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Journal of the Physical Society of Japan

Exactly Solvable Decorated-Kekulé-lattice Kitaev Model

Tomonari Mizoguchi

Published Jan 23, 2026• 0 citations
Publisher

Abstract

We propose the decorated-Kekulé-lattice Kitaev model, which is exactly solvable at any temperature. Similar to the decorated honeycomb model proposed in our previous work [J. Phys. Soc. Jpn. 91, 124709 (2022)], the exact solvability arises from the fact that the spins on the decorated sites become conserved quantities (i.e., classical variables) and serve as an effective field acting on the remaining spins. For a given sector of the classical variables, the remaining spins are described by the Hamiltonian of ensemble of decoupled two-site transverse Ising models, with eigenenergies being independent of the configuration of the classical variables. This enables us to calculate the partition function exactly, and hence the thermodynamic quantities are obtained at any temperature. Our result indicates that the site decoration for the compass-type models is a ubiquitous method to obtain the models that are exactly solvable at finite temperature.

DOI
10.7566/jpsj.95.024710

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We propose the decorated-Kekulé-lattice Kitaev model, which is exactly solvable at any temperature. Similar to the decorated honeycomb model proposed in our previous work [J. Phys. Soc. Jpn. 91, 124709 (2022)], the exact solvability arises from the fact that the spins on the decorated sites become conserved quantities (i.e., classical variables) and serve as an effective field acting on the remaining spins. For a given sector of the classical variables, the remaining spins are described by the Hamiltonian of ensemble of decoupled two-site transverse Ising models, with eigenenergies being independent of the configuration of the classical variables. This enables us to calculate the partition function exactly, and hence the thermodynamic quantities are obtained at any temperature. Our result indicates that the site decoration for the compass-type models is a ubiquitous method to obtain the models that are exactly solvable at finite temperature.
We study a Kitaev-type compass model on a decorated honeycomb lattice, where the localized spins sit on both vertices and edges of the honeycomb lattice.Key features are that the spins on the edges of the honeycomb lattice are reduced to classical Ising variables, and that they serve as a local field acting on the spins on the vertices.The model is therefore exactly solvable, in that the partition function can be exactly obtained.We show that the ground state is a classical spin liquid, and that the spin-spin correlation is finite-ranged for arbitrary temperatures.We also consider its extension by adding a specific type of the next-nearest-neighbor exchange interaction in such a way that the spins on the edges of the honeycomb lattice remain to be classical Ising variables.We reveal characteristic features of the specific heat and the spin-spin correlation induced by the additional term.
In the prior work [J. Phys. Soc. Jpn. 91 , 124709 (2022)], we have introduced the spin-half decorated honeycomb Kitaev model, and have elucidated its exactly-solvable nature at any temperature. In this work, we extend the model to that with an arbitrary spin quantum number S . We show that the ground state is classical spin liquid whose degeneracy is independent of S . As for the fi nite-temperature properties, the partition function cannot be obtained analytically, unlike the spin-half case. We therefore perform the Monte Carlo simulation to reveal the fi nite-temperature properties, taking S = 1 and 3/2 cases as examples. We fi nd that the temperature dependence of the energy and the specific heat exhibits typical behaviors of the classical spin liquid. In particular, the broad peak of the specific heat is accounted for by the configuration of the classical variables on the edge of the hexagons.
A new decoration method is presented which allows the quenched randomly diluted site-bond spin-1 Ising model on a regular lattice along a line in the plane of exchange interaction parameters versus temperature to be mapped onto a certain class of mixed-spin decorated-lattice problem. Approximate results for the quenched, randomly diluted site-bond, spin-1 Ising model on the honeycomb lattice are obtained through the use of the exact solution of the annealed model on the corresponding decorated lattice. The critical temperature and the magnetisation of the diluted system as functions of site and bond concentrations are calculated in detail.
An effective model in the isolated dimer limit of the Kitaev-type model on a decorated honeycomb lattice is investigated at finite temperature. The ground state of this model is exactly shown to be a chiral spin liquid with spontaneous breaking of time reversal symmetry. We elab- orate the finite-temperature phase diagram by using the mean-field approximation and Monte Carlo simulation. We find that the phase transition between the high-temperature paramagnetic phase and the low-temperature chiral spin liquid phase is always of second order in the Monte Carlo results, although a tricritical point appears in the mean-field phase diagram. The finite-size scaling analysis of the Monte Carlo data suggests that the phase transition belongs to the two-dimensional Ising universality class.
The exactly solvable Kitaev honeycomb lattice model is realized as the low-energy effect Hamiltonian of a spin-1/2 model with spin rotation and time-reversal symmetry. The mapping to low-energy effective Hamiltonian is exact without truncation errors in traditional perturbation series expansions. This model consists of a honeycomb lattice of clusters of four spin-1/2 moments and contains short-range interactions up to six-spin (or eight-spin) terms. The spin in the Kitaev model is represented not as these spin-1/2 moments but as pseudospin of the two-dimensional spin-singlet sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin correlations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this model. This exact construction is quite general and can be used to make other interesting spin-1/2 models from spin-rotation invariant Hamiltonians. We discuss two possible routes to generate the high-order spin interactions from more natural couplings, which involves perturbative expansions thus breaks the exact mapping, although in a controlled manner.
We present a modification of the exactly solvable spin-(1/)2 Kitaev model on the decorated honeycomb lattice, with a ground state of "spin metal" type. The model is diagonalized in terms of Majorana fermions; the latter form a 2D gapless state with a Fermi circle whose size depends on the ratio of exchange couplings. Low-temperature heat capacity C(T) and dynamic spin susceptibility χ(ω,T) are calculated in the case of small Fermi circle. Whereas, C(T)∼T at low temperatures as it is expected for a Fermi liquid, spin excitations are gapped and χ(ω,T) demonstrates unusual behavior with a power-law peak near the resonance frequency. The corresponding exponent as well as the peak shape are calculated.