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Exact twist-field correlators of the Yang–Lee minimal model on the lattice: two critical realizations and a three-level test

Rongzhen Dai

Published Jul 24, 2026• 0 citations
Publisher

Abstract

The Rényi entropies of a non-unitary critical system are not fixed by a single power of the central charge but by the full four-point function of twist fields in a cyclic orbifold, constrained by null-vector equations. For the Yang–Lee universality class—the minimal model with —Dupic, Estienne and Ikhlef (DEI) computed several of these twist correlators in closed form. We provide an independent lattice realization and calibration of those exact correlators on two distinct critical lattice models: the critical Restricted Solid-On-Solid (RSOS) model with and the quantum Ising chain in an imaginary longitudinal field. Both are diagonalised exactly, and the lattice-to-continuum correspondence is certified on three independent objects: the low-lying spectrum reproduces the tower with effective central charge and no exceptional-point degeneracy on the RSOS side; the single-interval vacuum and ground-state twist correlators reproduce the exact two- and four-point functions, including a quantitative determination of the sub-leading fusion channel; and the vacuum two-interval four-twist function is reproduced with a crossing self-consistency of . Along the way we correct a typographical error in the DEI Temperley–Lieb Hamiltonian. We define precisely the boundary-height-sector-resolved trace functional that the lattice certifies, and we do not address the microscopic-realization dependence of the physical two-interval mutual information, which involves a choice of observable subalgebra and is left to a separate study.

DOI
10.5281/zenodo.21523087

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The Rényi entropies of a non-unitary critical system are not fixed by a single power of the central charge but by the full four-point function of twist fields in a cyclic orbifold, constrained by null-vector equations. For the Yang–Lee universality class—the minimal model with —Dupic, Estienne and Ikhlef (DEI) computed several of these twist correlators in closed form. We provide an independent lattice realization and calibration of those exact correlators on two distinct critical lattice models: the critical Restricted Solid-On-Solid (RSOS) model with and the quantum Ising chain in an imaginary longitudinal field. Both are diagonalised exactly, and the lattice-to-continuum correspondence is certified on three independent objects: the low-lying spectrum reproduces the tower with effective central charge and no exceptional-point degeneracy on the RSOS side; the single-interval vacuum and ground-state twist correlators reproduce the exact two- and four-point functions, including a quantitative determination of the sub-leading fusion channel; and the vacuum two-interval four-twist function is reproduced with a crossing self-consistency of . Along the way we correct a typographical error in the DEI Temperley–Lieb Hamiltonian. We define precisely the boundary-height-sector-resolved trace functional that the lattice certifies, and we do not address the microscopic-realization dependence of the physical two-interval mutual information, which involves a choice of observable subalgebra and is left to a separate study.
We test whether the infrared fixed point of a non-unitary “parent” model — the object whose categorical data we certify — is the Yang–Lee minimal model M(2,5), by direct lattice measurement on the A₄ RSOS(4,3) chain. Three results are reported. (1) The effective central charge is measured by ground-state energy-level scaling, c_eff = 0.40 ± 0.02, with the error bar certified by the deviations of two known baselines (unitary M(4,5) → 0.70, non-unitary M(3,5) → 0.60). (2) The lowest scaling dimension is measured from the excited-gap scaling, h_φ = −1/5, with the topological spin θ_φ = e^{−2πi/5} following as a one-step corollary of the standard spin–weight relation (cross-checked against the Itzykson–Saleur–Zuber relation); this is done with no unitary oracle. (3) A channel-layer observable λ* proposed as a second carrier of the categorical data is shown, robustly, to sit ≈ 0 — at least 20× below the CP-lock target 1/φ² ≈ 0.382 across the raw, filtered, and far-edge extractions — a negative result classified as an observable degeneracy: the carrier dies, while the categorical identity is carried independently by (1) and (2). Honest-history sections (a withdrawn spectral-matching claim; a discarded unitary-oracle route; the data-cut dependence of the λ* decay law) are retained on purpose. Posture: realize, not decide — the parent IR being M(2,5) is measured; whether the cosmic charge sector equals (2,5) is out of scope and belongs to the companion framework/holography line. Grading convention used throughout. We keep four evidence levels distinct and never weld them into one: measured (extracted from lattice data), analytic (closed-form / CAS), structural self-check (algebraic consistency of the categorical data), and one-step corollary via the standard spin–weight relation.
On a constrained lattice realization of the non-unitary Yang–Lee / minimal model, the two-interval entanglement of a region is not fixed until one specifies the algebra of local observables retained inside that region. We show that on the restricted solid-on-solid (RSOS) chain the constraint-preserving local algebra of a region carries a non-trivial center: the height variable on the region’s boundary commutes with every interior generator, and this is forced by locality together with the adjacency constraint rather than chosen. Fixing the boundary height selects the electric-center algebra; absorbing it selects the trivial-center algebra. The two choices are the lattice counterparts of the center choices of Casini, Huerta and Rosabal for gauge theories, and the electric-center block decomposition is the constrained-lattice analog of the anyonic charge-sector decomposition standard in anyonic entanglement. Because the biorthogonal reduced density matrix of a non-unitary model is not positive, the weights of the center sectors form a signed normalized distribution: individual sector weights can be negative while their sum is exactly one. We report this distribution as a reproducible trace-level fact and trace each negative weight to a negative Boltzmann weight of the non-unitary theory. A single model then furnishes distinct two-interval Rényi mutual informations for its two center functionals, and, at fixed center, for two different twist operators. We read these numbers as the values of distinct well-posed quantities rather than as competing estimates of one quantity. A positive-definite entropy for the non-unitary center decomposition is left open.
In this paper we study the simplest massive 1+1 dimensional integrable quantum field theory which can be described as a perturbation of a non-unitary minimal conformal field theory: the Lee-Yang model. We are particularly interested in the features of the bi-partite entanglement entropy for this model and on building blocks thereof, namely twist field form factors. Non-unitarity selects out a new type of twist field as the operator whose two-point function (appropriately normalized) yields the entanglement entropy. We compute this two-point function both from a form factor expansion and by means of perturbed conformal field theory. We find good agreement with CFT predictions put forward in a recent work involving the present authors. In particular, our results are consistent with a scaling of the entanglement entropy given by ceff3log⁡ℓ\frac{c_{\text{eff}}}{3}\log \ell3ceff​​logℓ where ceffc_{\text{eff}}ceff​ is the effective central charge of the theory (a positive number related to the central charge) and ℓ\ellℓ is the size of the region. Furthermore the form factor expansion of twist fields allows us to explore the large region limit of the entanglement entropy and find the next-to-leading order correction to saturation. We find that this correction is very different from its counterpart in unitary models. Whereas in the latter case, it had a form depending only on few parameters of the model (the particle spectrum), it appears to be much more model-dependent for non-unitary models.
The critical curve with central charge c=-22/5 of the Ising quantum chain in an imaginary longitudinal field is calculated by finite-size extrapolation of the Yang-Lee square root branch point position. Various power laws are found. The field dependence of the off-criticality mass scale and the four lowest off-criticality scaling functions are calculated. The latter turn out to be universal along the critical line and agree very well with the results of the truncated Hilbert space method for the c=-22/5 continuum theory. The author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic Bethe ansatz result. The scattering phaseshift calculated by Luscher's method (1988) reproduces the Cardy-Mussardo minimal solution (1989) for the S-matrix. If a boundary defect is introduced, the only non-trivial primary field decouples, so that the spectrum consists only of the tower of the unit operator. In the antiferromagnetic regime with a non-staggered imaginary field the author finds boundary-dependent lines where the gap vanishes.
Abstract Using a corner transfer matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester–Baxter RSOS models (Bianchini et al 2015 J. Stat. Mech. P03010 ) in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point (Bianchini et al 2015 J. Phys. A: Math. Theor. 48 04FT01 ), showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton (2012 J. Stat. Mech. P09014 ) are put forward.