Entanglement-spectrum fingerprint of a non-invertible symmetry: the Kramers-Wannier duality defect on the lattice

Abstract

A non-invertible topological line does not by itself determine a physical reduced density matrix in a spatially twisted sector. We close this gap for the Kramers-Wannier defect of the critical Ising chain by fixing its orientation, charge sector, zero-mode prescription, and complete-prefix spin algebra. We first take L to infinity at fixed subsystem size, then enlarge the subsystem. In this ordered limit, we prove directly on the lattice that the defect entropy is Delta S_KW = (1/2) log 2 for the canonical mixed prescription and both pure completions, and the matched invertible eta line gives zero. This macroscopic result has a microscopic origin: the spatial Hamiltonian decomposes into an isolated spectator Majorana and an odd active cycle, and the spin and Gaussian RDMs are isospectral on complete prefixes. The mixed covariance has an exact entanglement level xi = 0 at finite L, whereas each pure completion satisfies the fixed-block bound xi_min <= 2 artanh sqrt[(4 ell - 1)/(2L - 1)] = O(L^(-1/2)). We evaluate the entropy through an odd-even critical-Majorana Toeplitz increment. Our exact finite-size results for the energy and modified momentum recover the well-known Ising CFT duality-defect sector, whereas the temporal fusion law D_sigma^2 = T^(-1)(1 + D_psi) provides an independent algebraic proof of non-invertibility.