Entanglement, Replicas, and Thetas

Abstract

We compute the single-interval Renyi entropy (replica partition function) for free fermions in 1+1d at finite temperature and finite spatial size by two methods: (i) using the higher-genus partition function on the replica Riemann surface, and (ii) using twist operators on the torus. We compare the two answers for a restricted set of spin structures, leading to a non-trivial proposed equivalence between higher-genus Siegel $\Theta$-functions and Jacobi $\theta$-functions. We exhibit this proposal and provide substantial evidence for it. The resulting expressions can be elegantly written in terms of Jacobi forms. Thereafter we argue that the correct Renyi entropy for modular-invariant free-fermion theories, such as the Ising model and the Dirac CFT, is given by the higher-genus computation summed over all spin structures. The result satisfies the physical checks of modular covariance, the thermal entropy relation, and Bose-Fermi equivalence.

34 pages

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Cited by in corpus (12)

• Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks
• Monodromy methods for torus conformal blocks and entanglement entropy at large central charge
• Rényi entanglement entropies for the compactified massless boson with open boundary conditions
• Rényi entropy and subsystem distances in finite size and thermal states in critical XY chains
• Free Fermion Cyclic/Symmetric Orbifold CFTs and Entanglement Entropy
• A perturbative expansion for entanglement entropy in string theory
• The operator algebra of cyclic orbifolds
• Monstrous entanglement
• Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium
• Thermal Correction to Entanglement Spectrum for Conformal Field Theories
• Total and Symmetry resolved Entanglement spectra in some Fermionic CFTs from the BCFT approach
• Rényi entropy of single-character CFTs on the torus