arXiv:2111.07054 · Journal of the Physical Society of Japan

First Order Transitions Between the Gapped Spin-Liquid and Ferrimagnetic Phases in $(1/2,1/2,1)$ Mixed Diamond Chains with Bond Alternation

Kazuo Hida

Published Jan 31, 2022• 1 citation

Abstract

The ground-state phases of mixed diamond chains with bond alternation $δ$, and ($S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)$, where $S$ is the magnitude of vertex spins, and $τ^{(1)}$ and $τ^{(2)}$ are those of apical spins, are investigated. The two apical spins in each unit cell are connected by an exchange coupling $λ$. The exchange couplings between the apical spins and the vertex spins take the values $1+δ$ and $1-δ$ alternatingly. This model has an infinite number of local conservation laws. For large $λ$ and $δ\neq 0$, the ground state is equivalent to that of the spin $1/2$ chain with bond alternation. Hence, the ground state is a gapped spin liquid. This energy gap vanishes for $δ=0$. With the decrease of $λ$, the ground state undergoes a transition at $λ=λ_{\rm c0}(δ)$ to a series of ferrimagnetic phases with a spontaneous magnetization $m_{\rm sp}=1/p$ per unit cell where $p$ is a positive integer. It is found that this transition is a first order transition for $δ\neq 0$ with a discontinuous change in $m_{\rm sp}$, while no discontinuity is found for $δ=0$. The critical behaviors of $m_{\rm sp}$ and $λ_{\rm c0}(δ)$ around the critical point $(δ,λ) =(0, λ_{\rm c0}(δ))$ are also discussed analytically.

DOI: 10.7566/jpsj.91.024706

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Infinite Series of Ferrimagnetic Phases Emergent from the Gapless Spin Liquid Phase of Mixed Diamond Chains

Kazuo Hida

The ground-state phases of mixed diamond chains with ($S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)$, where $S$ is the magnitude of vertex spins, and $τ^{(1)}$ and $τ^{(2)}$ are those of apical spins, are investigated. The apical spins $τ^{(1)}$ and $τ^{(2)}$ are connected with each other by an exchange coupling $λ$. Other exchange couplings are set equal to unity. This model has an infinite number of local conservation laws. For large $λ$, the ground state is equivalent to that of the uniform spin $1/2$ chain. Hence, the ground state is a gapless spin liquid. For $λ\leq 0$, the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization $m_{\rm sp}=1$ per unit cell. For intermediate $λ$, we find a series of ferrimagnetic phases with $m_{\rm sp}=1/p$ where $p$ takes positive integer values. The phases with $p \geq 2$ are accompanied by the spontaneous breakdown of the $p$-fold translational symmetry. It is suggested that the phase with arbitrarily large $p$, namely infinitesimal spontaneous magnetization, is allowed as $λ$ approaches the transition point to the gapless spin liquid phase.

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Kazuo Hida

The ground-state phases of mixed diamond chains with ($S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)$, where $S$ is the magnitude of vertex spins, and $τ^{(1)}$ and $τ^{(2)}$ are those of apical spins, are investigated with the single-site anisotropy $D$ on the $τ^{(2)}$-site. The two apical spins in each unit cell are coupled by an exchange coupling $λ$. The vertex spins are coupled with the top and bottom apical spins by exchange couplings $1+δ$ and $1-δ$, respectively. The ground-state phase diagram is determined using the numerical exact diagonalization and DMRG method in addition to the analytical approximations in various limiting cases. The phase diagram consists of a Néel ordered phase, a nonmagnetic Tomonaga-Luttinger liquid phase, and quantized and partial ferrimagnetic phases. A region with anisotropy inversion is found where the Ising-like Néel phase is realized for the easy-plane anisotropy $D >0$ and the XY-like Tomonaga-Luttinger liquid phase is realized for the easy-axis anisotropy $D <0$ on the $S=1$ sites.

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Ground-State Phases of Alternating-Bond S = 1 Diamond Chains

Kazuo Hida

arXiv:1909.11026Journal of the Physical Society of JapanJan 30, 20206 citations

Quantum Phase Transitions in Alternating-Bond Mixed Diamond Chains with Spins 1 and 1/2

Kazuo Hida, Ken'ichi Takano, Hidenori Suzuki

arXiv:0911.3838Journal of the Physical Society of JapanMar 25, 201014 citations

Ground-State and Thermodynamic Properties of the Quantum Mixed Spin-1/2-1/2-1-1 Chain

Takashi Tonegawa, Toshiya Hikihara, Makoto Kaburagi +3 more

arXiv:cond-mat/9712296Journal of the Physical Society of JapanMar 15, 199831 citations

arXiv:2111.07054 · Journal of the Physical Society of Japan

First Order Transitions Between the Gapped Spin-Liquid and Ferrimagnetic Phases in $(1/2,1/2,1)$ Mixed Diamond Chains with Bond AlternationFirst Order Transitions Between the Gapped Spin-Liquid and Ferrimagnetic Phases in $(1/2,1/2,1)$ Mixed Diamond Chains with Bond Alternation

Kazuo Hida

Published Jan 31, 2022• 1 citation

arXiv HTML PDF

Abstract

The ground-state phases of mixed diamond chains with bond alternation

δ

, and (

S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)

, where

S

is the magnitude of vertex spins, and

τ^{(1)}

and

τ^{(2)}

are those of apical spins, are investigated. The two apical spins in each unit cell are connected by an exchange coupling

λ

. The exchange couplings between the apical spins and the vertex spins take the values

1+δ

and

1-δ

alternatingly. This model has an infinite number of local conservation laws. For large

λ

and

δ\neq 0

, the ground state is equivalent to that of the spin

1/2

chain with bond alternation. Hence, the ground state is a gapped spin liquid. This energy gap vanishes for

δ=0

. With the decrease of

λ

, the ground state undergoes a transition at

λ=λ_{\rm c0}(δ)

to a series of ferrimagnetic phases with a spontaneous magnetization

m_{\rm sp}=1/p

per unit cell where

p

is a positive integer. It is found that this transition is a first order transition for

δ\neq 0

with a discontinuous change in

m_{\rm sp}

, while no discontinuity is found for

δ=0

. The critical behaviors of

m_{\rm sp}

and

λ_{\rm c0}(δ)

around the critical point

(δ,λ) =(0, λ_{\rm c0}(δ))

are also discussed analytically.

DOI: 10.7566/jpsj.91.024706

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Infinite Series of Ferrimagnetic Phases Emergent from the Gapless Spin Liquid Phase of Mixed Diamond Chains

Kazuo Hida

The ground-state phases of mixed diamond chains with ( $S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)$ , where $S$ is the magnitude of vertex spins, and $τ^{(1)}$ and $τ^{(2)}$ are those of apical spins, are investigated. The apical spins $τ^{(1)}$ and $τ^{(2)}$ are connected with each other by an exchange coupling $λ$ . Other exchange couplings are set equal to unity. This model has an infinite number of local conservation laws. For large $λ$ , the ground state is equivalent to that of the uniform spin $1/2$ chain. Hence, the ground state is a gapless spin liquid. For $λ\leq 0$ , the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization $m_{\rm sp}=1$ per unit cell. For intermediate $λ$ , we find a series of ferrimagnetic phases with $m_{\rm sp}=1/p$ where $p$ takes positive integer values. The phases with $p \geq 2$ are accompanied by the spontaneous breakdown of the $p$ -fold translational symmetry. It is suggested that the phase with arbitrarily large $p$ , namely infinitesimal spontaneous magnetization, is allowed as $λ$ approaches the transition point to the gapless spin liquid phase.The ground-state phases of mixed diamond chains with ($S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)$, where $S$ is the magnitude of vertex spins, and $τ^{(1)}$ and $τ^{(2)}$ are those of apical spins, are investigated. The apical spins $τ^{(1)}$ and $τ^{(2)}$ are connected with each other by an exchange coupling $λ$. Other exchange couplings are set equal to unity. This model has an infinite number of local conservation laws. For large $λ$, the ground state is equivalent to that of the uniform spin $1/2$ chain. Hence, the ground state is a gapless spin liquid. For $λ\leq 0$, the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization $m_{\rm sp}=1$ per unit cell. For intermediate $λ$, we find a series of ferrimagnetic phases with $m_{\rm sp}=1/p$ where $p$ takes positive integer values. The phases with $p \geq 2$ are accompanied by the spontaneous breakdown of the $p$-fold translational symmetry. It is suggested that the phase with arbitrarily large $p$, namely infinitesimal spontaneous magnetization, is allowed as $λ$ approaches the transition point to the gapless spin liquid phase.

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Kazuo Hida, Ken'ichi Takano, Hidenori Suzuki

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Takashi Tonegawa, Toshiya Hikihara, Makoto Kaburagi +3 more

We investigate both analytically and numerically the ground-state and thermodynamic properties of the quantum mixed spin-1/2-1/2-1-1 chain described by the Hamiltonian $H=\sum_{\ell=1}^{N/4} (J_1\vecs_{4\ell-3}\cdot \vecs_{4\ell-2}+J_2\vecs_{4\ell-2}\cdot\vecS_{4\ell-1}+J_3\vecS_{4\ell-1}\cdot \vecS_{4\ell}+J_2\vecS_{4\ell}\cdot\vecs_{4\ell+1})$, where two $S=1/2$ spins ($\vecs_{4\ell-3}$ and $\vecs_{4\ell-2}$) and two S=1 spins ($\vecS_{4\ell-1}$ and $\vecS_{4\ell}$) are arranged alternatively. Inseveral limiting cases of $J_1$ , $J_2$ , and $J_3$ we apply the Wigner-Eckart theorem and carry out a perturbation calculation to examine the behavior of the massless lines where the energy gap vanishes. Performing a quantum Monte Carlo calculation without global flips at a sufficiently low temperature for the case where $J_1=J_3=1.0$ and $J_2>0$ , we find that the ground state of the present system in this case undergoes a second-order phase transition accompanying the vanishing of the energy gap at $J_2=J_{2c$ with $J_{2 c} = 0.77 \pm 0.01$ . We also find that the ground states for both $J_2 < J_{2 c$ and $J_2 > J_{2c$ can be understood by means of the valence-bond-solid picture. A quantum Monte Carlo calculation which takes the global flips along the Trotter direction into account is carried out to elucidate the temperature dependences of the specific heat and the magnetic susceptibility. In particular, it is found that the susceptibility per unit cell for $J_2=0.77$ with $J_1=J_3=1.0$ takes a finite value at absolute zero temperature and that the specific heat per unit cell versus temperature curve for $J_2=5.0$ with $J_1=J_3=1.0$ has a double peak.We investigate both analytically and numerically the ground-state and thermodynamic properties of the quantum mixed spin-1/2-1/2-1-1 chain described by the Hamiltonian $H=\sum_{\ell=1}^{N/4} (J_1\vecs_{4\ell-3}\cdot \vecs_{4\ell-2}+J_2\vecs_{4\ell-2}\cdot\vecS_{4\ell-1}+J_3\vecS_{4\ell-1}\cdot \vecS_{4\ell}+J_2\vecS_{4\ell}\cdot\vecs_{4\ell+1})$, where two $S=1/2$ spins ($\vecs_{4\ell-3}$ and $\vecs_{4\ell-2}$) and two S=1 spins ($\vecS_{4\ell-1}$ and $\vecS_{4\ell}$) are arranged alternatively. Inseveral limiting cases of $J_1$, $J_2$, and $J_3$ we apply the Wigner-Eckart theorem and carry out a perturbation calculation to examine the behavior of the massless lines where the energy gap vanishes. Performing a quantum Monte Carlo calculation without global flips at a sufficiently low temperature for the case where $J_1=J_3=1.0$ and $J_2>0$, we find that the ground state of the present system in this case undergoes a second-order phase transition accompanying the vanishing of the energy gap at $J_2=J_{2c$ with $J_{2 c} = 0.77 \pm 0.01$. We also find that the ground states for both $J_2 < J_{2 c$ and $J_2 > J_{2c$ can be understood by means of the valence-bond-solid picture. A quantum Monte Carlo calculation which takes the global flips along the Trotter direction into account is carried out to elucidate the temperature dependences of the specific heat and the magnetic susceptibility. In particular, it is found that the susceptibility per unit cell for $J_2=0.77$ with $J_1=J_3=1.0$ takes a finite value at absolute zero temperature and that the specific heat per unit cell versus temperature curve for $J_2=5.0$ with $J_1=J_3=1.0$ has a double peak.

arXiv:cond-mat/9712296Journal of the Physical Society of JapanMar 15, 199831 citations

First Order Transitions Between the Gapped Spin-Liquid and Ferrimagnetic Phases in

(1/2,1/2,1)

Mixed Diamond Chains with Bond Alternation

The ground-state phases of mixed diamond chains with bond alternation

δ

, and (

S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)

, where

S

is the magnitude of vertex spins, and

τ^{(1)}

and

τ^{(2)}

are those of apical spins, are investigated. The two apical spins in each unit cell are connected by an exchange coupling

λ

. The exchange couplings between the apical spins and the vertex spins take the values

1+δ

and

1-δ

alternatingly. This model has an infinite number of local conservation laws. For large

λ

and

δ\neq 0

, the ground state is equivalent to that of the spin

1/2

chain with bond alternation. Hence, the ground state is a gapped spin liquid. This energy gap vanishes for

δ=0

. With the decrease of

λ

, the ground state undergoes a transition at

λ=λ_{\rm c0}(δ)

to a series of ferrimagnetic phases with a spontaneous magnetization

m_{\rm sp}=1/p

per unit cell where

p

is a positive integer. It is found that this transition is a first order transition for

δ\neq 0

with a discontinuous change in

m_{\rm sp}

, while no discontinuity is found for

δ=0

. The critical behaviors of

m_{\rm sp}

and

λ_{\rm c0}(δ)

around the critical point

(δ,λ) =(0, λ_{\rm c0}(δ))

are also discussed analytically.

The ground-state phases of mixed diamond chains with (

S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)

, where

S

is the magnitude of vertex spins, and

τ^{(1)}

and

τ^{(2)}

are those of apical spins, are investigated. The apical spins

τ^{(1)}

and

τ^{(2)}

are connected with each other by an exchange coupling

λ

. Other exchange couplings are set equal to unity. This model has an infinite number of local conservation laws. For large

λ

, the ground state is equivalent to that of the uniform spin

1/2

chain. Hence, the ground state is a gapless spin liquid. For

λ\leq 0

, the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization

m_{\rm sp}=1

per unit cell. For intermediate

λ

, we find a series of ferrimagnetic phases with

m_{\rm sp}=1/p

where

p

takes positive integer values. The phases with

p \geq 2

are accompanied by the spontaneous breakdown of the

p

-fold translational symmetry. It is suggested that the phase with arbitrarily large

p

, namely infinitesimal spontaneous magnetization, is allowed as

λ

approaches the transition point to the gapless spin liquid phase.

Ground-State Phase Diagram of

(1/2,1/2,1)

Mixed Diamond Chains

The ground-state phases of mixed diamond chains with (

S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)

, where

S

is the magnitude of vertex spins, and

τ^{(1)}

and

τ^{(2)}

are those of apical spins, are investigated. The two apical spins in each unit cell are coupled by an exchange coupling

λ

. The vertex spins are coupled with the top and bottom apical spins by exchange couplings

1+δ

and

1-δ

, respectively. Although this model has an infinite number of local conservation laws for

δ=0

, they are lost for finite

δ

. The ground-state phase diagram is determined using the numerical exact diagonalization and DMRG method in addition to the analytical approximations in various limiting cases. The phase diagram consists of a nonmagnetic phase and several kinds of ferrimagnetic phases. We find two different ferrimagnetic phases without spontaneous translational symmetry breakdown. It is also found that the quantized ferrimagnetic phases with large spatial periodicities present for

δ=0

are easily destroyed by small

δ

and replaced by a partial ferrimagnetic phase. The nonmagnetic phase is considered to be a gapless Tomonaga-Luttinger liquid phase based on the recently extended Lieb-Schultz-Mattis theorem to the site-reflection invariant spin chains and numerical diagonalization results.

Ground-State Phase Diagram of

(1/2,1/2,1)

Mixed Diamond Chains with Single-Site Anisotropy

The ground-state phases of mixed diamond chains with (

S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)

, where

S

is the magnitude of vertex spins, and

τ^{(1)}

and

τ^{(2)}

are those of apical spins, are investigated with the single-site anisotropy

D

on the

τ^{(2)}

-site. The two apical spins in each unit cell are coupled by an exchange coupling

λ

. The vertex spins are coupled with the top and bottom apical spins by exchange couplings

1+δ

and

1-δ

, respectively. The ground-state phase diagram is determined using the numerical exact diagonalization and DMRG method in addition to the analytical approximations in various limiting cases. The phase diagram consists of a Néel ordered phase, a nonmagnetic Tomonaga-Luttinger liquid phase, and quantized and partial ferrimagnetic phases. A region with anisotropy inversion is found where the Ising-like Néel phase is realized for the easy-plane anisotropy

D >0

and the XY-like Tomonaga-Luttinger liquid phase is realized for the easy-axis anisotropy

D <0

on the

S=1

sites.

m=1/6

and 1/3 per site. The ferrimagnetic state with

m=1/6

is accompanied by a spontaneous translational symmetry breakdown. This phase vanishes for strong bond alternation.

S=1/2

spins ($\vecs_{4\ell-3}$ and $\vecs_{4\ell-2}$) and two S=1 spins ($\vecS_{4\ell-1}$ and $\vecS_{4\ell}$) are arranged alternatively. Inseveral limiting cases of

J_1

J_2

, and

J_3

we apply the Wigner-Eckart theorem and carry out a perturbation calculation to examine the behavior of the massless lines where the energy gap vanishes. Performing a quantum Monte Carlo calculation without global flips at a sufficiently low temperature for the case where

J_1=J_3=1.0

and

J_2>0

, we find that the ground state of the present system in this case undergoes a second-order phase transition accompanying the vanishing of the energy gap at $J_2=J_{2c$ with

J_{2 c} = 0.77 \pm 0.01

. We also find that the ground states for both $J_2 < J_{2 c$ and $J_2 > J_{2c$ can be understood by means of the valence-bond-solid picture. A quantum Monte Carlo calculation which takes the global flips along the Trotter direction into account is carried out to elucidate the temperature dependences of the specific heat and the magnetic susceptibility. In particular, it is found that the susceptibility per unit cell for

J_2=0.77

with

J_1=J_3=1.0

takes a finite value at absolute zero temperature and that the specific heat per unit cell versus temperature curve for

J_2=5.0

with

J_1=J_3=1.0

has a double peak.

First Order Transitions Between the Gapped Spin-Liquid and Ferrimagnetic Phases in \((1/2,1/2,1)\) Mixed Diamond Chains with Bond Alternation

Abstract

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