Multipartite nonlocality in a thermalized Ising spin chain

We study multipartite correlations and nonlocality in an isotropic Ising ring under transverse magnetic field at both zero and finite temperature. We highlight parity-induced differences between the multipartite Bell-like functions used in order to quantify the degree of nonlocality within a ring state and reveal a mechanism for the passive protection of multipartite quantum correlations against thermal spoiling effects that is clearly related to the macroscopic properties of the ring model.

Figure 1

Behavior of entanglement against $B$ and $J$ for (a) bipartite spin ring and (b) tripartite spin ring. For small values of $B$ the ground state is highly entangled. As the field strength is increased we see the entanglement asymptotically approach zero.

Figure 2

Comparison of the $N$ concurrence $\mathcal{C}$ corresponding to the ground state of a ring of $N=2,\hspace{0.5em}4$, and $6$ spins (curves from top to bottom) at zero temperature. The $N$ concurrence is plotted against $B$ and $J$.

Figure 3

(a) Multipartite nonlocality of the ground state of a ring of an even number of spins against the amplitude of the magnetic field $B$ and at $J=5$. The solid line is for $N=6$, the dashed line for $N=4$, and, finally, the dotted one is for $N=2$. (b) Same as in (a) but for an odd number of spins. The dotted line is for $N=3$, the dashed for $N=5$, while the squares show the results for $N=7$.

Figure 4

Energy spectra for a spin ring of $N=2$ (a), $3$ (b), and $4$ (c) against $B$. We have taken $J=5$.

Figure 5

Behavior of Svetlichny functions ${\mathcal{S}}_N$ for the state of a ring with $J=5$ and $T\ne 0$. We address the cases of $N=2,\hspace{0.5em}3$, and $4$ [(a), (b), and (c), respectively] as representative of the general trend exhibited by arbitrary $N$ values. Each curve in the same plot is for a specific choice of $B\in [1,10]$. Regardless of $N$, as $B$ increases the degree of violation of the appropriate Bell-like inequality decreases. However, the resilience to spoiling effects due to temperature increases.

Figure 6

Two-spin negativity against magnetization and temperature for $J=5$. We have taken $T=1,\dots ,20$ (increasing in unit steps). The entanglement for magnetization going toward $1$ stays at finite nonzero values only because we have considered $B\in [0,100]$. By enlarging this range, smaller final values of negativity would be reached. In fact, we have ${\mathcal{N}}_2\to 0$ as $\langle {\mathrm{M̂}}_2\rangle \to 1$.