Factorization and Criticality in Finite $XXZ$ Systems of Arbitrary Spin

We analyze ground state (GS) factorization in general arrays of spins $s_i$ with $XXZ$ couplings immersed in nonuniform fields. It is shown that an exceptionally degenerate set of completely separable symmetry-breaking GSs can arise for a wide range of field configurations, at a quantum critical point where all GS magnetization plateaus merge. Such configurations include alternating fields as well as zero-bulk field solutions with edge fields only and intermediate solutions with zero field at specific sites, valid for $d$-dimensional arrays. The definite magnetization-projected GSs at factorization can be analytically determined and depend only on the exchange anisotropies, exhibiting critical entanglement properties. We also show that some factorization-compatible field configurations may result in field-induced frustration and nontrivial behavior at strong fields.

Figure 1

Top left: The two solutions of Eq. (7) for ${\theta }_j$ vs ${\theta }_i$ (thick solid lines). For an arbitrary initial spin orientation ${\vartheta }_0$ at one site, successive application of Eq. (7) determines the possible orientation angles (indicated by the arrows) of the remaining spins in a factorized eigenstate $|\mathrm{\Theta }⟩$. Each sequence of angles leads to a different factorizing field configuration determined by Eq. (8), shown in the top right panels for three spins and in the bottom rows for the first six spins of a chain with uniform spin and couplings. Two extremal cases arise: an alternating solution (a) and a zero-bulk field solution with edge fields only (d). Solutions with intermediate zero fields are also feasible (b),(c). In a cyclic chain, the first field is $2h_s$.

Figure 2

GS magnetization diagram for alternating fields $h^{2i-1}=h_1$, $h^{2i}=h_2$ in an $N=8$ spin-1 $XXZ$ chain with $\mathrm{\Delta }=1.2$. All magnetization plateaus $M=Ns,\dots ,-Ns$ coalesce at the factorizing fields $h_1=-h_2=\pm 2h_s$. The inset indicates the mean field (MF) phases.

Figure 3

Exact pair negativities $N_{ij}$ between spins $i$ and $j$ in the exact GS of the spin-1 chain in Fig. 2, for fields $h_1$, $h_2$ of opposite sign and first (top left), second (top right), and third (bottom left) neighbors. Bottom right: The exact pair negativities at factorization ($h_1=-h_2=2h_s$) in the definite magnetization GSs, for identical $N=8$ spin-$s$ chains with $s=1/2,\dots ,4$. At this point there are just three distinct pair negativities: $N_{oe}$ (odd-even), $N_{oo}$, and $N_{ee}$, independent of the actual separation $|i-j|$ and dependent on $M$.

Figure 4

Exact GS magnetization diagram for distinct spin arrays and field configurations with $\mathrm{\Delta }=1.2$. Top: Cyclic $N=12$ spin-$1/2$ chain with next alternating fields (left) and a zero-bulk field (right). Bottom: Open $3\times 4$ spin-$1/2$ arrays with alternating (left) and zero-bulk (right) field configurations. All plateaus merge at the factorizing point, where the GS has the indicated angles. Field-induced frustration in configurations with zero fields leads to a reduced $M=0$ plateau.