis of Majoranafermions, by expressing spin operators using four Majorana operators. The transitionfrom the extended space L~\tilde{\mathcal{L}}L~ to the physical subspace L\mathcal{L}Lis achieved through a gauge transformation of the Z2\mathbb{Z}_2Z2​ group. We thenreformulate the Hamiltonian in the basis of complex bond and matter fermions, transitionto momentum space, and assume that the system lies in the zero-flux sector. Through theBogoliubov--de Gennes transformation, we arrive at the final diagonalized Hamiltonianin the quasiparticle basis. Finally, we represent the possible phases -- the gaplessphase (phase BBB) and the gapped phases (phase AαiA_{\alpha_i}Aαi​​) -- compute thecorresponding energy spectrum, and present the complete phase diagram. By employing thecorrelation function, we show that all phases are characterized by strong quantumentanglement and spin fractionalization, classifying the system as a quantum spin liquid.
We investigate the exact solution of the honeycomb model proposed by Kitaev and derive an explicit formula for the projector onto the physical subspace. The physical states are simply characterized by the parity of the total occupation of the fermionic eigenmodes. We consider a general lattice on a torus and show that the physical fermion parity depends in a nontrivial way on the vortex configuration and the choice of boundary conditions. In the vortex-free case with a constant gauge field we are able to obtain an analytical expression of the parity. For a general configuration of the gauge field the parity can be easily evaluated numerically, which allows the exact diagonalization of large spin models. We consider physically relevant quantities, as in particular the vortex energies, and show that their true value and associated states can be substantially different from the one calculated in the unprojected space, even in the thermodynamic limit.