arXiv:quant-ph/0508115 · Physical Review B

Entanglement and quantum phase transitions in quantum mixed spin chains

Shangbin Li, Zhaoxin Xu, Jianhui Dai, Jing‐Bo Xu

Published May 10, 2006• 16 citations

Abstract

The ground entanglement and thermal entanglement in quantum mixed spin chains consisting of two integer spins 1 and two half-integer spins $\frac{1}{2}$ arrayed as $\frac{1}{2}\text{\ensuremath{-}}\frac{1}{2}\text{\ensuremath{-}}1\text{\ensuremath{-}}1$ in a unit cell with antiferromagnetic nearest-neighbor couplings ${J}_{1}$ $({J}_{2})$ between the spins of equal (different) magnitudes are investigated by employing the logarithmic negativity as entanglement measure. The ground entanglement transition found here is closely related to the valence-bond-solid phase transition, and the thermal entanglement near the critical point is calculated and it is shown that two distinct behaviors exist in the nearest-neighbor same kind of spins and different kind of spins, respectively.

DOI: 10.1103/physrevb.73.184411

Related papers

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

Thermal entanglement properties of small spin clusters

Indrani Bose, Amit Tribedi

arXiv:cond-mat/0503170Physical Review AAug 15, 200557 citations

A Quantum Monte Carlo Study on Mixed-Spin Chains of 1／2-1／2-1-1 and 3／2-3／2-1-1

XUZhao-Xin, ZHANGJun, YINGHe-Ping

理论物理通讯：英文版Jan 1, 20030 citations

A Quantum Monte Carlo Study on Mixed-Spin Chains of

Xu, Zhao-Xin

理论物理通讯：英文版Jan 1, 20030 citations

Ground-State Phase Diagram of the Spin-$\frac{\bf 1}{\bf 2}$-$\frac{\bf 1}{\bf 2}$-1-1 Chain

Toshiya Hikihara, Takashi Tonegawa, Makoto Kaburagi +3 more

We investigate the ground-state phase diagram of the quantum mixed spin chain in which two $S\!=\!\frac{1}{2}$ and two S =1 spins are arranged alternatively and isotropic exchange interactions between nearest-neighboring pairs of spins are assumed. Guided by our previous results [J. Phys. Soc. Jpn. 67 (1998) 1000] for a few limiting cases of the exchange interactions, we carry out a density-matrix renormalization-group calculation to determine the ground-state phase diagram in detail. From the gapless points of the singlet-triplet energy gap, five boundary lines are determined, which divide the phase diagram into six regions. In three regions the ground states with the Haldane-phase character appear, while in the remaining three regions those with the dimer character appear. We also present a characterization of the phases concerning how the ground state in each phase is represented in the framework of the valence-bond-solid (VBS) picture. Making use of the matrix-product wave function, we classify the VBS pictures and interpret the ground-state phase diagram in terms of symmetries schematically related to the VBS pictures. We investigate the ground-state nearest-neighboring two-spin correlation functions and a kind of string order parameters, by which we confirm the validity of the characterization.

Journal of the Physical Society of JapanApr 15, 200010 citations

Topological phases in half-integer higher spin J 1 − J 2 Heisenberg chains

Sahinur Reja, Satoshi Nishimoto

arXiv:2406.04030Physical review. B./Physical review. BAug 23, 20244 citations

arXiv:quant-ph/0508115 · Physical Review B

Entanglement and quantum phase transitions in quantum mixed spin chains

Shangbin Li, Zhaoxin Xu, Jianhui Dai, Jing‐Bo Xu

Published May 10, 2006• 16 citations

arXiv HTML PDF

Abstract

The ground entanglement and thermal entanglement in quantum mixed spin chains consisting of two integer spins 1 and two half-integer spins

\frac{1}{2}

arrayed as

\frac{1}{2}\text{\ensuremath{-}}\frac{1}{2}\text{\ensuremath{-}}1\text{\ensuremath{-}}1

in a unit cell with antiferromagnetic nearest-neighbor couplings

{J}_{1}

({J}_{2})

between the spins of equal (different) magnitudes are investigated by employing the logarithmic negativity as entanglement measure. The ground entanglement transition found here is closely related to the valence-bond-solid phase transition, and the thermal entanglement near the critical point is calculated and it is shown that two distinct behaviors exist in the nearest-neighbor same kind of spins and different kind of spins, respectively.

DOI: 10.1103/physrevb.73.184411

Related papers

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

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

Thermal entanglement properties of small spin clusters

Indrani Bose, Amit Tribedi

Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude $\frac{1}{2}$ , in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength ${J}_{1}$ , ${J}_{2}$ , and $K$ , respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a $S=\frac{1}{2}$ antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with ${J}_{2}=0$ , $K=0$ , in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature ${T}_{c}$ below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron $({J}_{1}={J}_{2},K=0)$ ---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature ${T}_{E}$ below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude $\frac{1}{2}$, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength ${J}_{1}$, ${J}_{2}$, and $K$, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a $S=\frac{1}{2}$ antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with ${J}_{2}=0$, $K=0$, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature ${T}_{c}$ below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron $({J}_{1}={J}_{2},K=0)$---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature ${T}_{E}$ below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.

arXiv:cond-mat/0503170Physical Review AAug 15, 200557 citations

A Quantum Monte Carlo Study on Mixed-Spin Chains of 1／2-1／2-1-1 and 3／2-3／2-1-1

XUZhao-Xin, ZHANGJun, YINGHe-Ping

The ground-state and thermodynamic properties of quantum mixed-spin chains of1/2-1/2-1-1and 3/2-3/2-1-1are investigated by a quantum Monte Carlo simulation with the loop-cluster algorithm. For 1/2-1/2-1-1 chain, we find it has two phases separated by an energy-gap vanishing point in the ground-state. For 3/2-3/2-1-1 chain, the numerical results show two energy-gap vanishing points isolated by different phases in its ground-state. Our calculations indicate that all these ground state phases can be understood by means of valence-bond-solid picture, and the thermodynamic behavior at finite temperatures is continuous as a function of parameterα=J2/J1.The ground-state and thermodynamic properties of quantum mixed-spin chains of1/2-1/2-1-1and 3/2-3/2-1-1are investigated by a quantum Monte Carlo simulation with the loop-cluster algorithm. For 1/2-1/2-1-1 chain, we find it has two phases separated by an energy-gap vanishing point in the ground-state. For 3/2-3/2-1-1 chain, the numerical results show two energy-gap vanishing points isolated by different phases in its ground-state. Our calculations indicate that all these ground state phases can be understood by means of valence-bond-solid picture, and the thermodynamic behavior at finite temperatures is continuous as a function of parameterα=J2/J1.

理论物理通讯：英文版Jan 1, 20030 citations

A Quantum Monte Carlo Study on Mixed-Spin Chains of

Xu, Zhao-Xin

理论物理通讯：英文版Jan 1, 20030 citations

Ground-State Phase Diagram of the Spin- $\frac{\bf 1}{\bf 2}$ - $\frac{\bf 1}{\bf 2}$ -1-1 ChainGround-State Phase Diagram of the Spin-$\frac{\bf 1}{\bf 2}$-$\frac{\bf 1}{\bf 2}$-1-1 Chain

Toshiya Hikihara, Takashi Tonegawa, Makoto Kaburagi +3 more

We investigate the ground-state phase diagram of the quantum mixed spin chain in which two $S\!=\!\frac{1}{2}$ and two S =1 spins are arranged alternatively and isotropic exchange interactions between nearest-neighboring pairs of spins are assumed. Guided by our previous results [J. Phys. Soc. Jpn. 67 (1998) 1000] for a few limiting cases of the exchange interactions, we carry out a density-matrix renormalization-group calculation to determine the ground-state phase diagram in detail. From the gapless points of the singlet-triplet energy gap, five boundary lines are determined, which divide the phase diagram into six regions. In three regions the ground states with the Haldane-phase character appear, while in the remaining three regions those with the dimer character appear. We also present a characterization of the phases concerning how the ground state in each phase is represented in the framework of the valence-bond-solid (VBS) picture. Making use of the matrix-product wave function, we classify the VBS pictures and interpret the ground-state phase diagram in terms of symmetries schematically related to the VBS pictures. We investigate the ground-state nearest-neighboring two-spin correlation functions and a kind of string order parameters, by which we confirm the validity of the characterization.We investigate the ground-state phase diagram of the quantum mixed spin chain in which two $S\!=\!\frac{1}{2}$ and two S =1 spins are arranged alternatively and isotropic exchange interactions between nearest-neighboring pairs of spins are assumed. Guided by our previous results [J. Phys. Soc. Jpn. 67 (1998) 1000] for a few limiting cases of the exchange interactions, we carry out a density-matrix renormalization-group calculation to determine the ground-state phase diagram in detail. From the gapless points of the singlet-triplet energy gap, five boundary lines are determined, which divide the phase diagram into six regions. In three regions the ground states with the Haldane-phase character appear, while in the remaining three regions those with the dimer character appear. We also present a characterization of the phases concerning how the ground state in each phase is represented in the framework of the valence-bond-solid (VBS) picture. Making use of the matrix-product wave function, we classify the VBS pictures and interpret the ground-state phase diagram in terms of symmetries schematically related to the VBS pictures. We investigate the ground-state nearest-neighboring two-spin correlation functions and a kind of string order parameters, by which we confirm the validity of the characterization.

Journal of the Physical Society of JapanApr 15, 200010 citations

Topological phases in half-integer higher spin J 1 − J 2 Heisenberg chains

Sahinur Reja, Satoshi Nishimoto

We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from $S=\frac{3}{2}$ to $S=\frac{11}{2}$ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of $\frac{{J}_{2}}{{J}_{1}}$ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and $\frac{2S+1}{2}$ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with $S$ and converge to a single point at $\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}$ in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for $S=\frac{1}{2}$ , and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of $\frac{{J}_{2}}{{J}_{1}}$ values.We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from $S=\frac{3}{2}$ to $S=\frac{11}{2}$ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of $\frac{{J}_{2}}{{J}_{1}}$ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and $\frac{2S+1}{2}$ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with $S$ and converge to a single point at $\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}$ in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for $S=\frac{1}{2}$, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of $\frac{{J}_{2}}{{J}_{1}}$ values.

arXiv:2406.04030Physical review. B./Physical review. BAug 23, 20244 citations

The ground entanglement and thermal entanglement in quantum mixed spin chains consisting of two integer spins 1 and two half-integer spins

\frac{1}{2}

arrayed as

\frac{1}{2}\text{\ensuremath{-}}\frac{1}{2}\text{\ensuremath{-}}1\text{\ensuremath{-}}1

in a unit cell with antiferromagnetic nearest-neighbor couplings

{J}_{1}

({J}_{2})

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.

Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude

\frac{1}{2}

, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength

{J}_{1}

{J}_{2}

, and

K

, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a

S=\frac{1}{2}

antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with

{J}_{2}=0

K=0

, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature

{T}_{c}

below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron

({J}_{1}={J}_{2},K=0)

---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature

{T}_{E}

below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.

Ground-State Phase Diagram of the Spin-

\frac{\bf 1}{\bf 2}

\frac{\bf 1}{\bf 2}

-1-1 Chain

We investigate the ground-state phase diagram of the quantum mixed spin chain in which two

S\!=\!\frac{1}{2}

and two S =1 spins are arranged alternatively and isotropic exchange interactions between nearest-neighboring pairs of spins are assumed. Guided by our previous results [J. Phys. Soc. Jpn. 67 (1998) 1000] for a few limiting cases of the exchange interactions, we carry out a density-matrix renormalization-group calculation to determine the ground-state phase diagram in detail. From the gapless points of the singlet-triplet energy gap, five boundary lines are determined, which divide the phase diagram into six regions. In three regions the ground states with the Haldane-phase character appear, while in the remaining three regions those with the dimer character appear. We also present a characterization of the phases concerning how the ground state in each phase is represented in the framework of the valence-bond-solid (VBS) picture. Making use of the matrix-product wave function, we classify the VBS pictures and interpret the ground-state phase diagram in terms of symmetries schematically related to the VBS pictures. We investigate the ground-state nearest-neighboring two-spin correlation functions and a kind of string order parameters, by which we confirm the validity of the characterization.

We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from

S=\frac{3}{2}

S=\frac{11}{2}

using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of

\frac{{J}_{2}}{{J}_{1}}

containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and

\frac{2S+1}{2}

valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with

S

and converge to a single point at

\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}

in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for

S=\frac{1}{2}

, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of

\frac{{J}_{2}}{{J}_{1}}

values.