Symmetry-enforced minimal entanglement and minimal correlation length in integer spin-J chains beyond matrix product states

An integer spin-J chain (J a positive integer) whose SO(3) spin-rotation and one-site lattice-translation symmetries are neither explicitly nor spontaneously broken cannot be arbitrarily weakly entangled. Li and Zou (SciPost Phys. 19, 020 (2025)) proved, within the class of translation-invariant matrix product states (uMPS), that the asymptotic Renyi-alpha entropy of a long segment is at least $\min\{\ln(J+1),f_\alpha(J)\}$ per entanglement cut, where $f_\alpha(J)=-\frac{1}{\alpha-1}\ln(2^{-\alpha}(1+(2J+1)^{-(\alpha-1)}))$, and that no symmetric uMPS has a vanishing correlation length. Whether these floors hold for all physically admissible symmetric states rests on two unproven working assumptions. The open problem is threefold: (i) prove or disprove that the thermodynamic Renyi-entropy limits exist for every $\alpha>0$ for all bounded-entropy translation-symmetric states (currently proven only for $\alpha=1$); (ii) prove that the uMPS entropy minima are global over all symmetric zero-momentum states and settle whether nonzero-momentum symmetric states must have strictly larger asymptotic entropies, or exhibit counterexamples; (iii) determine the exact minimal correlation length among symmetric states and prove or refute its divergence as $J\to\infty$.

Background

Consider a one-dimensional chain of $L$ sites in which each site carries a $(2J+1)$-dimensional Hilbert space transforming as the spin-$J$ irreducible representation of the rotation group $SO(3)$, with $J$ a positive integer. The states of interest are pure states that are invariant under global $SO(3)$ rotations, compatible with one-site lattice translation, and break neither symmetry spontaneously. Bipartite entanglement is quantified by the Renyi entropy $S_\alpha(\rho)=\frac{1}{1-\alpha}\ln\mathrm{Tr}(\rho^\alpha)$ of the reduced density matrix of a segment of $N$ contiguous sites with $1\ll N\ll L$, which becomes the von Neumann entropy at $\alpha=1$. In a closed (periodic) chain the segment has two entanglement cuts and its reduced density matrix is denoted $\rho_{\mathrm{t.c.}}(N,L)$; in an infinitely long open chain a segment of $N$ contiguous sites adjacent to the boundary has a single entanglement cut, with reduced density matrix denoted $\rho_{\mathrm{o.c.}}(N,L)$. The relevant quantities are the double limits $\lim_{N\to\infty}\lim_{L\to\infty}S_\alpha(\rho_{\mathrm{t.c.}}(N,L))$ and $\lim_{N\to\infty}\lim_{L\to\infty}S_\alpha(\rho_{\mathrm{o.c.}}(N,L))$, taken with $L\to\infty$ first.

The backdrop is the family of Lieb-Schultz-Mattis (LSM) constraints, which state that a quantum spin chain whose symmetries are not spontaneously broken must be long-range entangled (either symmetry-breaking order or violation of the entanglement area law, i.e. unbounded segment entropy) whenever the spin per unit cell is half-integer. For integer $J$ the area law is compatible with the symmetries, so a finite entanglement floor exists; nevertheless, semiclassical and LSM-type reasoning suggests that this floor, and the minimal correlation length, should still grow without bound as $J$ increases, so that large-$J$ integer-spin chains resemble their half-integer counterparts.

Li and Zou attacked this problem within the framework of matrix product states. A matrix product state (MPS) represents a many-body wavefunction through local rank-3 tensors contracted along the chain; a uniform MPS (uMPS) uses the same tensor at every site and is translation invariant. Its correlation structure is governed by the transfer matrix $T[A]=\sum_i A^i\otimes\bar{A}^i$; for an injective uMPS (one whose tensors define an injective map from the bond space to the physical space) the correlation length is $\xi=-1/\ln|\lambda_2|$, where $\lambda_2$ is the second-largest eigenvalue of $T[A]$ in magnitude. Symmetric tensors decompose into Clebsch-Gordan blocks labeled by pairs of virtual spin sectors, a structure fixed by $SO(3)$ symmetry following Sanz, Wolf, Perez-Garcia and Cirac. Within this framework Li and Zou proved that the minimal asymptotic Renyi-$\alpha$ entropy is $2\min\{\ln(J+1),f_\alpha(J)\}$ for two cuts and $\min\{\ln(J+1),f_\alpha(J)\}$ for one cut, with $f_\alpha(J)=-\frac{1}{\alpha-1}\ln(2^{-\alpha}(1+(2J+1)^{-(\alpha-1)}))$ and $f_1(J)=\ln(2\sqrt{2J+1})$; in particular the von Neumann floors are $\min\{2\ln(J+1),\ln(4(2J+1))\}$ and $\min\{\ln(J+1),\ln(2\sqrt{2J+1})\}$, which diverge as $J\to\infty$. The minima are saturated by explicit gapped-parent-Hamiltonian ground states: type-I states with a single virtual spin-$J/2$ sector (for $J=1$ the Affleck-Kennedy-Lieb-Tasaki state) and type-II states with virtual sectors $0\oplus J$, the winner depending on $J$ and $\alpha$. They further proved that no $SO(3)$-symmetric uMPS of an integer spin-$J$ chain has exactly vanishing correlation length, i.e. the transfer matrix cannot be rank one, while numerical optimization shows that the AKLT state does not minimize $\xi$: symmetric $J=1$ uMPS exist with $|\lambda_2|\approx0.106$ and $\approx0.076$, below the AKLT value $1/3$.

All minimality statements, however, are established only within the uMPS class. Their extension to arbitrary symmetric states rests on two assumptions the authors state explicitly but cannot prove: first, that for any translation-symmetric state whose segment Renyi entropies remain bounded under the limit procedure, the double limits exist for every $\alpha>0$ (this is proven only for $\alpha=1$, where strong subadditivity of the von Neumann entropy, $S(\rho_{AB})+S(\rho_{BC})\ge S(\rho_{ABC})+S(\rho_B)$, forces convergence, as shown by Wolf, Verstraete, Hastings and Cirac; Renyi entropies with $\alpha\ne1$ obey no comparable inequality); and second, that the asymptotic entropy minima and the minimal correlation length are attained within the uMPS class. Related results exist in adjacent settings: Gioia and Wang proved that nonzero momentum requires long-range entanglement in a filling-based setting, and a parallel program studies symmetry-enforced entanglement in mixed states with Abelian symmetries (Moharramipour et al.; Negari, Lessa and Sahu), but none of these addresses the pure-state $SO(3)$-symmetric chain questions above.

Problem Statement

Overarching objective: determine the exact symmetry-enforced floors of bipartite entanglement and of correlation length for all pure states of an integer spin-$J$ chain ($J\in\mathbb{Z}^+$) whose $SO(3)$ spin-rotation and one-site lattice-translation symmetries are neither explicitly nor spontaneously broken, thereby removing the uniform-matrix-product-state restriction under which these floors are currently established. The problem has three linked parts, each of which, together with every sub-claim it states, is a separately required deliverable that its own proof or disproof resolves. Part (2) is quantified over the states for which the limits in part (1) exist, and part (3) is independent of part (1); a disproof of part (1) is therefore a complete negative resolution of that part, showing that the Li-Zou floors are not universal, and leaves parts (2) and (3) to be settled on the states they cover. All three parts jointly decide whether the Li-Zou minima are universal properties of symmetric states.

(1) Existence of the asymptotic entropy limits. For a translation-symmetric pure state of the chain, prove or disprove that, for every $\alpha\in\mathbb{R}^+$ for which the Renyi entropies $S_\alpha(\rho_{\mathrm{t.c.}}(N,L))$ and $S_\alpha(\rho_{\mathrm{o.c.}}(N,L))$ of a segment of $N$ contiguous sites ($1\ll N\ll L$) remain bounded when $L\to\infty$ is taken first and $N\to\infty$ second, the double limits

$$\lim_{N\to\infty}\lim_{L\to\infty}S_\alpha(\rho_{\mathrm{t.c.}}(N,L)),\qquad \lim_{N\to\infty}\lim_{L\to\infty}S_\alpha(\rho_{\mathrm{o.c.}}(N,L))$$

exist. Existence is currently proven only for $\alpha=1$ (via strong subadditivity) and for states representable by uMPS (for all $\alpha$).

(2) Globality of the minimal entropies. Writing

$$f_\alpha(J)=-\frac{1}{\alpha-1}\ln\bigl(2^{-\alpha}(1+(2J+1)^{-(\alpha-1)})\bigr),\qquad f_1(J)=\ln(2\sqrt{2J+1}),$$

prove that the minima of $\lim_{N\to\infty}\lim_{L\to\infty}S_\alpha(\rho_{\mathrm{t.c.}}(N,L))$ and $\lim_{N\to\infty}\lim_{L\to\infty}S_\alpha(\rho_{\mathrm{o.c.}}(N,L))$ over all $SO(3)$-symmetric zero-momentum states for which the limits in part (1) exist equal the Li-Zou values $2\min\{\ln(J+1),f_\alpha(J)\}$ and $\min\{\ln(J+1),f_\alpha(J)\}$ respectively, or exhibit such a symmetric state with a strictly smaller limit. Furthermore, prove or refute that $SO(3)$-symmetric states which are eigenstates of one-site translation with nonzero momentum (and are therefore not strictly translation invariant) must, when the limits of part (1) exist, have strictly larger values of these limits than the corresponding Li-Zou minima.

(3) Minimal correlation length. Determine the exact minimal correlation length among $SO(3)$-symmetric states of the integer spin-$J$ chain, where for injective uMPS the correlation length is $\xi=-1/\ln|\lambda_2|$ with $\lambda_2$ the second-largest eigenvalue in magnitude of the transfer matrix, and for general states it is defined through the slowest decay rate among connected two-point correlation functions of local operators (i.e. a supremum over pairs of local operators, since different pairs may decay at different rates). In particular: determine the exact minimal achievable $\xi$ for each $J$, exactly or through two-sided bounds that determine its value (already unknown for $J=1$, where numerical minima with $|\lambda_2|\approx0.076$ are not proven optimal), and prove or refute the expectation, motivated by semiclassical analysis combined with Lieb-Schultz-Mattis-type constraints, that the minimal correlation length diverges as $J\to\infty$.

References

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