Entanglement, Replicas, and Thetas

Understand

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.