Separability and parity transitions in $XYZ$ spin systems under nonuniform fields

We examine the existence of completely separable ground states (GS) in finite spin-$s$ arrays with anisotropic $XYZ$ couplings, immersed in a nonuniform magnetic field along one of the principal axes. The general conditions for their existence are determined. The analytic expressions for the separability curve in field space, and for the ensuing factorized state and GS energy, are then derived for alternating solutions, valid for any spin and size. They generalize results for uniform fields and show that nonuniform fields can induce GS factorization also in systems which do not exhibit this phenomenon in a uniform field. It is also shown that such a curve corresponds to a fundamental $S_z$-parity transition of the GS, present for any spin and size, and that two different types of GS parity diagrams can emerge, according to the anisotropy. The role of factorization in the magnetization and entanglement of these systems is also analyzed, and analytic expressions at the borders of the factorizing curve are provided. Illustrative examples for spin pairs and chains are as well discussed.

Figure 1

The ground state factorization curves (solid lines) in the field plane $(h_1,h_2)$ determined by Eq. (11), for $j_y=j_x/2>0$ and different values of $j_z/j_x$. At these curves a GS $S_z$-parity transition takes place. For a spin $1/2$ pair, the lighter (darker) colored sectors separated by these curves correspond to positive (negative) GS parity $P_z=\pm 1$. The same factorization curves remain, nevertheless, valid for a general spin-$s$ pair as well as for a spin-$s$ chain or lattice (see Fig. 3).

Figure 2

The angles ${\theta }_{1\left(2\right)}$ that characterize the separable GS along the factorization curves depicted in Fig. 1, as a function of the scaled magnetic field $h_1$ (with $j_y=j_x/2>0$). Dashed vertical lines indicate the asymptotes $h_1=\mp j_z$ where $h_2\to \pm \infty$ and ${\theta }_2\to 0$ or $\pi$. For $j_z=j_y$ (bottom left panel) all angles approach $\pi /2$ at zero field.

Figure 3

GS spin-parity phase diagram in the $(h_1,h_2)$ field space for a pair of spins 1 (left) and for a spin $1/2$ chain of eight spins (right), for $j_y=0.5j_x$ and three values of $j_z/j_x$. Solid lines depict the factorizing curves (the same as those of Fig. 1) while dashed lines the remaining GS parity transitions. Signs denote the GS parity in each sector, with dark colored regions indicating negative parity. For $j_z=j_y$ (central panels) all curves and GS parity sectors meet at the origin.

Figure 4

Examples of spin arrays with first neighbor anisotropic $XYZ$ couplings under a nonuniform field (schematic representation), which possess an alternating separable GS for any spin $s$ when the indicated fields $h_1$ and $h_2$ satisfy Eq. (11): (a) Spin pair, (b) open chain, (c) cyclic chain, (d) square lattice, and (e) ladder with nonuniform couplings. Here the factorizing fields are $r{h}_{1\left(2\right)}$ ($r^{\prime }{h}_{1\left(2\right)}$) in the lower (upper) row, with $r=2\alpha +\beta$ ($r^{\prime }=2\gamma +\beta$).

Figure 5

Ground state magnetization $M=\langle S_1^z+S_2^z\rangle$ (top) and entanglement (bottom), measured through the concurrence $C$, of a spin $1/2$ pair as a function of the scaled magnetic fields $h_i/j_x$ at each spin. The $XYZ$ couplings are $j_y=0.5j_x$ and $j_z=0.25j_x\left(0.75j_x\right)$ on the left (right) panels. Solid lines indicate the side limits at the factorizing curves, which determine the GS parity transitions (both $C$ and $M$ are dimensionless).

Figure 6

Ground state magnetization (top) and entanglement (bottom), measured through $C=\sqrt{S_2\left({\rho }_i\right)}$, for a spin 1 pair as a function of the scaled magnetic fields $h_i/j_x$ for $j_y=0.5j_x$ and $j_z=0.25j_x$ (left) and $0.75j_x$ (right). Details are similar to those of Fig. 5.

Figure 7

Ground state magnetization $M=\langle {\sum }_i{S}_i^z\rangle$ of a spin $1/2$ cyclic chain of eight spins and $XYZ$ Heisenberg couplings with $j_y=0.5j_x$ and $j_z=0.25j_x\left(0.75j_x\right)$ on the left (right) panels, as a function of the scaled magnetic fields $h_i/j_x$ at each site. Solid lines depict the magnetization at the factorization curves.

Figure 8

Ground state pairwise concurrence for first (top), second (center), and third (bottom) neighbors for $j_y=0.5j_x$ and $j_z=0.25j_x$ (left) and $0.75j_x$ (right), for the same chain of Fig. 7, as a function of the scaled magnetic fields $h_i/j_x$. Solid lines depict the side limits at the factorizing fields.