Entanglement, replicas, and Thetas

Sunil Mukhi, Sameer Murthy*, Jie Qiang Wu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)
216 Downloads (Pure)


We compute the single-interval Rényi 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 Θ-functions and Jacobi θ-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 Rényi 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.

Original languageEnglish
Article number5
Number of pages32
JournalJournal of High Energy Physics
Issue number1
Publication statusPublished - 2 Jan 2018


  • Conformal and W Symmetry
  • Conformal Field Theory
  • Field Theories in Lower Dimensions


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