Parity-induced logarithmic term in the min-entropy of the critical XXZ chain: prove or refute that the prefactor equals $\alpha=\pi R^{2}=1/(4K)$ throughout $-1<\Delta\leq 1$

For the spin-1/2 XXZ chain with periodic boundary conditions, the $n=\infty$ Shannon-Rényi entropy (min-entropy) $S_{\infty}=-\log p_{\max}$ of the ground state in the $\sigma^{z}$ basis equals the single-copy entanglement of the associated two-dimensional six-vertex Rokhsar-Kivelson ground state on an infinite cylinder. Pradhan, Cobos, Rico, and Sierra (arXiv:2502.19152; Phys. Rev. B 113, 094438 (2026)) discovered that throughout the critical regime $-1<\Delta\leq 1$ an odd chain length $L$ produces a logarithmic term in $S_{\infty}$ that is absent for even $L$, traced to the frustration of the Néel-type maximal configuration by an unpaired moving spinon (a spin-$1/2$ excitation). They conjectured, from a conditional exact derivation at $\Delta=1/2$ (resting on Razumov-Stroganov combinatorial conjectures) and high-precision numerics at the free-fermion point $\Delta=0$, exact-diagonalization fits for $L\leq 23$, a variational infinite matrix product state ansatz, and a heuristic conformal-field-theory argument, that the prefactor is universal: $b=\alpha=\pi R^{2}=1/(4K)$, where $\Delta=-\cos(2\pi\alpha)$ and $R$ (equivalently $K$) is the compactification radius (Luttinger parameter) of the low-energy $c=1$ theory. The open problem is to turn this conjecture into a rigorous derivation valid for the whole critical interval -- discharging the two unproven steps of the authors' route, namely the asymptotic constancy of the odd/even ratio of Gaudin's discrete Coulomb-gas partition functions and an exact conformal-field-theory representation of the Néel maximal states with their overlaps with the spinon primary -- or to refute it by exhibiting an anisotropy where the prefactor differs.

Background

The spin-1/2 XXZ chain with periodic boundary conditions (PBC) is the Hamiltonian $H_{\mathrm{XXZ}}=\sum_{i=1}^{L}(\sigma_i^x\sigma_{i+1}^x+\sigma_i^y\sigma_{i+1}^y+\Delta\,\sigma_i^z\sigma_{i+1}^z)$, with anisotropy $\Delta$. For $-1<\Delta\leq 1$ the chain is critical (massless): its low-energy description is a free compactified boson, a conformal field theory (CFT) with central charge $c=1$ characterized by a compactification radius $R=\sqrt{\frac{1}{2\pi}-\frac{1}{2\pi^{2}}\cos^{-1}\Delta}$, or equivalently by the Luttinger parameter $K=1/(4\pi R^{2})$ that governs the power-law decay of spin correlations. The point $\Delta=0$ maps via a Jordan-Wigner transformation to free spinless fermions; $\Delta=1$ is the Heisenberg point and the endpoint of a Berezinskii-Kosterlitz-Thouless transition (a topological transition that, in the bosonic language, is driven by vortex binding and separates the critical phase from the gapped Néel-ordered phase); $\Delta=1/2$ is tied to deep combinatorial results through the Razumov-Stroganov conjectures relating the ground-state components of the $\Delta=\pm 1/2$ chain to enumeration of alternating sign matrices (square matrices of $0,\pm 1$ whose row and column sums are $1$).

A Rokhsar-Kivelson (RK) wavefunction is a superposition $|\mathrm{RK}\rangle=\mathcal{Z}^{-1/2}\sum_{c}e^{-E(c)/2}|c\rangle$ over all configurations $c$ satisfying local constraints, weighted by the square root of a classical energy; such states are exact ground states of specially constructed two-dimensional quantum Hamiltonians at conformal quantum critical points (Ardonne, Fendley, Fradkin). The quantum six-vertex model on a cylinder is the archetype: arrows on lattice links obey the six-vertex "ice rule" (two arrows in, two out at each vertex), and the amplitudes of its RK ground state in a cut basis coincide with the ground-state amplitudes of the XXZ chain in the $\sigma^{z}$ basis.

Stéphan, Furukawa, Misguich and Pasquier showed that for such a cylinder bipartition the Schmidt eigenvalues of the 2D RK state equal the probabilities $p_i=|\langle i|\mathrm{gs}\rangle|^{2}$ of the 1D chain ground state over transfer-matrix basis states. Hence the $n$-th Renyi entanglement entropy of the 2D wavefunction equals the $n$-th Shannon-Rényi entropy $S_n=\frac{1}{1-n}\log\sum_i p_i^{n}$ of the 1D state. At $n=\infty$ this is the min-entropy $S_{\infty}=-\log p_{\max}$, which in entanglement language is the single-copy entanglement (Eisert, Cramer). Stéphan, Misguich and Pasquier additionally showed that $S_n$ undergoes a "locking" phase transition in the Renyi order $n$: beyond a critical $n_c$ the entropy is dominated by the maximal-probability states, which makes $S_{\infty}$ the natural object of the theory.

For the critical XXZ chain the most probable basis state is the Néel state (perfectly alternating up/down spins). With PBC and odd $L$, perfect alternation is impossible: one pair of neighboring spins is necessarily parallel, which acts as a spinon; the ground state then carries total $S^{z}=\pm 1/2$, is doubly degenerate, and the maximal configurations acquire an extensive positional degeneracy (the spinon can sit in $L$ inequivalent places).

Against this backdrop, Pradhan, Cobos, Rico and Sierra found that for odd $L$ the min-entropy acquires a logarithmic term surviving the thermodynamic limit, $S_{\infty}^{(\mathrm{o})}(L)=a(\Delta)L+b\log L+c(\Delta)+\cdots$, while for even $L$ the scaling is linear-plus-constant. Their evidence: (i) at $\Delta=1/2$, $b=1/3$ follows from the Razumov-Stroganov combinatorial conjectures, themselves only partially proven through quantum Knizhnik-Zamolodchikov (qKZ) equation methods; (ii) at $\Delta=0$, $b=1/4$ is supported by high-precision numerics on the exact Slater-determinant probabilities up to $L=51$ ($b=0.2566\pm 0.0006$), though no closed asymptotic derivation exists; (iii) exact-diagonalization fits for 20 values of $\Delta$ at $L=7,\dots,23$ track the curve $\Delta=-\cos(2\pi b)$, with finite-size effects acknowledged as still relevant; (iv) a variational infinite matrix product state (iMPS; a matrix product state on infinitely many sites built from CFT vertex operators, in the sense of Cirac and Sierra) with parameter $\alpha$, $\Delta=-\cos(2\pi\alpha)$, reproduces the probabilities with over $99\%$ fidelity for $L\leq 20$ and reduces the entropy difference to $S_{\infty}^{(\mathrm{o})}-S_{\infty}^{(\mathrm{e})}=-4\alpha\log\frac{p_{\max}^{(\mathrm{o})}}{p_{\max}^{(\mathrm{e})}}\big|_{\Delta=0}-4\alpha N\log(1+\frac{1}{2N})+\log\frac{Z_{\alpha}^{(\mathrm{o})}(N)}{Z_{\alpha}^{(\mathrm{e})}(N)}$, where $Z_{\alpha}$ is the partition function of Gaudin's discrete Coulomb gas -- $N$ particles on an $L$-site circle with logarithmic pair interaction, exactly evaluated by Gaudin only at inverse temperatures $\beta=8\alpha\in\{0,1,2,4\}$ (the superscripts $(\mathrm{o})$ and $(\mathrm{e})$ denote evaluation at odd and even chain length, $L=2N+1$ and $L=2N$ respectively, in the source's notation); and (v) a heuristic CFT argument attributing the term to the spinon: the odd ground state is not the CFT vacuum but a spin-$1/2$ excitation created by a primary field (an operator whose conformal weight, the exponent governing its scaling with system size, is fixed by the operator content of the XXZ universality class) of weight $\Delta_{1/2}$, so that $p_{\max}^{(\mathrm{o})}\simeq(2\pi/L)^{2\Delta_{1/2}}p_{\max}^{(\mathrm{e})}$ and $S_{\infty}^{(\mathrm{o})}-S_{\infty}^{(\mathrm{e})}\simeq 2\Delta_{1/2}\log L$, consistent with the conjecture if $\Delta_{1/2}=\alpha/2$.

This yields the central conjecture, stated as Eq. (13) of the source and left open by it: throughout the critical interval, $S_{\infty}^{(\mathrm{o})}(L)-S_{\infty}^{(\mathrm{e})}(L)\sim\alpha\log L+\mathrm{const},\qquad b=\alpha=\pi R^{2}=\frac{1}{4K},\qquad \Delta=-\cos(2\pi\alpha).$ The claim is physically sharp: it would make the parity of the system size a direct, coefficient-exact probe of the compactification radius (equivalently the Luttinger parameter) of the underlying $c=1$ CFT. Known universal logarithmic corrections to entanglement at two-dimensional conformal quantum critical points depend on central charge, topology and boundary smoothness (Zaletel, Bardarson, Moore), and known one-dimensional parity effects concern oscillatory corrections to block entanglement governed by $K$ and the Fermi momentum (Calabrese, Campostrini, Essler, Nienhuis). Parity effects with respect to the length of a cylindrical subregion of two-dimensional RK states were observed by Ju et al. and by Stéphan, Ju, Fendley and Melko; the source paper explicitly calls those "orthogonal" to the present effect, which concerns the parity of the cylinder circumference. No earlier work states the circumference-parity min-entropy law, and its prefactor is established at no point of the interval except -- modulo unproven combinatorial conjectures -- $\Delta=1/2$. The source additionally verified numerically that no parity oddity occurs at the transverse-field Ising point (a special point of the eight-vertex model, which relaxes the six-vertex ice rule by also allowing the all-in and all-out vertices), indicating that the effect is specific to the six-vertex/XXZ setting rather than automatic for related lattice Hamiltonians.

Problem Statement

For the spin-1/2 XXZ chain $H_{\mathrm{XXZ}}=\sum_{i=1}^{L}(\sigma_i^x\sigma_{i+1}^x+\sigma_i^y\sigma_{i+1}^y+\Delta\,\sigma_i^z\sigma_{i+1}^z)$ with periodic boundary conditions, in the $\sigma^{z}$ basis, taking the odd-$L$ ground state within the $S^{z}=\pm 1/2$ doublet and the even-$L$ ground state with $S^{z}=0$: prove or refute that throughout the critical regime $-1<\Delta\leq 1$ the min-entropy $S_{\infty}=-\log\max_{i}|\langle i|\mathrm{gs}\rangle|^{2}$ satisfies

$S_{\infty}^{(\mathrm{o})}(L)-S_{\infty}^{(\mathrm{e})}(L)=\alpha\log L+O(1),\qquad \alpha=\frac{1}{2\pi}\cos^{-1}(-\Delta)=\pi R^{2}=\frac{1}{4K},$

equivalently $S_{\infty}^{(\mathrm{o})}(L)=a(\Delta)L+\frac{1}{4K}\log L+c(\Delta)+o(1)$. A complete resolution must, by proof, controlled approximation, or certified computation, decide the law $b=\alpha$ throughout $-1<\Delta\leq 1$; the two unproven steps (a) and (b) below are the route the source derivation leaves open, a negative outcome on either step refutes only that route step rather than the law itself, and the derivation may instead be replaced by an alternative controlled argument:

(a) show that the ratio of discrete Coulomb-gas partition functions $Z_{\alpha}^{(\mathrm{o})}(N)/Z_{\alpha}^{(\mathrm{e})}(N)$ -- where $Z_{\alpha}^{(\mathrm{o})}(N)=Z_{\alpha}(2N+1,N)$ and $Z_{\alpha}^{(\mathrm{e})}(N)=Z_{\alpha}(2N,N)$ are, in the source's notation, the partition functions of Gaudin's discrete Coulomb gas ($N$ particles on an $L$-site circle with logarithmic pair interaction, at inverse temperature $\beta=8\alpha$) evaluated at odd and even chain length -- tends to a constant as $N\to\infty$ for all $0<\alpha\leq 1/2$ (currently established only at $\alpha=1/4$ and $\alpha=1/2$ from Gaudin's exact solutions at $\beta=8\alpha=2,4$), or identify any exceptional $\alpha$ where it does not;

(b) construct a precise representation of the Néel-type maximal states as states of the compact-boson CFT and evaluate exactly their overlaps with the spinon primary of conformal weight $\Delta_{1/2}$, confirming or correcting the identification $\Delta_{1/2}=\alpha/2$ that fixes the prefactor.

References

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