Nontransverse factorizing fields and entanglement in finite spin systems

We determine the conditions for the existence of nontransverse factorizing magnetic fields in general spin arrays with anisotropic $XYZ$ couplings of arbitrary range. It is first shown that a uniform, maximally aligned, completely separable eigenstate can exist just for fields ${\mathit{h}}_s$ parallel to a principal plane and forming four straight lines in the field space, with the alignment direction different from that of ${\mathit{h}}_s$ and determined by the anisotropy. Such a state always becomes a nondegenerate ground state for sufficiently strong (yet finite) fields along these lines, in both ferromagnetic and antiferromagnetic-type systems. In antiferromagnetic chains, this field coexists with the nontransverse factorizing field ${\mathit{h}}_s^{\prime }$ associated with a degenerate Néel-type separable ground state, which is shown to arise at a level crossing in a finite chain. It is also demonstrated for arbitrary spin that pairwise entanglement reaches full range in the vicinity of both ${\mathit{h}}_s$ and ${\mathit{h}}_s^{\prime }$, vanishing at ${\mathit{h}}_s$ but approaching small yet finite side limits at ${\mathit{h}}_s^{\prime }$, which are analytically determined. The behavior of the block entropy and entanglement spectrum in their vicinity is also analyzed.

Figure 1

Schematic representation of the uniform solution.

Figure 2

Factorizing fields for ferromagnetic (FM; left) and antiferromagnetic (AFM; right) $XYZ$ chains in the $xz$ principal plane of the field space. Solid straight lines depict fields determining a uniform ground state (UGS); dashed straight lines, fields determining a uniform excited eigenstate (UES). The ellipse depicts fields corresponding to a Néel-type ground state (NGS; solid lines) or excited eigenstate (NES; dashed line). The plot corresponds to $J_z=0$ and $J_x>0 \left(J_x<0\right)$ in the FM (AFM) case, with $0<J_y/J_x<1$. The arrow indicates a direction of the external field along which one (FM) or two (AFM) GS factorizing fields are encountered as its magnitude increases. The field direction ${\mathit{n}}_{\gamma }$ differs from the spin alignment direction ${\mathit{n}}_{\theta }$. Insets: Decomposition, (18), of the nontransverse factorizing field for the UGS in both diagrams, with the dashed arrow indicating the transverse factorizing field ${\mathit{h}}_{zs}$.

Figure 3

Factorization diagram (top panels) and factorizing fields (middle and bottom panels) for $XYZ$ couplings satisfying Eq. (14), with $J_x>0 \left(J_x<0\right)$ in the left (right) panels and $J_y$ of the same sign as $J_x$. The different combinations of couplings are indicated (see also the text). For $J_x<J_y<0$, the Néel separable eigenstate ceases to be the GS when $J_z\ge -J_y$, although the uniform separable eigenstate remains the GS for appropriate fields, as shown in the top- and bottom-right panels. Equation (25) may determine a hyperboloid when the couplings have different signs, as shown in the top and bottom panels.

Figure 4

Concurrences $C_l$ between spin $i$ and spin $i+l$ (top) in an FM (left) and an AFM (right) finite spin-1/$2 XY$ chain with $\chi =J_y/J_x=1/2$, as a function of the magnitude $h=\left|\mathit{h}\right|$ of the nontransverse field at a fixed orientation ${\mathit{n}}_{\gamma }=(sin\gamma ,cos\gamma )$ in the $xz$ plane, with $\gamma =0.02\pi$ (FM) and $\gamma =0.36\pi$ (AFM). For these values there is a single factorizing field, $|{\mathit{h}}_s|\approx 0.76J_x$ [Eq. (19)], in the FM case, determining a UGS and two factorizing fields, $|{\mathit{h}}_s^{\prime }|\approx 1.06|J_x|$ [Eq. (26)] and $|{\mathit{h}}_s|\approx 1.43|J_x|$, in the AFM case, corresponding to an NGS and a UGS, respectively. Insets: Details in the vicinity of these fields, showing that all $C_l$'s vanish linearly at ${\mathit{h}}_s$ [Eq. (30)] and approach the finite $l$-independent side limits, (33), at ${\mathit{h}}_s^{\prime }$. All pairs are entangled in the vicinity of ${\mathit{h}}_s$ and ${\mathit{h}}_s^{\prime }$, remaining so between both fields in the AFM case considered. All labels are dimensionless.

Figure 5

Block entropies $S\left(m\right)=S\left[\rho \right(m\left)\right]$ of $m$ contiguous spins (top) and eigenvalues $p_i$ (entanglement spectrum) of the corresponding reduced states $\rho \left(m\right)$ for $m=5$, in the same FM (left) and AFM (right) systems as in Fig. 4. At ${\mathit{h}}_s$ (UGS), $S\left(m\right)$ and all but one $\left(p_1\right)$ of the eigenvalues of $\rho \left(m\right)$ vanish, while for $\mathit{h}\to {{\mathit{h}}^{\prime }}_s^{\pm }$ (NGS), $S\left(m\right)$ approaches the finite side limits, (37) (indicated for $m=n/2$), and two eigenvalues $(p_1$ and $p_2$) remain nonzero, with $p_1$ approaching the indicated side limits, (36). Insets: Details in the vicinity of the factorizing fields.

Figure 6

Scaled intensive magnetizations $m_{\mu }={\sum }_{i}2\langle S_i^{\mu }\rangle /n$ and $m\equiv \left|\mathit{m}\right|$ (top) and scaled energy gap between the ground and the first two excited states (bottom) in the same FM (left) and AFM (right) systems as in Fig. 4. Note that $m=1$ at the NTFF ${\mathit{h}}_s$ determining the UGS, as shown in the insets. The energy gap shows that the UGS is well separated from the first excited state, while the NGS is twofold degenerate. All labels are dimensionless.

Figure 7

Concurrences $C_l$ between spin $i$ and spin $i+l$ in FM and AFM $XYZ$ chains with $J_y/J_x=1/2,J_z=0.2\left|J_x\right|$, and $J_x>0 \left(J_x<0\right)$ in the left (right) panel. The orientation of the applied magnetic field in each panel is the same as that in Figs. 4, 5, 6. The factorizing fields are now $|{\mathit{h}}_s|\approx 0.52J_x$ in the FM case, where $\chi =0.375$, and $|{\mathit{h}}_s|\approx 1.39J_x,|{\mathit{h}}_s^{\prime }|\approx 0.84J_x$ in the AFM case, where $\chi =0.583$. The side limits, (33), at ${\mathit{h}}_s^{\prime }$ and the linear vanishing of all concurrences at ${\mathit{h}}_s$ are again verified.