arXiv:1803.08385 · Journal of the Physical Society of Japan

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

Akihisa Koga, Hiroyuki Tomishige, Joji Nasu

Published May 2, 2018• 73 citations

Abstract

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 $S\le 2$.

DOI: 10.7566/jpsj.87.063703

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arXiv:1803.08385 · Journal of the Physical Society of Japan

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

Akihisa Koga, Hiroyuki Tomishige, Joji Nasu

Published May 2, 2018• 73 citations

arXiv HTML PDF

Abstract

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

DOI: 10.7566/jpsj.87.063703

Related papers

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.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:1902.00165Physical review. B./Physical review. BMay 23, 201916 citations

Spin-1 Kitaev model in one dimension

Diptiman Sen, Ravi Shankar, Deepak Dhar +1 more

We study a one-dimensional version of the Kitaev model on a ring of size $N$ , in which there is a spin $S>1/2$ on each site and the Hamiltonian is $J{\ensuremath{\sum}}_{n}{S}_{n}^{x}{S}_{n+1}^{y}$. The cases where $S$ is integer and half-odd integer are qualitatively different. We show that there is a ${\mathbb{Z}}_{2}$ -valued conserved quantity ${W}_{n}$ for each bond $(n,n+1)$ of the system. For integer $S$ , the Hilbert space can be decomposed into ${2}^{N}$ sectors, of unequal sizes. The number of states in most of the sectors grows as ${d}^{N}$ , where $d$ depends on the sector. The largest sector contains the ground state, and for this sector, for $S=1$ , $d=(\sqrt{5}+1)/2$ . We carry out exact diagonalization for small systems. The extrapolation of our results to large $N$ indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term $\ensuremath{\lambda}{\ensuremath{\sum}}_{n}{W}_{n}$, and show that this has gapless excitations in the range ${\ensuremath{\lambda}}_{1}^{c}\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{2}^{c}$. We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points ${\ensuremath{\lambda}}_{1}^{c}$ and ${\ensuremath{\lambda}}_{2}^{c}$.We study a one-dimensional version of the Kitaev model on a ring of size $N$, in which there is a spin $S>1/2$ on each site and the Hamiltonian is $J{\ensuremath{\sum}}_{n}{S}_{n}^{x}{S}_{n+1}^{y}$. The cases where $S$ is integer and half-odd integer are qualitatively different. We show that there is a ${\mathbb{Z}}_{2}$-valued conserved quantity ${W}_{n}$ for each bond $(n,n+1)$ of the system. For integer $S$, the Hilbert space can be decomposed into ${2}^{N}$ sectors, of unequal sizes. The number of states in most of the sectors grows as ${d}^{N}$, where $d$ depends on the sector. The largest sector contains the ground state, and for this sector, for $S=1$, $d=(\sqrt{5}+1)/2$. We carry out exact diagonalization for small systems. The extrapolation of our results to large $N$ indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term $\ensuremath{\lambda}{\ensuremath{\sum}}_{n}{W}_{n}$, and show that this has gapless excitations in the range ${\ensuremath{\lambda}}_{1}^{c}\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{2}^{c}$. We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points ${\ensuremath{\lambda}}_{1}^{c}$ and ${\ensuremath{\lambda}}_{2}^{c}$.

arXiv:1009.2583Physical Review BNov 17, 201024 citations

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

Akihisa Koga, Joji Nasu

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.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.

arXiv:1906.04335Physical review. B./Physical review. BSep 9, 201915 citations

Thermodynamic classification of three-dimensional Kitaev spin liquids

Tim Eschmann, Petr A. Mishchenko, Kevin M. O'Brien +5 more

In the field of frustrated magnetism, Kitaev models provide a unique framework to study the phenomena of spin fractionalization and emergent lattice gauge theories in two and three spatial dimensions. Their ground states are quantum spin liquids, which can typically be described in terms of a Majorana band structure and an ordering of the underlying ${\mathbb{Z}}_{2}$ gauge structure. Here we provide a comprehensive classification of the ``gauge physics'' of a family of elementary three-dimensional Kitaev models, discussing how their thermodynamics and ground state order depends on the underlying lattice geometry. Using large-scale, sign-free quantum Monte Carlo simulations we show that the ground-state gauge order can generally be understood in terms of the length of elementary plaquettes---a result which extends the applicability of Lieb's theorem to lattice geometries beyond its original scope. At finite temperatures, the proliferation of (gapped) vison excitations destroys the gauge order at a critical temperature scale, which we show to correlate with the size of vison gap for the family of three-dimensional Kitaev models. We also discuss two notable exceptions where the lattice structure gives rise to ``gauge frustration'' or intertwines the gauge ordering with time-reversal symmetry breaking. In a more general context, the thermodynamic gauge transitions in such 3D Kitaev models are one of the most natural settings for phase transitions beyond the standard Landau-Ginzburg-Wilson paradigm.In the field of frustrated magnetism, Kitaev models provide a unique framework to study the phenomena of spin fractionalization and emergent lattice gauge theories in two and three spatial dimensions. Their ground states are quantum spin liquids, which can typically be described in terms of a Majorana band structure and an ordering of the underlying ${\mathbb{Z}}_{2}$ gauge structure. Here we provide a comprehensive classification of the ``gauge physics'' of a family of elementary three-dimensional Kitaev models, discussing how their thermodynamics and ground state order depends on the underlying lattice geometry. Using large-scale, sign-free quantum Monte Carlo simulations we show that the ground-state gauge order can generally be understood in terms of the length of elementary plaquettes---a result which extends the applicability of Lieb's theorem to lattice geometries beyond its original scope. At finite temperatures, the proliferation of (gapped) vison excitations destroys the gauge order at a critical temperature scale, which we show to correlate with the size of vison gap for the family of three-dimensional Kitaev models. We also discuss two notable exceptions where the lattice structure gives rise to ``gauge frustration'' or intertwines the gauge ordering with time-reversal symmetry breaking. In a more general context, the thermodynamic gauge transitions in such 3D Kitaev models are one of the most natural settings for phase transitions beyond the standard Landau-Ginzburg-Wilson paradigm.

arXiv:2006.07386Physical review. B./Physical review. BAug 18, 202036 citations

Finite-temperature properties of the anisotropic AFM Kitaev model in the magnetic field

S.L. Jia, Ya‐Min Quan, Hai‐Qing Lin +1 more

We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the $[001]$ magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with $K_{X}=K_{Y}$ , $K_{X}+K_{Y}+K_{Z}=-3K$ , and $K_{Z}<-K$ , with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at $h_{c1}=g\mu_{B}H_{z1}/K$ , to another gapless QSL with $C_{6}$ rotational symmetry at $h_{c2}$ , and to a new $U(1)$ gapless QSL between $h_{c3}$ and $h_{c4}$ , respectively. These indicate that magnetic field could first turn the anisotropic gapped or gapless QSL back into the isotropic $C_{6}$ gapless one and then make it to undergo the similar evolution as the isotropic case. Moreover, the critical magnetic fields $h_{c1}$ , $h_{c2}$ , $h_{c3}$ , and $h_{c4}$ come up monotonically with the increasing Kitaev coupling; this suggests that the magnetic field can be applied to the modulation of the anisotropic Kitaev materials.We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the $[001]$ magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with $K_{X}=K_{Y}$, $K_{X}+K_{Y}+K_{Z}=-3K$, and $K_{Z}<-K$, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at $h_{c1}=g\mu_{B}H_{z1}/K$, to another gapless QSL with $C_{6}$ rotational symmetry at $h_{c2}$, and to a new $U(1)$ gapless QSL between $h_{c3}$ and $h_{c4}$, respectively. These indicate that magnetic field could first turn the anisotropic gapped or gapless QSL back into the isotropic $C_{6}$ gapless one and then make it to undergo the similar evolution as the isotropic case. Moreover, the critical magnetic fields $h_{c1}$, $h_{c2}$, $h_{c3}$, and $h_{c4}$ come up monotonically with the increasing Kitaev coupling; this suggests that the magnetic field can be applied to the modulation of the anisotropic Kitaev materials.

Apr 27, 20210 citations

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.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:2111.11457Physical Review ResearchMar 16, 202215 citations

We study ground-state and thermodynamic properties of an

S=1

E/N\sim -0.65J

, where

J

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

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.

We study a one-dimensional version of the Kitaev model on a ring of size

N

, in which there is a spin

S>1/2

on each site and the Hamiltonian is $J{\ensuremath{\sum}}_{n}{S}_{n}^{x}{S}_{n+1}^{y}$. The cases where

S

is integer and half-odd integer are qualitatively different. We show that there is a

{\mathbb{Z}}_{2}

-valued conserved quantity

{W}_{n}

for each bond

(n,n+1)

of the system. For integer

S

, the Hilbert space can be decomposed into

{2}^{N}

sectors, of unequal sizes. The number of states in most of the sectors grows as

{d}^{N}

, where

d

depends on the sector. The largest sector contains the ground state, and for this sector, for

S=1

d=(\sqrt{5}+1)/2

. We carry out exact diagonalization for small systems. The extrapolation of our results to large

N

indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term $\ensuremath{\lambda}{\ensuremath{\sum}}_{n}{W}_{n}$, and show that this has gapless excitations in the range ${\ensuremath{\lambda}}_{1}^{c}\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{2}^{c}$. We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points ${\ensuremath{\lambda}}_{1}^{c}$ and ${\ensuremath{\lambda}}_{2}^{c}$.

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.

{\mathbb{Z}}_{2}

gauge structure. Here we provide a comprehensive classification of the ``gauge physics'' of a family of elementary three-dimensional Kitaev models, discussing how their thermodynamics and ground state order depends on the underlying lattice geometry. Using large-scale, sign-free quantum Monte Carlo simulations we show that the ground-state gauge order can generally be understood in terms of the length of elementary plaquettes---a result which extends the applicability of Lieb's theorem to lattice geometries beyond its original scope. At finite temperatures, the proliferation of (gapped) vison excitations destroys the gauge order at a critical temperature scale, which we show to correlate with the size of vison gap for the family of three-dimensional Kitaev models. We also discuss two notable exceptions where the lattice structure gives rise to ``gauge frustration'' or intertwines the gauge ordering with time-reversal symmetry breaking. In a more general context, the thermodynamic gauge transitions in such 3D Kitaev models are one of the most natural settings for phase transitions beyond the standard Landau-Ginzburg-Wilson paradigm.

We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the

[001]

magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with

K_{X}=K_{Y}

K_{X}+K_{Y}+K_{Z}=-3K

, and

K_{Z}<-K

, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at

h_{c1}=g\mu_{B}H_{z1}/K

, to another gapless QSL with

C_{6}

rotational symmetry at

h_{c2}

, and to a new

U(1)

gapless QSL between

h_{c3}

and

h_{c4}

, respectively. These indicate that magnetic field could first turn the anisotropic gapped or gapless QSL back into the isotropic

C_{6}

gapless one and then make it to undergo the similar evolution as the isotropic case. Moreover, the critical magnetic fields

h_{c1}

h_{c2}

h_{c3}

, and

h_{c4}

come up monotonically with the increasing Kitaev coupling; this suggests that the magnetic field can be applied to the modulation of the anisotropic Kitaev materials.

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.