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.
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 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.
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 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 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 (log2S) 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 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 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 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 this paper, using the infinite time-evolving block decimation (iTEBD) algorithm and Bell-type inequalities, we investigate multipartite quantum nonlocality in an infinite one-dimensional quantum spin-21XXZ system. High hierarchy of multipartite nonlocality can be observed in the gapless phase of the model, while only the lowest hierarchy of multipartite nonlocality is observed in most regions of the gapped antiferromagnetic phase. Thereby, Bell-type inequalities disclose different correlation structures in the two phases of the system. Furthermore, at the infinite-order quantum phase transition (QPT, or Kosterlitz-Thouless QPT) point of the model, the correlation measures always show a local minimum value, regardless of the length of the subchains. It indicates that relatively low hierarchy of multipartite nonlocality would be observed at the infinite-order QPT point in a Bell-type experiment. The result is in contrast to the existing results of the second-order QPT in the one-dimensional XY model, where multipartite nonlocality with the highest hierarchy has been observed. Thus, multipartite nonlocality provides an alternative perspective to distinguish between these two kinds of QPTs. Reliable clues for the existence of tripartite quantum entanglement have also been found.
This study investigates quantum Fisher information (QFI) with respect to a class of nonlocal operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}$ in one-dimensional quantum chains. Based upon the triangular matrix-product-operator theory, we establish a universal scaling behavior of QFI with respect to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}$ for general infinite matrix-product states (iMPSs): $\mathcal{F}\ensuremath{\sim}{\ensuremath{\varrho}}_{2}{N}^{2}+{\ensuremath{\varrho}}_{1}N+{\ensuremath{\varrho}}_{0}$, where the coefficients ${\ensuremath{\varrho}}_{i}$ can be efficiently identified by iMPS algorithms. Applied to a spin-1 Bilinear-Biquadratic model and an XXZ model, the scaling coefficients ${{\ensuremath{\varrho}}_{i}}$ distinguish various phases through distinct scaling behaviors: For instance, the topologically ordered Haldane phase exhibits characteristic quadratic scaling (${\ensuremath{\varrho}}_{2}\ensuremath{\ne}0$), while the trimer and dimer phases are featured with linear scaling (${\ensuremath{\varrho}}_{1}\ensuremath{\ne}0$). Notably, since the QFI scaling behavior undergoes fundamental changes between distinct phases, these coefficients provide signals at first-order, second-order, and Berezinskii-Kosterlitz-Thouless type quantum phase transitions in these models. Furthermore, finite bond-dimension (D) scaling analysis reveals a dichotomy. For gapped phases, the coefficients converge rapidly as D increases. Conversely, for critical systems, the coefficients exhibit D-scaling effects that depend on the specific model. By bypassing finite-size fittings with limited N, our approach accesses the thermodynamic-limit physics for QFI with respect to nonlocal operators, offering new avenues for characterizing exotic quantum phases in one-dimensional quantum systems.