The multipartite nonlocality of finite and infinite transverse‐field Ising model is investigated. For the finite chain, the exact diagonalization method is used to study the multipartite nonlocality at zero and finite temperatures. It is found that the logarithm of nonlocality S presents the linear relationship with the chain length of N at zero temperature. The multipartite nonlocality S shows a peak value near the phase transition point, which can be used to characterize the quantum phase transition of the Ising model. The inherent physical mechanics is that the nonlocality S captures the effect of low‐lying states with a small energy gap. The nonlocality S at the finite temperature is also observed to be scaled as log 2 S ≈ aN + b at any fixed magnetic field B , where a and b are fitting parameters. For the infinite chain, the thermal tensor network algorithm is used to examine the multipartite nonlocality. The behaviors of nonlocality S of the infinite chain at the finite temperature are similar to those of finite chain. Compared with the finite chain, the positions of the peak values of the multipartite nonlocality shift to the larger magnetic field, and the peak values are smaller than those of finite chain.
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.
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.
Tackling the critical transverse Ising ring with or without ring frustration, we establish the concept of nonlocality in a many-body system in the thermodynamic limit by calculating the nonlocal factors embedded in the factorizable correlation function. Through this exactly solvable prototype, we clearly show the intriguing difference between the periodic chains with odd and even numbers of lattice sites even in the thermodynamic limit. In the context of nonlocality, we also address the important application of finite-size scaling analysis by numerically working out the nonlocal factors of the isotropic XY model and the spin-1/2 Heisenberg model.
The spin-1/2 Ising-Heisenberg trimerized chain in a magnetic field is revisited with the aim to explore the quantum entanglement and non-locality within the exactly solved spin system, which exhibits in a low-temperature magnetization curve two intermediate plateaux at zero and one-third of the saturation magnetization.The groundstate phase diagram involves two quantum (antiferromagnetic, ferrimagnetic I) and two classical (ferrimagnetic II, saturated paramagnetic) phases.We have rigorously calculated the concurrence and Bell function in order to quantify the quantum entanglement and non-locality at zero as well as non-zero temperatures.It is demonstrated that the entanglement can be thermally induced also above the classical ground states unlike the quantum nonlocality, which means that the thermal entanglement is indispensable for a violation of the locality principle.