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.
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.
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.
Entanglement entropy in SU(2) lattice gauge theory using a dual loop formulation
One must exercise care in calculating entanglement entropy of gauge theories, because the local Gauss' laws prevent a tensor-product decomposition of the Hilbert space into spatial regions. In this work, we study entanglement entropy in lattice SU(2) gauge theory using the loop formulation together with the Casini-Huerta-Rosabal (henceforth, abbreviated as CHR) algebraic entropy approach. The loop formulations involve a set of canonical transformations that trade the Kogut-Susskind link variables and the local Gauss' laws for the physical loop variables and a single global Gauss' law. Within this framework, we reformulate the CHR entropy directly in the physical Hilbert space. We identify the regional algebras and the center, which, in the loop formulation, is not generated by boundary-localized operators, but instead reduces to a global Casimir of the regional algebra. This structure reveals that the area law of entanglement entropy cannot be determined solely from the gauge constraints. The gauge constraints decompose the reduced density matrix into superselection sectors, and the locality of correlations in the physical states gives rise to the area law. Thus, the analysis separates the algebraic structure dictated by Gauss' constraints from the geometric origin of the area law.
In gauge theories the presence of constraints can obstruct expressing the global Hilbert space as a tensor product of the Hilbert spaces corresponding to degrees of freedom localized in complementary regions. In algebraic terms, this is due to the presence of a center---a set of operators which commute with all others---in the gauge invariant operator algebra corresponding to a finite region. A unique entropy can be assigned to algebras with a center, giving a place to a local entropy in lattice gauge theories. However, ambiguities arise on the correspondence between algebras and regions. In particular, it is always possible to choose (in many different ways) local algebras with a trivial center, and hence a genuine entanglement entropy, for any region. These choices are in correspondence with maximal trees of links on the boundary, which can be interpreted as partial gauge fixings. This interpretation entails a gauge fixing dependence of the entanglement entropy. In the continuum limit, however, ambiguities in the entropy are given by terms local on the boundary of the region, in such a way relative entropy and mutual information are finite, universal, and gauge independent quantities.
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.
A non-invertible topological line fixes global defect data but does not select a reduced density matrix (RDM) in a spatially twisted sector. For the Kramers-Wannier defect of the critical Ising chain, we choose the equal-weight incoherent mixture of the two charge-sector ground states and reduce it to the ordinary full spin algebra of a complete non-wrapping prefix anchored at the defect endpoint. The finite-size RDM is the convex average of two charge-sector RDMs rather than a sign-zero Gaussian proxy, and an exact descended antiunitary enforces many-body Kramers pairing without determining the entropy. At fixed prefix, the RDMs converge in trace norm to a common Gaussian state as the circumference grows. Convergence is uniform over all ground-doublet preparations, including coherent pure states. The subsequent large-prefix entropy limit follows from the odd-even Toeplitz/Fisher-Hartwig endpoint, yielding half the natural logarithm of two above the homogeneous chain. The matched spin-flip defect has zero excess. The exact joint energy-modified-translation character of the same twisted Hamiltonian records the finite-size roots and their multiplicities, retains chirality in the marked scaling limit, and, using the standard Ising character identities, resolves the four Virasoro towers of the Ising duality-twisted sector.
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 3cefflogℓ where ceff is the effective central charge of the theory (a positive number related to the central charge) and ℓ 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.