For the critical restricted solid-on-solid (RSOS) models of Andrews, Baxter and Forrester, Westerfeld, Großpietsch, Kakuschke and Frahm (J. Stat. Mech. 2023) proved, within the topological sector containing the ground state, that the reduced density matrix of a short segment factorizes into a physical part built from a single nearest-neighbour two-point function ω and parameter-independent structure functions in the Temperley–Lieb algebra, yielding exact multi-point local height probabilities and su(2)_k anyon correlation functions. The paper itself leaves open a coherent cluster of extensions: other topological sectors, where the physical part is expected to involve two distinct nearest-neighbour functions; completion of the four-site Temperley–Lieb decomposition; factorization in the massive regimes III and IV; rigorous proofs of the asymptotic identity (3.8) and the discrete functional equation (3.10) that entered as conjectural inputs; and closed-form ω_r for general r (coefficients tabulated only for r ≤ 10). No later work resolving any of these items is known: the only engagements transfer the method to a different face model while keeping factorization as an ansatz, or cite the paper for context. All five items remain open.

Background

The restricted solid-on-solid (RSOS) models introduced by Andrews, Baxter and Forrester (ABF) are two-dimensional exactly solved lattice models of statistical mechanics in which every site of a square lattice carries an integer 'height' taking the values 1, …, r−1 (r ≥ 3), with nearest-neighbour heights constrained to differ by exactly one unit. At their critical points these models realize the unitary minimal conformal field theories, and they are the prototypical interaction-round-a-face (IRF) representatives of the Temperley–Lieb universality class shared with the spin-1/2 Heisenberg chain. The natural correlation functions of the RSOS models are the local height probabilities (LHPs), the joint probabilities of observing a prescribed pattern of heights on a chosen set of sites.

A compact packaging of all LHPs on a segment of n adjacent sites is the reduced density matrix (RDM) of the segment, obtained by tracing the ground-state density matrix over all other sites; its matrix elements are precisely the multi-point LHPs on the segment. For the spin-1/2 Heisenberg chain a remarkable structure was observed for such segment RDMs: they 'factorize' into a physical part — a small set of scalar two-point functions carrying the entire dependence on system size, state and model parameters — multiplied by parameter-independent structure functions built from representations of the Temperley–Lieb (TL) algebra, the algebra with generators e_i obeying e_i² = δ e_i and the braid relations e_i e_{i±1} e_i = e_i, realized here through the fusion hierarchy of the RSOS transfer matrix. On a periodic chain the spectrum of the RSOS model decomposes into invariant topological sectors labelled by a quantum number j, each carrying a 'quantum dimension' d_q(j) = sin(π(2j+1)/r)/sin(π/r) that gives the asymptotic degeneracy of the sector's states; the ground state lies in the trivial sector with d_q = 1. Westerfeld, Großpietsch, Kakuschke and Frahm (J. Stat. Mech. 2023) established the analogous factorization for the critical RSOS models in this trivial sector containing the ground state, with a single nearest-neighbour function ω as the physical ingredient.

The proof strategy, developed for general integrable face models by Frahm and Westerfeld (SciPost Phys. 2021), goes through a reduced qKZ (quantum Knizhnik–Zamolodchikov) equation: a functional equation for the segment RDM in the spectral parameters attached to the segment, obtained from the qKZ-type equations for finite-chain density matrices after projection onto the sector. Within this framework the 2023 paper computes ω for the infinite-system ground state (as the thermodynamic-limit conjecture ω_r(x), with coefficients tabulated only for r ≤ 10), evaluates multi-point LHPs on up to four adjacent sites, and translates the results into three-point correlation functions of non-Abelian su(2)k anyons — quasiparticles attached to the RSOS heights whose fractional exchange statistics are governed by the quantum group su(2) at level k. Two relations, however, entered as conjectural inputs supported by numerical evidence: an asymptotic identity (Eq. (3.8) of the paper) stating that the N-site RDM D_N reduces to the (N−1)-site RDM D{N−1}, projected onto the last site through the one-site density matrix D_1, when the last spectral parameter λ_N is sent to i∞; and a discrete functional equation (Eq. (3.10)) for ω, checked by the authors for r up to 12.

The paper is explicit that this completes only the first step of the program. The analysis is restricted to one topological sector; in the other sectors preliminary evidence suggests the physical part involves two distinct nearest-neighbour functions rather than one. Only a single matrix element of the four-site decomposition was verified, with the full expansion of the structure-function blocks F_{4;I,J} (labelled by their Temperley–Lieb representation content I, J) in the Temperley–Lieb basis TL_4 explicitly deferred to future work. Only the critical (massless) regimes are treated, whereas the ABF models also possess massive regimes III and IV — the two non-critical phases of the height model, in which correlations decay exponentially — where RDMs are currently accessible only through multiple-integral representations. In addition, the factorization is not unique in special cases: for r = 4 a linear identity among the ω terms (sin(2λ₁₂) ω(λ₁,λ₂) + sin(2λ₂₃) ω(λ₂,λ₃) = sin(2λ₁₃) ω(λ₁,λ₃) in the source's spectral-parameter notation, with λ_kl = λ_k − λ_l) implies that one structure function does not enter the decomposition, and the source resolves this non-uniqueness by choosing that function to be zero. Finally, the source notes that extending the procedure to other eigenstates of the RSOS transfer matrix in this sector is straightforward once the additional poles of ω in its analyticity strips S₀ = {z ∈ ℂ : −λ/2 ≲ Re z ≲ 3λ/2} (the physical strip) and S₁ = {z ∈ ℂ : −π + 3λ/2 ≲ Re z ≲ −λ/2} are taken into account, where λ = π/r is the crossing parameter of the ABF weights.

Problem Statement

Complete the factorization program for reduced density matrices of the critical RSOS models, replacing the sector-restricted and partly conjectural statements of Westerfeld, Großpietsch, Kakuschke and Frahm (J. Stat. Mech. 2023) by complete and rigorous ones. The common scientific objective uniting the requested results is an exact, model-wide description of the segment density matrices of the ABF RSOS models in factorized form — valid across topological sectors and system sizes, and resting on proved identities rather than numerically supported conjectures. Concretely, in the notation of that paper: (1) Topological sectors: extend the factorization analysis to the other topological sectors of the critical RSOS models — determine whether the physical part involves two distinct nearest-neighbour functions (as expected by analogy with the XXZ chain) and, if so, derive the algebraic decomposition and determine the additional physical functions in those sectors. (2) Four-site decomposition: complete the expansion of all blocks F_{4;I,J} in the Temperley–Lieb basis TL_4, that is, the coefficients for every block and matrix element, currently verified only for a single matrix element. (3) Massive regimes: determine whether factorized representations of the RDMs exist in the massive regimes III and IV, and construct them, providing an alternative to the existing multiple-integral formulas. (4) Rigorous foundations: prove the asymptotic identity (3.8) — the statement that D_N reduces to D_{N−1} projected through the one-site density matrix D_1 as the last spectral parameter λ_N is sent to i∞ — and the discrete functional equation (3.10), which were used as conjectural inputs, or refute them by explicit counterexample. In explicit form, equation (3.10) asserts that, for λ = π/r and v an inhomogeneity parameter of the transfer matrix,

ω(u,v+λ)=sin⁡2λcos⁡2(u−v)−cos⁡2λ−cos⁡(2(u−v−λ))−cos⁡2λcos⁡(2(u−v))−cos⁡2λ ω(u,v),\omega(u, v+\lambda) = \frac{\sin^2\lambda}{\cos 2(u-v) - \cos 2\lambda} - \frac{\cos(2(u-v-\lambda)) - \cos 2\lambda}{\cos(2(u-v)) - \cos 2\lambda}\,\omega(u, v),

for the nearest-neighbour function ω(u, v) in the ground state of the thermodynamic limit. (5) Closed forms: prove the thermodynamic-limit conjecture ω_r(x) and extend it to general r with closed-form coefficient formulas (the source's Table 1 reaches only r ≤ 10). Items (1)–(3) are the open directions explicitly proposed by the source paper (conclusion; §IV.C for the four-site deferral), while items (4)–(5) concern the relations the paper explicitly leaves as numerically supported conjectures (Eqs. (3.8) and (3.10), and the thermodynamic-limit expression ω_r(x) with coefficients tabulated only for r ≤ 10); they are linked as instances of a single mechanism — the reduced-qKZ/factorization ansatz for face-model density matrices. The five items are independently standing, separately attackable open sub-questions, but the objective stated here is their conjunction: the program is complete only when all of (1)–(5) are resolved, and a result that settles only some of them is a partial resolution confined to those items. For the determine-whether items (1) and (3), a well-supported negative determination is itself a resolution of that item. For item (4), an explicit counterexample to either identity (3.8) or (3.10) resolves the item on its own; absent such a refutation, the item requires both identities to be proved.

References

  1. D. Westerfeld, M. Großpietsch, H. Kakuschke, H. Frahm, 'Factorization of density matrices in the critical RSOS models', J. Stat. Mech. (2023) 083104, DOI 10.1088/1742-5468/aceeef, arXiv:2303.15252, https://arxiv.org/abs/2303.15252
  2. H. Frahm, D. Westerfeld, 'Density matrices in integrable face models', SciPost Phys. 11, 057 (2021), DOI 10.21468/SciPostPhys.11.3.057, arXiv:2104.06269, https://arxiv.org/abs/2104.06269
  3. K. Fukai, R. Kleinemühl, B. Pozsgay, E. Vernier, 'On correlation functions in models related to the Temperley-Lieb algebra', SciPost Phys. 16, 003 (2024), DOI 10.21468/SciPostPhys.16.1.003, arXiv:2309.07472, https://arxiv.org/abs/2309.07472
  4. T. S. Tavares, G. A. P. Ribeiro, 'Correlation functions of the six-vertex IRF model and its quantum spin chain', Nucl. Phys. B 1008, 116715 (2024), DOI 10.1016/j.nuclphysb.2024.116715, arXiv:2409.05814, https://arxiv.org/abs/2409.05814
  5. T. S. Tavares, G. A. P. Ribeiro, 'Finite temperature properties of an integrable zigzag ladder chain', Nucl. Phys. B 995, 116333 (2023), DOI 10.1016/j.nuclphysb.2023.116333, arXiv:2307.11279, https://arxiv.org/abs/2307.11279
  6. M. Sinha, T. S. Tavares, A. Roy, H. Saleur, 'Integrability and lattice discretizations of all topological defect lines in minimal CFTs', Nucl. Phys. B 1023, 117308 (2026), DOI 10.1016/j.nuclphysb.2026.117308, arXiv:2509.04257, https://arxiv.org/abs/2509.04257
  7. M. Blakeney, L. Corcoran, M. de Leeuw, B. Pozsgay, E. Vernier, 'Temperley-Lieb integrable models and fusion categories', JHEP 04 (2026) 165, DOI 10.1007/JHEP04(2026)165, arXiv:2510.19902, https://arxiv.org/abs/2510.19902
  8. H. Topchyan, 'The ZN×3\mathbb{Z}_N^{\times 3} symmetry protected boundary modes in two-dimensional Potts paramagnets', arXiv:2604.00910 (2026), https://arxiv.org/abs/2604.00910