Persistence of logarithmic corrections for smooth entangling boundaries at the (2+1)D Ising critical point

Determine whether a logarithmic contribution persists in the scaling of the disorder-operator expectation value in the two-replica ensemble at the (2+1)D Ising conformal critical point for a subregion with a smooth entangling boundary; specifically, ascertain whether the fitted term b ln L in the scaling -ln⟨X⟩_{Z_A^{(2)}} = a L + b ln L + c for the disorder operator X = ∏_{i∈A} σ_i^x and subregion A of size (L/2)×L remains in the thermodynamic limit.

Background

The paper introduces a QMC method to compute symmetry-resolved Rényi entropies via charged moments measured as disorder-operator expectation values on replica manifolds. For the 2D transverse-field Ising model at its (2+1)D critical point, the single-replica ensemble shows a pure area law for -ln⟨X⟩, while the two-replica ensemble is well described by an area law plus a logarithmic correction.

The entangling region used is a smooth, corner-free cylinder of size (L/2)×L. In (2+1)D conformal field theories, smooth entangling boundaries are generally not expected to yield logarithmic corrections, raising the question of whether the observed b ln L term is a genuine asymptotic feature or a finite-size effect. The authors note that clarifying this point requires further investigation.