Modular invariance and entanglement entropy

Abstract

We study the Rényi and entanglement entropies for free 2d CFT's at finite temperature and finite size, with emphasis on their properties under modular transformations of the torus. We address the issue of summing over fermion spin structures in the replica trick, and show that the relation between entanglement and thermal entropy determines two different ways to perform this sum in the limits of small and large interval. Both answers are modular covariant, rather than invariant. Our results are compared with those for a free boson at unit radius in the two limits and complete agreement is found, supporting the view that entanglement respects Bose-Fermi duality. We extend our computations to multiple free Dirac fermions having correlated spin structures, dual to free bosons on the Spin(2d) weight lattice.

Entanglement, replicas, and Thetas

“…A strong form of the thermal entropy relation, as explained in [6,8], says that the n th replica partition function Z n (z 12 , τ ) should reduce in the limit z 12 ∼ 0 to Z 1 (τ ) n (up to the prefactor that encodes the singularity induced by the collision of the two twist operators), while in the limit z 12 ∼ 1 it should go over to Z 1 (nτ ). What was shown in [12] is that the former prescription only agrees with this prediction at small intervals z 12 ∼ 0, while 1 Physically the length of the entangling interval is real, and the modular transformation S transforms z12 from real to imaginary values. Thus the modular transformation of entanglement entropy is a "temporal" version of it.…”

“…The twist-operator computation exhibits an ambiguity in whether the spin structure should be summed over before or after taking the product of θ-functions across replicas. As shown in [12], neither way of performing the sum satisfies the thermal entropy relation. A strong form of the thermal entropy relation, as explained in [6,8], says that the n th replica partition function Z n (z 12 , τ ) should reduce in the limit z 12 ∼ 0 to Z 1 (τ ) n (up to the prefactor that encodes the singularity induced by the collision of the two twist operators), while in the limit z 12 ∼ 1 it should go over to Z 1 (nτ ).…”

“…The correct answer appears in [19] (building on previous work in [20], in turn based on the orbifold twist-operator computations of [21]). In [19] the answer was shown to satisfy the thermal entropy relation, while in [12] it was shown to also be modular covariant. This result is essentially the higher-genus partition function of the free boson theory on a Riemann surface of genus n, where n is the number of replicas.…”

“…Thus it is natural to ask if the result is modular covariant. 1 In this context the following puzzle was raised in [12]. Firstly it was observed that, although the replica partition function for any fixed spin structure is not modular covariant with respect to the full modular group of the torus, one can obtain a modular covariant answer by summing over all four torus spin-structures for the fermions.…”

“…This means that the twist-operator computation, at least in the form currently known, is inadequate to compute the (finite size, finite temperature) Rényi entropy of the free Majorana (Ising)/free Dirac CFT. Subsequent to [12], issues regarding summing over fermion spin structures when computing Rényi entropies have been discussed in different contexts in [14][15][16][17][18].…”