Pair entanglement in dimerized spin-$s$ chains

We examine the pair entanglement in the ground state of finite dimerized spin-$s$ chains interacting through anisotropic $XY$ couplings immersed in a transverse magnetic field by means of a self-consistent pair mean-field approximation. The approach, which makes no a priori assumptions on the pair states, predicts, for sufficiently low coupling between pairs, $2s$ distinct dimerized phases for increasing fields below the pair factorizing field, separated by spin-parity-breaking phases. The dimerized phases lead to approximate magnetization and pair entanglement plateaus, while the parity-breaking phases are characterized by weak pair entanglement but non-negligible entanglement of the pair with the rest of the system. These predictions are confirmed by the exact results obtained in finite $s=1$ and $s=3/2$ chains. It is also shown that for increasing values of the spin $s$, the entanglement of an isolated pair, as measured by the negativity, rapidly saturates in the anisotropic $XY$ case but increases as $s^{1/2}$ in the $XX$ case, reflecting a distinct single-spin entanglement spectrum.

Figure 1

Negativity of the spin-1 pair GS for an anisotropic $XY$ coupling with $\chi =J_y/J_x=0.75$, as a function of the scaled transverse field $B/j_x$, with $j_x=2J_{x}s$. The quantity $s-\left|m\right|$, with $m$ the intensive magnetization $\langle S_1^z+S_2^z\rangle /2$, is also depicted. The inset depicts the lowest energy levels $E_{\pm }$ for each parity, which cross at $B_{c1}$ and $B_{c2}=B_s$ (pair separability field) and lead to the negativity steps (all labels dimensionless in all figures).

Figure 2

Top: Pair mean-field (GMF) phase diagram of the spin-1 dimerized cyclic chain in the $\alpha$-field plane for $\chi =J_y/J_x=0.75$. Colored sectors depict dimerized definite $S_z$ parity phases $\left(\langle S^x\rangle =0\right)$, whereas the white sector corresponds to the parity-breaking phase $[{\alpha }_c$ indicates the critical value (19) at zero field]. The dashed line depicts the separability field (7), entirely contained in the parity-breaking phase, which determines the last parity transition of the exact GS. The dotted line indicates the conventional mean-field critical field $B_c^{\mathrm{mf}}=J_{x}s(1+\alpha )$. Bottom: The parity-breaking parameter $\langle S^x\rangle$ for increasing fields at different fixed $\alpha$'s (0.025, 0.05, 0.1, 0.5, and 1). Its behavior reflects the phases of the top panel, showing a nonmonotonous field dependence at low $\alpha$, with “deaths and revivals” if $\alpha <{\alpha }_c$. For $\alpha =1$ it lies close to the conventional MF result.

Figure 3

Top panels: Exact and GMF results for the negativity of a strongly coupled pair in the dimerized spin-1 chain, as a function of the magnetic field, for $\chi =0.75$ and coupling factors $\alpha =0.05$ (left) and $\alpha =0.1$ (right). The quantity $s-\left|m\right|$, with $m=\langle {\sum }_i{S}_i^z\rangle /2n$ the intensive magnetization, is also depicted. Bottom panels: The corresponding exact (solid lines) and GMF (dashed lines) results for the entanglement entropies of a strongly coupled spin pair $\left(S_2\right)$ and a single spin $\left(S_1\right)$ with the rest of the chain, for the same values of $\alpha$ and $s$. The inset shows the discontinuities in the exact $S_2$ stemming from the GS parity transitions, which occur within the parity-breaking phases of the GMF approach.

Figure 4

Top: The lowest exact excitation energies in the $2n=8$ dimerized spin-1 chain for $\alpha =0.05$ and $\chi =0.75$. The dashed line depicts the (scaled) GMF parity-breaking parameter $\langle S^x\rangle$. All parity transitions of the GS are seen to take place within the GMF parity-breaking phases, where the lowest excitation energy becomes small (and the first band of excitation energies minimum). Bottom: The negativity of a strongly coupled pair as a function of the coupling factor $\alpha$ at different fields for the same $\chi$, according to exact and GMF results.

Figure 5

Top panel: The GMF phase diagram of the spin-3/2 dimerized chain in the $\alpha$-field plane for anisotropy $\chi =J_y/J_x=0.75$. There are now three definite parity-dimerized phases below $B_s$ (colored sectors) if $\alpha$ is sufficiently small. Remaining details as in Fig. 2. Bottom panel: The corresponding parity-breaking parameter $\langle S^x\rangle$ for increasing fields at different fixed values of $\alpha$ ($(0.01,0.02,0.05,0.1,0.5,$ and $1)$. Its behavior reflects the phases of the top panel, exhibiting a nonmonotonous variation at low $\alpha$, with “deaths and revivals” if $\alpha \le {\alpha }_c\left(0\right)$.

Figure 6

Top: Exact and GMF results for the negativity in the dimerized spin-3/2 chain as a function of the (scaled) magnetic field, for two different values of the coupling factor $\alpha$. Again, $m=\langle S_z\rangle /n$ denotes the intensive magnetization. Bottom: The corresponding exact (solid lines) and GMF (dashed lines) results for the entanglement entropies of a strongly coupled spin pair $\left(S_2\right)$ and a single spin $\left(S_1\right)$, with the rest of the chain, as a function of the (scaled) magnetic field in the $s=3/2$ chain for the previous values of $\alpha$.

Figure 7

Top: The entanglement spectrum of the GS of a spin-$s$ pair as a function of the applied field for $s=5$ and $\chi =J_y/J_x=0.75$ (left) and 1 (right). In the anisotropic case it is formed essentially by just two degenerate eigenvalues below the factorizing field [as described by Eq. (20)], whereas in the $XX$ case (right) there are several nonvanishing eigenvalues (twofold degenerate), in agreement with the Gaussian profile (25) [shown in the bottom-right panel at $B=0$ for $s=5$ and 20, together with the exact results, indistinguishable from (25)]. Bottom: The negativity of the pair as a function of the (scaled) magnetic field for different spin values and $\chi =0.75$ (left), where $s=1/2,1,\frac{3}{2},2,5,10,$ and $\infty$ [bosonic limit, Eq. (21)], and $\chi =1$ (right), where $s=1/2,1,\frac{3}{2},2,5,10,20,$ and 50. Notice the different scales.

Figure 8

Negativity of a pair for increasing spin $s$ at zero field for different anisotropies $\chi =J_y/J_x$. For $\chi <1$ they saturate, approaching the limit values (21), while for $\chi =1$ they increase as $\sqrt{s}$, as given by Eq. (26) (indistinguishable from the exact result for $s\ge 1$ on this scale).