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.
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.
Multipartite quantum nonlocality and Bell-type inequalities are used to characterize quantum correlations in an infinite-size bond-alternating spin-
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 to characterize the finite-temperature nonlocality in the large-
n limit. We find that in high-temperature regions,
K is reduced linearly as the temperature rises.
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.
In this paper, we investigate Quantum Phase Transitions (QPTs) in the long-range XXZ spin chain using two information-theoretic probes: The half-chain entanglement entropy and a witness of genuine multipartite nonlocality. Both quantities effectively identify the phase boundary, but exhibit distinct scaling behaviors: the entanglement entropy grows linearly with system size N, while the multipartite nonlocality increases exponentially, with [Formula: see text]. By independently tuning the power-law decay exponents of the transverse ([Formula: see text]) and longitudinal ([Formula: see text]) couplings, we uncover contrasting trends in the critical point [Formula: see text]: Increasing [Formula: see text] shifts it to larger values, while increasing [Formula: see text] shifts it to smaller ones. These monotonic and saturating dependencies are captured by an empirical exponential model, offering a compact and physically transparent description of the critical behavior. Our results highlight the crucial role of interaction range in driving criticality, with implications for engineered quantum systems and long-range entanglement control.
We investigate both bipartite and multipartite nonlocality in theJ1-J2Heisenberg model. Bipartite nonlocality is measured by the Clauser-Horne-Shimony-Holt inequality, while multipartite nonlocality is explored through Bell-type inequalities. Our findings reveal that neither ground-state nor full thermal-state nonlocality reliably characterizes quantum phase transitions (QPTs). However, we uncover that the mixed-state nonlocality of the ground and first excited states exhibits distinctive characteristics applicable to both bipartite and multipartite scenarios. We also demonstrate how mixed-state quantum correlation behaviors depend on varying temperature regimes. In the bipartite case, we observe a phenomenon known as 'correlation reversal' with increasing temperature, a previously unreported occurrence in other models. For the multipartite case, the ability to signify phase transitions is significantly enhanced as the temperature rises. Furthermore, we discover a linear scaling effect that provides valuable insights for extrapolating QPTs in the thermodynamic limit asN→∞. Additionally, we identify the critical temperature at which mixed-state nonlocality becomes a reliable indicator of phase transitions.
Multipartite nonlocality has been extensively investigated within one-dimensional quantum lattices. Previous research has primarily focused on the nonlocality measure
S, which quantifies the violation of Bell-type inequalities. However, the optimal nonlocal operators, which are related to specific experimental settings required to achieve the violation, often remain elusive. In this work, we employ a string-like nonlocal operator
S^N, characterized by a core single-site operator
p^, to investigate the optimal measurement setting in translationally invariant quantum chains. By analyzing the infinite-size transverse-field Ising, Cluster-Ising, and extended Ising models, we uncover two general results. First, for typical ground states, we find that the optimal single-site operator
p^ possesses an intrinsic mirror symmetry. Second, the optimal nonlocal operator
S^(p^) exhibits remarkable robustness: for a specific model, as the Hamiltonian parameter changes, the structure of
p^ remains stable and persists across distinct quantum phases. These findings not only redefine the numerical optimization paradigm for multipartite nonlocality, but also significantly simplify the experimental requirements by identifying fixed measurement bases. This structural stability provides practical guidance for implementing macroscopic Bell tests in large-scale quantum simulators, making it highly compatible with modern efficient measurement protocols.