Factorization in spin systems under general fields and separable ground-state engineering

We discuss ground-state factorization schemes in spin $S$ arrays with general quadratic couplings under general magnetic fields, not necessarily uniform or transverse. It is shown that, given arbitrary spin alignment directions at each site, nonzero $XYZ$ couplings between any pair and fields at each site always exist such that the ensuing Hamiltonian has an exactly separable eigenstate with the spins pointing along the specified directions. Furthermore, by suitable tuning of the fields this eigenstate can always be cooled down to a nondegenerate ground state. It is also shown that in open one-dimensional systems with fixed arbitrary first-neighbor couplings at least one separable eigenstate compatible with an arbitrarily chosen spin direction at one site is always feasible if the fields at each site can be tuned. We demonstrate as well that in the vicinity of factorization, i.e., for small perturbations in the fields or couplings, pairwise entanglement reaches full range. Some noticeable examples of factorized eigenstates are unveiled. The present results open the way for separable ground-state engineering. A notation to quantify the complexity of a given type of solution according to the required control on the system couplings and fields is introduced.

Figure 1

Left column: Schematic representation of the spin alignment directions ${\mathit{n}}_i$ and ${\mathit{n}}_j$ and the perpendicular factorizing fields ${\mathit{h}}_{\perp }^{ij}$ and ${\mathit{h}}_{\perp }^{ji}$ [Eq. (6)]. Right column: Exchange couplings in the exchange couplings space compatible with the spin alignment configuration. Top row: ${\mathit{U}}^{ij}$ and ${\mathit{V}}^{ij}$ are linearly independent and Eqs. (8) define a line [Eq. (9)] of possible exchange couplings. Bottom row: ${\mathit{U}}^{ij}$ and ${\mathit{V}}^{ij}$ are linearly dependent (as ${\mathit{n}}_i$ and ${\mathit{n}}_j$ belong to a principal plane) and Eqs. (8) define a plane of possible exchange couplings [Eq. (A4)].

Figure 2

Left: Typical contour plot of Eqs. (8) for fixed ${\mathit{J}}^{ij}$ and ${\mathit{n}}_j$ (${\mathit{J}}^{ij}=J(1,0.75,-0.2),{\theta }_j=\pi /3,{\varphi }_j=\pi /5$). The two intersection points correspond to the two solutions ${\mathit{n}}_i^{\pm }$ for ${\mathit{n}}_i$ given by Eq. (10). Right: Schematic representation of the $2^{N-1}$ configurations of $|\mathrm{\Theta }\rangle$ compatible with the exchange vectors ${\mathit{J}}^{ij}$ for a spin chain of $N$ spins and an initial alignment direction at site $j$.

Figure 3

Schematic representation of the spin spiral separable eigenstate in a spin chain. The alignment direction of the spin at site $i$ is determined by ${\theta }_i=\theta$ and ${\varphi }_i={\varphi }_1+(i-1)\mathrm{\Delta }\varphi \forall i$. The top (bottom) panel corresponds to a top (side) view.

Figure 4

Left panel: Scaled energy spectrum of a finite spin-$1/2$ chain with first-neighbor $XYZ$ couplings and eight spins when a factorizing field ${\mathit{h}}^i={\mathit{h}}_{\perp }^i+h_{\parallel }{\mathit{n}}^i$ with ${\mathit{h}}_{\perp }^i$ fixed and $h_{\parallel }\ge 0$ is applied. The decreasing straight line represents the energy $E_{\mathrm{\Theta }}$ of the separable eigenstate $|\mathrm{\Theta }\rangle$, which becomes GS for $h_{\parallel }>h_{\parallel c}$ (dashed line). Right panel: GS concurrences $C_{ij}$ between a central spin and first and second neighbors, showing that for $h_{\parallel }<h_{\parallel c}$ the GS is entangled whereas for $h_{\parallel }>h_{\parallel c}$ it is completely separable. All labels are dimensionless.

Figure 5

GS concurrences $C_{ij}$ between spins $i$ and $j$ in the chain of Fig. 4 in the vicinity of factorization (point at $h_{\parallel }/\left|\mathit{J}\right|=1$ on Fig. 6). In the top panel, we have set the couplings as $J_{\mu }^{ij}+\mathrm{\Delta }J \forall i,j$, with $J_{\mu }^{ij}$ the factorizing values, such that $C_{ij}=0 \forall i,j$ if $\mathrm{\Delta }J=0$. In the bottom panel, the couplings are fixed at the factorizing values but the fields are now ${\mathit{h}}^i+\mathrm{\Delta }h{\mathit{n}}_{i\perp }$, with ${\mathit{n}}_{i\perp }\propto {\mathit{h}}_{\perp }^i$, such that $C_{ij}=0 \forall i,j$ at $\mathrm{\Delta }h=0$. $C_{ij}$ is verified to reach full range in the vicinity of the present general factorization point (nonuniform couplings and fields). Right panels show the same quantities of the left panels at a smaller scale. All labels are dimensionless.

Figure 6

Schematic representation of the separable state with arbitrary spin alignment directions ${\mathit{n}}_i$ at each site (left panel) and with a Néel-type configuration (right panel). (a) The alignment directions at each site are specified. Then, for each pair the (b) exchange couplings and (c) perpendicular fields are determined. (d) The perpendicular factorizing field at each site is ${\mathit{h}}_{\perp }^i={\sum }_j{\mathit{h}}_{\perp }^{ij}$. (e) The uniform factorizing field ${\mathit{h}}_s^{\prime }$ is determined by Lemma 4.

Figure 7

Schematic representation of a spin chain with nonuniform couplings (left panel) and with uniform anisotropic couplings leading to a uniform separable state (right panel). (a) The exchange couplings are fixed. In the left panel one of the spin alignment directions ${\mathit{n}}_i$ is specified. (b) Left: Using Lemma 3 the spin alignment directions for the spins at sites $i\pm 1$ can be determined. Right: $\mathit{n}$ belongs to the principal plane $\mu \nu$ determined by the exchange couplings. (c,d) Left: By successive applications of Lemma 3 the spin alignments of the remaining spins are determined.