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Spin-S Decorated Honeycomb Kitaev Model with S ≥ 1

Tomonari Mizoguchi

Published Nov 5, 2024• 0 citations
OpenAlex

Abstract

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.

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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.
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.
We study the thermodynamic behavior of modified spin-S Kitaev models introduced by Baskaran, Sen, and Shankar [Phys. Rev. B 78, 115116 (2008)PRBMDO1098-012110.1103/PhysRevB.78.115116]. These models have the property that for half-odd-integer spins their eigenstates map on to those of spin-1/2 Kitaev models, with well-known highly entangled quantum spin-liquid states and Majorana fermions. For integer spins, the Hamiltonian is made out of commuting local operators. Thus, the eigenstates can be chosen to be completely unentangled between different sites, though with a significant degeneracy for each eigenstate. For half-odd-integer spins, the thermodynamic properties can be related to the spin-1/2 Kitaev models apart from an additional degeneracy. Hence we focus here on the case of integer spins. We use transfer matrix methods, high-temperature expansions, and Monte Carlo simulations to study the thermodynamic properties of ferromagnetic and antiferromagnetic models with spin S=1 and S=2. Apart from large residual entropies, which all the models have, we find that they can have a variety of different behaviors. Transfer matrix calculations show that for the different models, the correlation lengths can be finite as T→0, become critical as T→0, or diverge exponentially as T→0. The Z_{2} flux variable associated with each hexagonal plaquette saturates at the value +1 as T→0 in all models except the S=1 antiferromagnet where the mean flux remains zero as T→0. We provide qualitative explanations for these results.
Here we revisit the thermodynamics of the Kitaev quantum spin liquid realized on the honeycomb lattice. We address two main questions: First, we investigate whether there are observable thermodynamic signatures of the topological Majorana boundary modes of the Kitaev honeycomb model. We argue that for the time-reversal invariant case the residual low-temperature entropy is the primary thermodynamic signature of these Majorana edge modes, and verify using large-scale Monte Carlo simulations that this residual entropy is present in the full Kitaev model. When time-reversal symmetry is broken, the Majorana edge modes are potentially observable in more direct thermodynamic measurements such as the specific heat, though only at temperatures well below the bulk gap. % Second, we study the energetics, and the corresponding thermodynamic signatures, of the flux excitations in the Kitaev model. Specifically, we study the flux interactions on both cylinder and torus geometries numerically, and quantify their impact on the thermodynamics of the Kitaev spin liquid by using a polynomial fit for the average flux energy as a function of flux density and extrapolating it to the thermodynamic limit. By comparing this model to Monte Carlo simulations, we find that flux interactions have a significant quantitative impact on the shape and the position of the low-temperature peak in the specific heat.
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.
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.