Physical Review A

Magnetic-field-induced oscillation of multipartite nonlocality in spin ladders

Hong-Guang Cheng, Meng Li, Yuying Wu, Mei Wang, Duo Zhang, Jia Bao, Bin Guo, Zhao‐Yu Sun

Published May 22, 2020• 16 citations

Abstract

Spin-$\frac{1}{2}$ two-leg ladder models under a magnetic field have a well-known phase diagram. In this paper, we use multipartite nonlocality (a measure of multipartite quantum correlations) to characterize the quantum correlations in the ladder models at zero temperature. Both finite-size and infinite-size ladders are considered. We investigate the global nonlocality ${\mathcal{S}}_{g}=\mathcal{S}(|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle})$, which describes quantum correlations of the ground states $|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle}$ of the entire lattices, and the partial nonlocality ${\mathcal{S}}_{p}=\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n})$, which describes quantum correlations of the reduced states ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}$ of some sublattices in the ladders. We find that as the magnetic field $\ensuremath{\lambda}$ increases, the global nonlocality ${\mathcal{S}}_{g}$ presents a single-peak curve. Moreover, the logarithmic measure $ln{\mathcal{S}}_{g}$ changes dramatically at the two critical fields ${\ensuremath{\lambda}}_{{c}_{1}}$ and ${\ensuremath{\lambda}}_{{c}_{2}}$ of the models and thus signals the quantum phase transitions in the models. For the partial nonlocality ${\mathcal{S}}_{p}$, in the regions ${\ensuremath{\lambda}}_{{c}_{1}}<\ensuremath{\lambda}<{\ensuremath{\lambda}}_{{c}_{2}}$, we observe that the ${\mathcal{S}}_{p}(\ensuremath{\lambda})$ curve shows an oscillation. Numerical results reveal that the underling mechanism is the ``major component transitions'' in the reduced states ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}$ of the sublattices. More importantly, the oscillation of the partial nonlocality ${\mathcal{S}}_{p}$ is modulated by the single-peak curve of the global nonlocality ${\mathcal{S}}_{g}$. The result provides valuable clues about the relation between partial nonlocality and global nonlocality in low-dimensional quantum models.

DOI: 10.1103/physreva.101.052116

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Physical Review A

Magnetic-field-induced oscillation of multipartite nonlocality in spin ladders

Hong-Guang Cheng, Meng Li, Yuying Wu, Mei Wang, Duo Zhang, Jia Bao, Bin Guo, Zhao‐Yu Sun

Published May 22, 2020• 16 citations

Publisher

Abstract

Spin-

\frac{1}{2}

two-leg ladder models under a magnetic field have a well-known phase diagram. In this paper, we use multipartite nonlocality (a measure of multipartite quantum correlations) to characterize the quantum correlations in the ladder models at zero temperature. Both finite-size and infinite-size ladders are considered. We investigate the global nonlocality

{\mathcal{S}}_{g}=\mathcal{S}(|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle})

, which describes quantum correlations of the ground states

|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle}

of the entire lattices, and the partial nonlocality

{\mathcal{S}}_{p}=\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n})

, which describes quantum correlations of the reduced states

{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}

of some sublattices in the ladders. We find that as the magnetic field

\ensuremath{\lambda}

increases, the global nonlocality

{\mathcal{S}}_{g}

presents a single-peak curve. Moreover, the logarithmic measure

ln{\mathcal{S}}_{g}

changes dramatically at the two critical fields

{\ensuremath{\lambda}}_{{c}_{1}}

and

{\ensuremath{\lambda}}_{{c}_{2}}

of the models and thus signals the quantum phase transitions in the models. For the partial nonlocality

{\mathcal{S}}_{p}

, in the regions

{\ensuremath{\lambda}}_{{c}_{1}}<\ensuremath{\lambda}<{\ensuremath{\lambda}}_{{c}_{2}}

, we observe that the

{\mathcal{S}}_{p}(\ensuremath{\lambda})

curve shows an oscillation. Numerical results reveal that the underling mechanism is the ``major component transitions'' in the reduced states

{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}

of the sublattices. More importantly, the oscillation of the partial nonlocality

{\mathcal{S}}_{p}

is modulated by the single-peak curve of the global nonlocality

{\mathcal{S}}_{g}

. The result provides valuable clues about the relation between partial nonlocality and global nonlocality in low-dimensional quantum models.

DOI: 10.1103/physreva.101.052116

Related papers

Multipartite nonlocality and boundary conditions in one-dimensional spin chains

Zhao‐Yu Sun, Mei Wang, Yu-Yin Wu +1 more

In quantum lattice models, in the large- $N$ limit, boundary conditions have little effect upon local observables for sites in the centers of the lattices. In this paper, we will study the boundary effects upon multipartite nonlocality (a kind of multipartite quantum correlation associated with Bell-type inequalities) in one-dimensional finite-size spin chains, both for zero temperature and for finite temperatures. We define a quantity $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ to characterize the boundary effects, where $\mathcal{S}$ is a measure of global multipartite nonlocality of the entire lattice, and $\ensuremath{\delta}\mathcal{S}$ is the difference of the measure induced by changing the boundary conditions. We find $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ does not vanish in the large- $N$ limit. Instead, at zero temperature, with the increase of $N$ , $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ would increase steadily in the vicinity of the quantum phase transition point of the models, and converge to a nonzero constant in noncritical regions. It shows clearly that boundary effects generally exist, in the form of multipartite correlations, in long chains. The boundary effects are explained by the competition between the two orders of the models. In addition, based on these numerical results, we construct a Bell inequality, which is violated by chains with periodic (closed) boundary conditions and not violated by chains with open boundary conditions. Furthermore, we study $\frac{\ensuremath{\delta}{\mathcal{S}}_{T}}{{\mathcal{S}}_{T}}$ of finite-size chains at finite temperatures, and show that boundary effects survive in finite temperature regions.In quantum lattice models, in the large-$N$ limit, boundary conditions have little effect upon local observables for sites in the centers of the lattices. In this paper, we will study the boundary effects upon multipartite nonlocality (a kind of multipartite quantum correlation associated with Bell-type inequalities) in one-dimensional finite-size spin chains, both for zero temperature and for finite temperatures. We define a quantity $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ to characterize the boundary effects, where $\mathcal{S}$ is a measure of global multipartite nonlocality of the entire lattice, and $\ensuremath{\delta}\mathcal{S}$ is the difference of the measure induced by changing the boundary conditions. We find $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ does not vanish in the large-$N$ limit. Instead, at zero temperature, with the increase of $N$, $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ would increase steadily in the vicinity of the quantum phase transition point of the models, and converge to a nonzero constant in noncritical regions. It shows clearly that boundary effects generally exist, in the form of multipartite correlations, in long chains. The boundary effects are explained by the competition between the two orders of the models. In addition, based on these numerical results, we construct a Bell inequality, which is violated by chains with periodic (closed) boundary conditions and not violated by chains with open boundary conditions. Furthermore, we study $\frac{\ensuremath{\delta}{\mathcal{S}}_{T}}{{\mathcal{S}}_{T}}$ of finite-size chains at finite temperatures, and show that boundary effects survive in finite temperature regions.

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Physical Review AJan 14, 202031 citations

Multipartite nonlocality in one-dimensional quantum spin chains at finite temperatures

Zhao‐Yu Sun, Meng Li, Long-Hui Sheng +1 more

Multipartite nonlocality is an important measure of multipartite quantum correlations. In this paper, we show that the nonlocal $n$ -site Mermin-Klyshko operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}}_{n}$ can be exactly expressed as a matrix product operator with a bond dimension $D=2$ , and then the calculation of nonlocality measure $\mathcal{S}$ can be simplified into standard one-dimensional (1D) tensor networks. With the help of this technique, we analyze finite-temperature multipartite nonlocality in several typical 1D spin chains, including an $XX$ model, an $XXZ$ model, and a Kitaev model. For the $XX$ model and the $XXZ$ model, in a finite-temperature region, the logarithm measure of nonlocality ( ${log}_{2}\mathcal{S}$ ) is a linear function of the temperature $T$ , i.e., ${log}_{2}\mathcal{S}\ensuremath{\sim}\ensuremath{-}aT+b$. It provides us with an intuitive picture about how thermodynamic fluctuations destroy multipartite nonlocality in 1D quantum chains. Moreover, in the $XX$ model $\mathcal{S}$ presents a magnetic-field-induced oscillation at low temperatures. This behavior has a nonlocal nature and cannot be captured by local properties such as the magnetization. Finally, for the Kitaev model, we find that in the limit $T\ensuremath{\rightarrow}0$ and $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the nonlocality measure may be used as an alternative order parameter for the topological-type quantum phase transition in the model.Multipartite nonlocality is an important measure of multipartite quantum correlations. In this paper, we show that the nonlocal $n$-site Mermin-Klyshko operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}}_{n}$ can be exactly expressed as a matrix product operator with a bond dimension $D=2$, and then the calculation of nonlocality measure $\mathcal{S}$ can be simplified into standard one-dimensional (1D) tensor networks. With the help of this technique, we analyze finite-temperature multipartite nonlocality in several typical 1D spin chains, including an $XX$ model, an $XXZ$ model, and a Kitaev model. For the $XX$ model and the $XXZ$ model, in a finite-temperature region, the logarithm measure of nonlocality (${log}_{2}\mathcal{S}$) is a linear function of the temperature $T$, i.e., ${log}_{2}\mathcal{S}\ensuremath{\sim}\ensuremath{-}aT+b$. It provides us with an intuitive picture about how thermodynamic fluctuations destroy multipartite nonlocality in 1D quantum chains. Moreover, in the $XX$ model $\mathcal{S}$ presents a magnetic-field-induced oscillation at low temperatures. This behavior has a nonlocal nature and cannot be captured by local properties such as the magnetization. Finally, for the Kitaev model, we find that in the limit $T\ensuremath{\rightarrow}0$ and $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the nonlocality measure may be used as an alternative order parameter for the topological-type quantum phase transition in the model.

Physical Review AMay 11, 202115 citations

Global multipartite nonlocality and Bell-type inequalities in infinite-size quantum spin chains

Zhao‐Yu Sun, Bin Guo, Hailin Huang

In this paper, we characterize the global multipartite nonlocality in one-dimensional infinite-size spin-1/2 chains. For this purpose, we calculate the multipartite nonlocality of $n$ -site subchains, and take the large- $n$ limit. As $n$ increases from 2 to 100, we find that the average revenue and the marginal revenue of nonlocality converge gradually, thus they can be used to characterize the global multipartite nonlocality in infinite-size chains. Furthermore, in the vicinity of the quantum phase transition (QPT) points of the transverse-field Ising model and the $\mathit{XXZ}$ model, in the large- $n$ limit, multipartite nonlocality becomes extremely sensitive to external perturbation, that is, arbitrarily small perturbation away from the QPT points would lead to a drastic change in the global nonlocality. This signal of QPTs is easily distinguishable, and can be used to detect QPTs. We also draw a vivid physical picture of multipartite correlations and QPTs by analogy with magnetic domains in ferromagnets.In this paper, we characterize the global multipartite nonlocality in one-dimensional infinite-size spin-1/2 chains. For this purpose, we calculate the multipartite nonlocality of $n$-site subchains, and take the large-$n$ limit. As $n$ increases from 2 to 100, we find that the average revenue and the marginal revenue of nonlocality converge gradually, thus they can be used to characterize the global multipartite nonlocality in infinite-size chains. Furthermore, in the vicinity of the quantum phase transition (QPT) points of the transverse-field Ising model and the $\mathit{XXZ}$ model, in the large-$n$ limit, multipartite nonlocality becomes extremely sensitive to external perturbation, that is, arbitrarily small perturbation away from the QPT points would lead to a drastic change in the global nonlocality. This signal of QPTs is easily distinguishable, and can be used to detect QPTs. We also draw a vivid physical picture of multipartite correlations and QPTs by analogy with magnetic domains in ferromagnets.

Physical Review AAug 24, 201526 citations

Spin-

\frac{1}{2}

{\mathcal{S}}_{g}=\mathcal{S}(|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle})

, which describes quantum correlations of the ground states

|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle}

of the entire lattices, and the partial nonlocality

{\mathcal{S}}_{p}=\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n})

, which describes quantum correlations of the reduced states

{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}

of some sublattices in the ladders. We find that as the magnetic field

\ensuremath{\lambda}

increases, the global nonlocality

{\mathcal{S}}_{g}

presents a single-peak curve. Moreover, the logarithmic measure

ln{\mathcal{S}}_{g}

changes dramatically at the two critical fields

{\ensuremath{\lambda}}_{{c}_{1}}

and

{\ensuremath{\lambda}}_{{c}_{2}}

of the models and thus signals the quantum phase transitions in the models. For the partial nonlocality

{\mathcal{S}}_{p}

, in the regions

{\ensuremath{\lambda}}_{{c}_{1}}<\ensuremath{\lambda}<{\ensuremath{\lambda}}_{{c}_{2}}

, we observe that the

{\mathcal{S}}_{p}(\ensuremath{\lambda})

curve shows an oscillation. Numerical results reveal that the underling mechanism is the ``major component transitions'' in the reduced states

{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}

of the sublattices. More importantly, the oscillation of the partial nonlocality

{\mathcal{S}}_{p}

is modulated by the single-peak curve of the global nonlocality

{\mathcal{S}}_{g}

. The result provides valuable clues about the relation between partial nonlocality and global nonlocality in low-dimensional quantum models.

In quantum lattice models, in the large-

N

limit, boundary conditions have little effect upon local observables for sites in the centers of the lattices. In this paper, we will study the boundary effects upon multipartite nonlocality (a kind of multipartite quantum correlation associated with Bell-type inequalities) in one-dimensional finite-size spin chains, both for zero temperature and for finite temperatures. We define a quantity $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ to characterize the boundary effects, where

\mathcal{S}

is a measure of global multipartite nonlocality of the entire lattice, and $\ensuremath{\delta}\mathcal{S}$ is the difference of the measure induced by changing the boundary conditions. We find $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ does not vanish in the large-

N

limit. Instead, at zero temperature, with the increase of

N

, $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ would increase steadily in the vicinity of the quantum phase transition point of the models, and converge to a nonzero constant in noncritical regions. It shows clearly that boundary effects generally exist, in the form of multipartite correlations, in long chains. The boundary effects are explained by the competition between the two orders of the models. In addition, based on these numerical results, we construct a Bell inequality, which is violated by chains with periodic (closed) boundary conditions and not violated by chains with open boundary conditions. Furthermore, we study $\frac{\ensuremath{\delta}{\mathcal{S}}_{T}}{{\mathcal{S}}_{T}}$ of finite-size chains at finite temperatures, and show that boundary effects survive in finite temperature regions.

Multipartite quantum nonlocality and Bell-type inequalities are used to characterize quantum correlations in an infinite-size bond-alternating spin-

\frac{1}{2}

Heisenberg chain with next-nearest-neighbor interactions. With the help of powerful tensor-network algorithms, both zero and finite temperatures are considered. First, at zero temperature, both in the even-Haldane phase and in the odd-Haldane phase, a high hierarchy of multipartite nonlocality is observed. Nevertheless, in the two phases, the spread of the multipartite nonlocality among the lattice is different. Thereby, the nonlocality measures are relatively large in one phase and vanish in the other phase, and provide quite sharp signals for the topological quantum phase transitions (QPTs) between these two phases. The influence of the next-nearest-neighbor coupling $\ensuremath{\alpha}$ upon the multipartite nonlocality in the model is also discussed. Second, we find that the footprints of the QPTs survive at low temperatures. Third, based upon the scaling behavior of the finite-temperature nonlocality measure, we propose a quantity

\mathcal{K}

to characterize the finite-temperature nonlocality in the large-

n

limit. We find that in high-temperature regions,

\mathcal{K}

is reduced linearly as the temperature rises.

n

-spin subsystem) in the QPT of an infinite

XY

chain. An efficient numerical optimization procedure is proposed to figure out the violation measure

{M}_{n}

of the inequalities. For $n\ensuremath{\le}7$, the magnetic-field ($\ensuremath{\lambda}$) dependence of

{M}_{n}

is studied. We find the derivative of

{M}_{n}

is divergent exactly at the QPT point ${\ensuremath{\lambda}}_{c}=1$ for any

n

. In addition, with the increase of

n

{M}_{n}

converges quickly for $\ensuremath{\lambda}<{\ensuremath{\lambda}}_{c}$ and converges very slowly for $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}$, which can be regarded as another signal for the QPT. Furthermore, in the vicinity of ${\ensuremath{\lambda}}_{c}$, high-order Bell-type inequalities will be violated as long as

n

is large enough. This indicates that high-level multipartite correlation will be present when the system is in the vicinity of the QPT point. Nevertheless, genuine

n

-partite entanglement or genuine

n

-partite nonlocality is not observed in the QPT.

Lipkin-Meshkov-Glick (LMG) models describe a set of

N

qubits which are located at the corners of an

N

-dimensional simplex and embedded in an external magnetic field. Because of its high coordinate number, quantum correlations in the models are quite nontrivial. For instance, in the large-

N

limit, two-site entanglement concurrence vanishes for any magnetic field. Multisite global entanglement and generalized global entanglement also tend to vanish in some quantum regions. We characterize the quantum correlations in the LMG model with multipartite nonlocality. In LMG models with anisotropy in the $x\ensuremath{-}y$ plane, for any fixed magnetic field, the ground-state nonlocality (denoted by

\mathcal{S}

) is found to scale as ${log}_{10}\mathcal{S}\ensuremath{\sim}aN+b$, with

a

and

b

the fitting parameters. For LMG models which are isotropic in the $x\ensuremath{-}y$ plane, nevertheless, the nonlocality for each Dicke state scales as ${log}_{10}\mathcal{S}\ensuremath{\sim}a\frac{1}{N}+b$, with

b>0

. Signals for the quantum phase transitions of the models will also be discussed. These results indicate that multipartite nonlocality captures some key ingredient of the ground-state quantum correlations in the models. In addition to the ground states, we also study multipartite nonlocality in the LMG models at finite temperatures. In the anisotropic models, the thermal stability of the nonlocality at low temperatures is determined by the energy gap. For the isotropic models, nevertheless, as the temperature increases, the nonlocality presents a sharp valley which is followed by a round peak. The behavior is far beyond the above-mentioned ``thermal stability vs energy gap'' picture. The underlying mechanics is the ``contribution inversion'' of low-lying energy states in the thermal-state nonlocality $\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{T})$; that is, in a low-temperature region, the first-excited state ``defeats'' the ground state and plays a dominant role in thermal-state nonlocality.

Multipartite nonlocality is an important measure of multipartite quantum correlations. In this paper, we show that the nonlocal

n

-site Mermin-Klyshko operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}}_{n}$ can be exactly expressed as a matrix product operator with a bond dimension

D=2

, and then the calculation of nonlocality measure

\mathcal{S}

can be simplified into standard one-dimensional (1D) tensor networks. With the help of this technique, we analyze finite-temperature multipartite nonlocality in several typical 1D spin chains, including an

XX

model, an

XXZ

model, and a Kitaev model. For the

XX

model and the

XXZ

model, in a finite-temperature region, the logarithm measure of nonlocality (

{log}_{2}\mathcal{S}

) is a linear function of the temperature

T

, i.e., ${log}_{2}\mathcal{S}\ensuremath{\sim}\ensuremath{-}aT+b$. It provides us with an intuitive picture about how thermodynamic fluctuations destroy multipartite nonlocality in 1D quantum chains. Moreover, in the

XX

model

\mathcal{S}

presents a magnetic-field-induced oscillation at low temperatures. This behavior has a nonlocal nature and cannot be captured by local properties such as the magnetization. Finally, for the Kitaev model, we find that in the limit $T\ensuremath{\rightarrow}0$ and $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the nonlocality measure may be used as an alternative order parameter for the topological-type quantum phase transition in the model.

In this paper, we characterize the global multipartite nonlocality in one-dimensional infinite-size spin-1/2 chains. For this purpose, we calculate the multipartite nonlocality of

n

-site subchains, and take the large-

n

limit. As

n

increases from 2 to 100, we find that the average revenue and the marginal revenue of nonlocality converge gradually, thus they can be used to characterize the global multipartite nonlocality in infinite-size chains. Furthermore, in the vicinity of the quantum phase transition (QPT) points of the transverse-field Ising model and the

\mathit{XXZ}

model, in the large-

n

limit, multipartite nonlocality becomes extremely sensitive to external perturbation, that is, arbitrarily small perturbation away from the QPT points would lead to a drastic change in the global nonlocality. This signal of QPTs is easily distinguishable, and can be used to detect QPTs. We also draw a vivid physical picture of multipartite correlations and QPTs by analogy with magnetic domains in ferromagnets.