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

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Multipartite nonlocality spectrum and quantum criticality in one-dimensional quantum chains

Fan-Qin Xu, Hui-Xin Wen, Zhao‐Yu Sun

Published Oct 22, 2024• 2 citations
Publisher

Abstract

Multipartite nonlocality is a measure of multipartite quantum correlations, and has been used to characterize quantum criticality in various low-dimensional quantum models. In this work, we propose the nonlocality spectrum ${|{\ensuremath{\lambda}}_{i}|}$ of the nonlocality transfer matrix as a tool to characterize multipartite nonlocality and quantum criticality in infinite-size translation-invariant quantum systems. The approach is used to analyze four typical quantum spin chain models, i.e., the transverse-field Ising model, an XXZ model, a three-body coupling cluster model and an XYZ model. In various types of quantum phase transitions in these models, it is found that at the critical points, the leading eigenvalue $|{\ensuremath{\lambda}}_{1}|$ presents identifiable signals, and the spectrum gap $\mathrm{\ensuremath{\Delta}}|\ensuremath{\lambda}|=|{\ensuremath{\lambda}}_{1}|\ensuremath{-}|{\ensuremath{\lambda}}_{2}|$ closes. Moreover, the nonlocality spectrum provides a unified mathematical framework to reveal the underlying mechanisms for various previously reported behaviors of multipartite nonlocality. Our results demonstrate the power of the nonlocality spectrum as a valuable tool for the comprehensive characterization of multipartite nonlocality and quantum criticality in one-dimensional quantum systems.

DOI
10.1103/physreva.110.042217

Related papers

Multipartite nonlocality spectrum and quantum criticality in the one-dimensional infinite quantum model

Jia Bao, Bin Guo, S. Q. Qu +2 more

Based on transfer matrix theory and the multipartite nonlocality spectrum [Formula: see text], we systematically investigate ground-state nonlocal correlations across three canonical infinite, translation-invariant one-dimensional spin models, including the XXZ model, the three-body coupled Cluster–Ising model, and the Kitaev ladder model, which cover both conventional and topological Quantum Phase Transitions (QPTs). We show that the emergence of multipartite nonlocal correlations can be quantitatively determined via the leading eigenvalue criterion [Formula: see text], offering a concise and reliable indicator for diagnosing nonlocal quantum features. Further numerical comparisons reveal that the leading eigenvalue [Formula: see text] is strongly model-dependent and lacks universal applicability for identifying quantum critical behaviors across different QPT types. In contrast, the nonlocality spectral gap [Formula: see text] exhibits prominent robustness against variations in system symmetry, coupling strength, and intrinsic transition mechanisms. It characterizes quantum criticality through three distinct response modes, thereby enabling a unified description of both conventional and topological QPTs. This work establishes the nonlocality spectrum as a versatile and powerful framework for the joint characterization of multipartite nonlocality and quantum criticality in one-dimensional many-body quantum systems.

Modern Physics Letters B•Sep 4, 2026•0 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.

Physical Review A•Aug 24, 2015•26 citations

Multipartite nonlocality in one-dimensional quantum chains: A transfer-matrix theory

Zhao‐Yu Sun, Hui-Xin Wen, Meng Li +1 more

We propose a transfer-matrix theory to reveal the hidden structure in the multipartite nonlocality operators in one-dimensional (1D) quantum chains. The theory offers a unified description for the scaling behaviors of multipartite quantum nonlocality in general 1D quantum lattices. Furthermore, in order to figure out the hierarchy of multipartite nonlocality for infinite-size chains, powerful transfer-matrix-based algorithms are proposed. In quantum critical regions, the algorithms converge much faster than the traditional approach.

Physical Review A•Jan 25, 2022•9 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.

Physical Review A•May 11, 2021•15 citations

Multipartite nonlocality and symmetry breaking in one-dimensional quantum chains

Zhao‐Yu Sun, Liang-Fu Liu, Hui-Xin Wen +3 more

Multipartite nonlocality is a measure of multipartite quantum correlations. In this paper we investigate the influence of symmetry-breaking perturbations upon ground-state nonlocality by considering several typical finite-size quantum models, including the transverse-field Ising chains, the $XY$($XX$) chains, and the $XXZ$ chains. We find that even a slight perturbation can reshape the nonlocality curve dramatically. For instance, in the $XY$ chains, a perturbation can induce an oscillation behavior in the nonlocality curve. A clear microscopic mechanism for the results is proposed. Furthermore, we also connect the behaviors to the macroscopic symmetry properties of the chains and make some predictions on general models.

Physical Review A•May 24, 2023•5 citations

Multipartite quantum nonlocality and Bell-type inequalities in a spin-1/2 model with topological quantum phase transitions

Zhao‐Yu Sun, Hui-Xin Wen, Meng Li +1 more

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.

Physical Review A•Nov 2, 2021•11 citations
Multipartite nonlocality is a measure of multipartite quantum correlations, and has been used to characterize quantum criticality in various low-dimensional quantum models. In this work, we propose the nonlocality spectrum ∣\ensuremathλi∣{|{\ensuremath{\lambda}}_{i}|}∣\ensuremathλi​∣ of the nonlocality transfer matrix as a tool to characterize multipartite nonlocality and quantum criticality in infinite-size translation-invariant quantum systems. The approach is used to analyze four typical quantum spin chain models, i.e., the transverse-field Ising model, an XXZ model, a three-body coupling cluster model and an XYZ model. In various types of quantum phase transitions in these models, it is found that at the critical points, the leading eigenvalue ∣\ensuremathλ1∣|{\ensuremath{\lambda}}_{1}|∣\ensuremathλ1​∣ presents identifiable signals, and the spectrum gap \ensuremathΔ∣\ensuremathλ∣=∣\ensuremathλ1∣\ensuremath−∣\ensuremathλ2∣\mathrm{\ensuremath{\Delta}}|\ensuremath{\lambda}|=|{\ensuremath{\lambda}}_{1}|\ensuremath{-}|{\ensuremath{\lambda}}_{2}|\ensuremathΔ∣\ensuremathλ∣=∣\ensuremathλ1​∣\ensuremath−∣\ensuremathλ2​∣ closes. Moreover, the nonlocality spectrum provides a unified mathematical framework to reveal the underlying mechanisms for various previously reported behaviors of multipartite nonlocality. Our results demonstrate the power of the nonlocality spectrum as a valuable tool for the comprehensive characterization of multipartite nonlocality and quantum criticality in one-dimensional quantum systems.
Based on transfer matrix theory and the multipartite nonlocality spectrum [Formula: see text], we systematically investigate ground-state nonlocal correlations across three canonical infinite, translation-invariant one-dimensional spin models, including the XXZ model, the three-body coupled Cluster–Ising model, and the Kitaev ladder model, which cover both conventional and topological Quantum Phase Transitions (QPTs). We show that the emergence of multipartite nonlocal correlations can be quantitatively determined via the leading eigenvalue criterion [Formula: see text], offering a concise and reliable indicator for diagnosing nonlocal quantum features. Further numerical comparisons reveal that the leading eigenvalue [Formula: see text] is strongly model-dependent and lacks universal applicability for identifying quantum critical behaviors across different QPT types. In contrast, the nonlocality spectral gap [Formula: see text] exhibits prominent robustness against variations in system symmetry, coupling strength, and intrinsic transition mechanisms. It characterizes quantum criticality through three distinct response modes, thereby enabling a unified description of both conventional and topological QPTs. This work establishes the nonlocality spectrum as a versatile and powerful framework for the joint characterization of multipartite nonlocality and quantum criticality in one-dimensional many-body quantum systems.
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 nnn-site subchains, and take the large-nnn limit. As nnn 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 XXZ\mathit{XXZ}XXZ model, in the large-nnn 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.
We propose a transfer-matrix theory to reveal the hidden structure in the multipartite nonlocality operators in one-dimensional (1D) quantum chains. The theory offers a unified description for the scaling behaviors of multipartite quantum nonlocality in general 1D quantum lattices. Furthermore, in order to figure out the hierarchy of multipartite nonlocality for infinite-size chains, powerful transfer-matrix-based algorithms are proposed. In quantum critical regions, the algorithms converge much faster than the traditional approach.
Multipartite nonlocality is an important measure of multipartite quantum correlations. In this paper, we show that the nonlocal nnn-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=2D=2D=2, and then the calculation of nonlocality measure S\mathcal{S}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 XXXXXX model, an XXZXXZXXZ model, and a Kitaev model. For the XXXXXX model and the XXZXXZXXZ model, in a finite-temperature region, the logarithm measure of nonlocality (log2S{log}_{2}\mathcal{S}log2​S) is a linear function of the temperature TTT, 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 XXXXXX model S\mathcal{S}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 a measure of multipartite quantum correlations. In this paper we investigate the influence of symmetry-breaking perturbations upon ground-state nonlocality by considering several typical finite-size quantum models, including the transverse-field Ising chains, the XYXYXY(XXXXXX) chains, and the XXZXXZXXZ chains. We find that even a slight perturbation can reshape the nonlocality curve dramatically. For instance, in the XYXYXY chains, a perturbation can induce an oscillation behavior in the nonlocality curve. A clear microscopic mechanism for the results is proposed. Furthermore, we also connect the behaviors to the macroscopic symmetry properties of the chains and make some predictions on general models.
Multipartite quantum nonlocality and Bell-type inequalities are used to characterize quantum correlations in an infinite-size bond-alternating spin-12\frac{1}{2}21​ 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 K\mathcal{K}K to characterize the finite-temperature nonlocality in the large-nnn limit. We find that in high-temperature regions, K\mathcal{K}K is reduced linearly as the temperature rises.