Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1906.04335 · Physical review. B./Physical review. B

Residual entropy and spin fractionalizations in the mixed-spin Kitaev model

Akihisa Koga, Joji Nasu

Published Sep 9, 2019• 15 citations
arXivHTMLPDF

Abstract

We investigate the ground-state and finite-temperature properties of the mixed-spin $(s,S)$ Kitaev model. When one of the spins is a half integer and the other is an integer, we introduce two kinds of local symmetries, which results in a macroscopic degeneracy in each energy level. Applying the exact diagonalization to several clusters with $(s,S)=(1/2,1)$, we confirm the presence of this large degeneracy in the ground states, in contrast to conventional Kitaev models. By means of the thermal pure quantum state technique, we calculate the specific heat, entropy, and spin-spin correlations in the system. We find that in the mixed-spin Kitaev model with $(s,S)=(1/2,1)$, at least, the double-peak structure appears in the specific heat and the plateau in the entropy at intermediate temperatures, indicating the existence of spin fractionalization. Deducing the entropy in the mixed-spin system with $s,S\ensuremath{\le}2$ systematically, we clarify that the smaller spin $s$ is responsible for the thermodynamic properties at higher temperatures.

DOI
10.1103/physrevb.100.100404

Related papers

Insights into the anisotropic spin- S Kitaev chain

Jacob S. Gordon, Hae‐Young Kee

Recently, there has been a renewed interest in properties of the higher-spin Kitaev models, especially their low-dimensional analogs with additional interactions. These quasi-one-dimensional systems exhibit rich phase diagrams with symmetry-protected topological phases, Luttinger liquids, hidden order, and higher-rank magnetism. However, the nature of the pure spin-$S$ Kitaev chains is not yet fully understood. Earlier works found a unique ground state with short-ranged correlations for $S=1$ and an intriguing double-peak structure in the heat capacity associated with an entropy plateau. To understand the low-energy excitations and thermodynamics for general $S$, we study the anisotropic spin-$S$ Kitaev chain. Starting from the dimerized limit, we derive an effective low-energy Hamiltonian at finite anisotropy. For half-integer spins we find a trivial effective model, reflecting a nonlocal symmetry protecting the degeneracy, while for integer $S$ we find interactions among the flux degrees of freedom that select a unique ground state. The effective model for integer spins is used to predict the low-energy excitations and thermodynamics, and we make a comparison with the semiclassical limit through linear spin wave theory. Finally, we speculate on the nature of the isotropic limit.

arXiv:2111.11457•Physical Review Research•Mar 16, 2022•15 citations

Thermodynamic behavior of modified integer-spin Kitaev models on the honeycomb lattice

Owen Bradley, J. Oitmaa, Diptiman Sen +1 more

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.

arXiv:2005.01255•Physical review. E•Feb 5, 2021•1 citation

Thermal Fractionalization of Quantum Spins in a Kitaev Model: Coherent Transport of Majorana Fermions and $T$-linear Specific Heat

Joji Nasu, Masafumi Udagawa, Yukitoshi Motome

Finite-temperature ($T$) properties of a Kitaev model defined on a honeycomb lattice are investigated by a quantum Monte Carlo simulation, from the viewpoint of fractionalization of quantum $S=1/2$ spins into two types of Majorana fermions, itinerant and localized. In this system, the entropy is released successively at two well-separated $T$ scales, as a clear indication of the thermal fractionalization. We show that the high-$T$ crossover, which is driven by itinerant Majorana fermions, is closely related with the development of nearest-neighbor spin correlations. On the other hand, the low-$T$ crossover originates in thermal fluctuations of fluxes composed of localized Majorana fermions, by which the spectrum of itinerant Majorana fermions is significantly disturbed. As a consequence, in the intermediate-$T$ range, the system exhibits coherent transport of Majorana fermions and $T$-linear behavior in the specific heat, despite the Dirac semimetallic spectrum in the low-$T$ limit. We also show that the flux fluctuations affect the gapless-gapped phase boundary. Our results indicate that the fractionalization is experimentally observable in spin correlations, specific heat, and transport properties.

•Apr 6, 2015•0 citations

Ground-state and Thermodynamic Properties of an S = 1 Kitaev Model

Akihisa Koga, Hiroyuki Tomishige, Joji Nasu

We study ground-state and thermodynamic properties of an $S=1$ Kitaev model. We first clarify the existence of global parity symmetry in addition to the local symmetry on each plaquette, which enables us to perform the large scale calculations up to 24 sites. It is found that the ground state should be singlet and its energy is estimated as $E/N\sim -0.65J$, where $J$ is the Kitaev exchange coupling. We find that a lowest excited state belongs to the same subspace as the ground state and the gap monotonically decreases with increasing system size, which suggests that the ground state of the $S=1$ Kitaev model is gapless. By means of the thermal pure quantum states, we clarify finite temperature properties characteristic of the Kitaev models with $S\le 2$.

arXiv:1803.08385•Journal of the Physical Society of Japan•May 2, 2018•73 citations

Low-temperature properties in the bilayer Kitaev model

Hiroyuki Tomishige, Joji Nasu, Akihisa Koga

The ground state of the bilayer Kitaev model with the Heisenberg-type interlayer exchange interaction is investigated by means of the exact diagonalization. Calculating the ground-state energy, local quantity defined on each plaquette, and dynamical spin structure factor, we obtain results suggesting the existence of a quantum phase transition between the Kitaev quantum spin liquid (QSL) and dimer singlet states when the interlayer coupling is antiferromagnetic. On the other hand, increasing the ferromagnetic interlayer coupling, there exists no singularity in the physical quantities, suggesting that the $S=1/2$ Kitaev QSL state realized in each layer adiabatically connects to another QSL state realized in the $S=1$ Kitaev model. Thermodynamic properties are also studied by means of the thermal pure quantum state method.

arXiv:1902.00165•Physical review. B./Physical review. B•May 23, 2019•16 citations

Thermodynamics of three-dimensional Kitaev quantum spin liquids via tensor networks

Saeed S. Jahromi, Hadi Yarloo, Román Orús

We study the 3D Kitaev and Kitaev-Heisenberg models, respectively, on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph projected entangled-pair states (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high temperature leads to fractionalization of spins to Majorana fermions and gauge fields that occurs in two separate temperature regimes, leaving their fingerprint on specific heat as a double-peak feature as well as on other quantities such as the thermal entropy, spin-spin correlations, and bond entropy. Using the Majorana representation of the Kitaev model, we further show that the low-temperature thermal transition to the Kitaev quantum spin liquid (QSL) phase is associated with the nontrivial Majorana band topology and the presence of Weyl nodes, which manifests itself via nonvanishing Chern number and finite thermal Hall conductivity. Beyond the pure Kitaev limit, we study the 3D Kitaev-Heisenberg (KH) model on the hyperoctagon lattice and extract the full phase diagram for different Heisenberg couplings. We further explore the thermodynamic properties of the magnetically-ordered regions in the KH model and show that, in contrast to the QSL phase, here the thermal phase transition follows the standard Landau symmetry-breaking theory.

arXiv:2011.11577•Physical Review Research•Sep 2, 2021•13 citations
We investigate the ground-state and finite-temperature properties of the mixed-spin (s,S)(s,S)(s,S) Kitaev model. When one of the spins is a half integer and the other is an integer, we introduce two kinds of local symmetries, which results in a macroscopic degeneracy in each energy level. Applying the exact diagonalization to several clusters with (s,S)=(1/2,1)(s,S)=(1/2,1)(s,S)=(1/2,1), we confirm the presence of this large degeneracy in the ground states, in contrast to conventional Kitaev models. By means of the thermal pure quantum state technique, we calculate the specific heat, entropy, and spin-spin correlations in the system. We find that in the mixed-spin Kitaev model with (s,S)=(1/2,1)(s,S)=(1/2,1)(s,S)=(1/2,1), at least, the double-peak structure appears in the specific heat and the plateau in the entropy at intermediate temperatures, indicating the existence of spin fractionalization. Deducing the entropy in the mixed-spin system with s,S\ensuremath≤2s,S\ensuremath{\le}2s,S\ensuremath≤2 systematically, we clarify that the smaller spin sss is responsible for the thermodynamic properties at higher temperatures.
Recently, there has been a renewed interest in properties of the higher-spin Kitaev models, especially their low-dimensional analogs with additional interactions. These quasi-one-dimensional systems exhibit rich phase diagrams with symmetry-protected topological phases, Luttinger liquids, hidden order, and higher-rank magnetism. However, the nature of the pure spin-SSS Kitaev chains is not yet fully understood. Earlier works found a unique ground state with short-ranged correlations for S=1S=1S=1 and an intriguing double-peak structure in the heat capacity associated with an entropy plateau. To understand the low-energy excitations and thermodynamics for general SSS, we study the anisotropic spin-SSS Kitaev chain. Starting from the dimerized limit, we derive an effective low-energy Hamiltonian at finite anisotropy. For half-integer spins we find a trivial effective model, reflecting a nonlocal symmetry protecting the degeneracy, while for integer SSS we find interactions among the flux degrees of freedom that select a unique ground state. The effective model for integer spins is used to predict the low-energy excitations and thermodynamics, and we make a comparison with the semiclassical limit through linear spin wave theory. Finally, we speculate on the nature of the isotropic limit.
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.
Thermal Fractionalization of Quantum Spins in a Kitaev Model: Coherent Transport of Majorana Fermions and TTT-linear Specific Heat
Finite-temperature (TTT) properties of a Kitaev model defined on a honeycomb lattice are investigated by a quantum Monte Carlo simulation, from the viewpoint of fractionalization of quantum S=1/2S=1/2S=1/2 spins into two types of Majorana fermions, itinerant and localized. In this system, the entropy is released successively at two well-separated TTT scales, as a clear indication of the thermal fractionalization. We show that the high-TTT crossover, which is driven by itinerant Majorana fermions, is closely related with the development of nearest-neighbor spin correlations. On the other hand, the low-TTT crossover originates in thermal fluctuations of fluxes composed of localized Majorana fermions, by which the spectrum of itinerant Majorana fermions is significantly disturbed. As a consequence, in the intermediate-TTT range, the system exhibits coherent transport of Majorana fermions and TTT-linear behavior in the specific heat, despite the Dirac semimetallic spectrum in the low-TTT limit. We also show that the flux fluctuations affect the gapless-gapped phase boundary. Our results indicate that the fractionalization is experimentally observable in spin correlations, specific heat, and transport properties.
We study ground-state and thermodynamic properties of an S=1S=1S=1 Kitaev model. We first clarify the existence of global parity symmetry in addition to the local symmetry on each plaquette, which enables us to perform the large scale calculations up to 24 sites. It is found that the ground state should be singlet and its energy is estimated as E/N∼−0.65JE/N\sim -0.65JE/N∼−0.65J, where JJJ is the Kitaev exchange coupling. We find that a lowest excited state belongs to the same subspace as the ground state and the gap monotonically decreases with increasing system size, which suggests that the ground state of the S=1S=1S=1 Kitaev model is gapless. By means of the thermal pure quantum states, we clarify finite temperature properties characteristic of the Kitaev models with S≤2S\le 2S≤2.
The ground state of the bilayer Kitaev model with the Heisenberg-type interlayer exchange interaction is investigated by means of the exact diagonalization. Calculating the ground-state energy, local quantity defined on each plaquette, and dynamical spin structure factor, we obtain results suggesting the existence of a quantum phase transition between the Kitaev quantum spin liquid (QSL) and dimer singlet states when the interlayer coupling is antiferromagnetic. On the other hand, increasing the ferromagnetic interlayer coupling, there exists no singularity in the physical quantities, suggesting that the S=1/2S=1/2S=1/2 Kitaev QSL state realized in each layer adiabatically connects to another QSL state realized in the S=1S=1S=1 Kitaev model. Thermodynamic properties are also studied by means of the thermal pure quantum state method.
We study the 3D Kitaev and Kitaev-Heisenberg models, respectively, on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph projected entangled-pair states (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high temperature leads to fractionalization of spins to Majorana fermions and gauge fields that occurs in two separate temperature regimes, leaving their fingerprint on specific heat as a double-peak feature as well as on other quantities such as the thermal entropy, spin-spin correlations, and bond entropy. Using the Majorana representation of the Kitaev model, we further show that the low-temperature thermal transition to the Kitaev quantum spin liquid (QSL) phase is associated with the nontrivial Majorana band topology and the presence of Weyl nodes, which manifests itself via nonvanishing Chern number and finite thermal Hall conductivity. Beyond the pure Kitaev limit, we study the 3D Kitaev-Heisenberg (KH) model on the hyperoctagon lattice and extract the full phase diagram for different Heisenberg couplings. We further explore the thermodynamic properties of the magnetically-ordered regions in the KH model and show that, in contrast to the QSL phase, here the thermal phase transition follows the standard Landau symmetry-breaking theory.