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Journal of the Physical Society of Japan

Exactly Solvable Kitaev-Type Compass Model on a Decorated Honeycomb Lattice and Its Extension

Tomonari Mizoguchi

Published Nov 10, 2022• 7 citations
Publisher

Abstract

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.

DOI
10.7566/jpsj.91.124709

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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 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.
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 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.
Quantum spin models with spatially dependent interactions, known as compass models, play an important role in the study of frustrated quantum magnetism. One example is the Kitaev model on the honeycomb lattice with spin-liquid (SL) ground states and anyonic excitations. Another example is the geometrically frustrated quantum 120° model on the same lattice whose ground state has not been unambiguously established. To generalize the Kitaev model beyond the exactly solvable limit and connect it with other compass models, we propose a new model, dubbed 'the tripod model', which contains a continuum of compass-type models. It smoothly interpolates the Ising model, the Kitaev model, and the quantum 120° model by tuning a single parameter , the angle between the three legs of a tripod in the spin space. Hence it not only unifies three paradigmatic spin models, but also enables the study of their quantum phase transitions. We obtain the phase diagram of the tripod model numerically by tensor networks in the thermodynamic limit. We show that the ground state of the quantum 120° model has long-range dimer order. Moreover, we find an extended spin-disordered (SL) phase between the dimer phase and an antiferromagnetic phase. The unification and solution of a continuum of frustrated spin models as outline here may be useful to exploring new domains of other quantum spin or orbital models.
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.
Quantum spin models with spatially dependent interactions, known as compass models, play an important role in study of frustrated quantum magnetism and Mott insulators with orbital degrees of freedom. One example is Kitaev model on honeycomb lattice with spin liquid ground states and anyonic excitations. It is highly desirable to go beyond this exactly solvable limit by generalizing Kitaev model to connect with specific materials or other known models. A conventional way to achieve this is through hybridization with other models. For example, Kitaev-Heisenberg model was proposed to describe iridates. Here we pursue an alternative direction and propose a new dubbed the tripod model, which contains a continuum of compass-type models. It smoothly interpolates Ising Kitaev and quantum 120∘120^\circ120∘ model by tuning a single parameter θ′\theta'θ′, angle between three legs of a tripod in spin space. Not only does tripod model establish a natural connection between three paradigmatic models, but it also enables study of quantum phase transitions between them. We solve this model numerically by tensor networks in thermodynamic limit and find a spin-disordered (spin liquid) phase between an antiferromagnetic phase and a dimer phase. Our results unambiguously establish long-range order of geometrically frustrated 120∘120^\circ120∘ model and provide an intuitive picture for phase transition between spin-liquid phase and dimer phase. The tripod model proposed here could be realized in cold atoms experiments by introducing non-Abelian artificial gauge fields for fermionic Hubbard model.