Magnetic quantum correlations in the one-dimensional transverse-field $XXZ$ model

One-dimensional spin-$\frac{1}{2}$ systems are well-known candidates to study the quantum correlations between particles. In condensed matter physics, studies often are restricted to first-neighbor particles. In this work, we consider the one-dimensional $XXZ$ model in a transverse magnetic field (TF) which is not integrable except at specific points. Analytical expressions for quantum correlations (entanglement and quantum discord) between spin pairs at any distance are obtained for both zero and finite temperature by using the analytical approach proposed by Caux et al. [Phys. Rev. B 68, 134431 (2003)]. We compare the efficiency of the quantum discord (QD) with respect to the entanglement in the detection of critical points as the neighboring spin pairs go farther than the next-nearest neighbors. In the absence of the TF and at zero temperature, we show that the QD for spin pairs farther than the second neighbors is able to capture the critical points while the pairwise entanglement is absent. In contrast with the pairwise entanglement, two-site QD is effectively long range in the critical regimes where it decays algebraically with the distance between pairs. We also show that the thermal QD between neighbor spins possesses strong distinctive behavior at the critical point that can be seen at finite temperature and, therefore, spotlights the critical point while the entanglement fails in this task.

Figure 1

(a) Entanglement of formation and (b) quantum discord between the first-, second-, third-, fourth-, and fifth-nearest neighbors as a function of anisotropy at zero temperature and zero magnetic field. Inset shows scaling behavior of quantum discord at the critical point $\mathrm{\Delta }=1$ in terms of distances between spins pair. Parameters are dimensionless.

Figure 2

The three-dimensional panorama of thermal (a) entanglement between first-nearest neighbors, (b) quantum discord between first-nearest neighbors and (c) quantum discord between second-nearest-neighbor spin pairs at zero magnetic field. Parameters are dimensionless.

Figure 3

Three-dimensions of entanglement as a function of magnetic field and anisotropy $\mathrm{\Delta }$ at zero temperature between the (a) first- and (b) second-nearest-neighbor spins. (c) The critical entangled field as a function of anisotropy parameter $\mathrm{\Delta }$, at $T=0$. Parameters are dimensionless.

Figure 4

Three-dimensional view of quantum discord as a function of magnetic field and anisotropy $\mathrm{\Delta }$ at zero temperature, between the (a) first-, (b) second-, and (c) third-nearest-neighbors spins. Parameters are dimensionless.

Figure 5

Three-dimensional panorama of the entanglement as a function of magnetic field and anisotropy $\mathrm{\Delta }$ at $T=0.1$, between the (a) first- and (b) second-nearest-neighbors spins. (c) Three-dimensional view of the entanglement as a function of $h$ and $\mathrm{\Delta }$ between the second-nearest-neighbors spins at $T=0.2$. Parameters are dimensionless.

Figure 6

Three-dimensional view of quantum discord as a function of magnetic field and anisotropy $\mathrm{\Delta }$ at $T=0.1$, between the (a) first-, (b) second-, and (c) third-nearest-neighbor spins. Parameters are dimensionless.

Figure 7

(a) Thermal entanglement of formation between first-neighbor spin pairs, (b) thermal quantum discord between first-neighbor spin pairs, and (c) thermal quantum discord between third-neighbor spin pairs as a function of magnetic field and temperature for $\mathrm{\Delta }=0.5$. Parameters are dimensionless.