Finiteness of Entanglement Entropy in Collective Field Theory

TL;DR

This paper investigates the finiteness of entanglement entropy in holographic gravitational theories, demonstrating through explicit calculations that it is a non-perturbative effect, with the collective field theory providing a finite result consistent with fermionic models.

Contribution

The paper provides the first explicit collective field theory calculation showing the finiteness of entanglement entropy, confirming non-perturbative effects in holographic duals.

Findings

Entanglement entropy is finite in the non-critical string theory model.

Perturbative series are UV divergent but resummation yields a finite result.

Finiteness is a non-perturbative phenomenon linked to the Newton constant.

Abstract

We explore the question of finiteness of the entanglement entropy in gravitational theories whose emergent space is the target space of a holographic dual. In the well studied duality of two-dimensional non-critical string theory and $c=1$ matrix model, this question has been studied earlier using fermionic many-body theory in the space of eigenvalues. The entanglement entropy of a subregion of the eigenvalue space, which is the target space entanglement in the matrix model, is finite, with the scale being provided by the local Fermi momentum. The Fermi momentum is, however, a position-dependent string coupling, as is clear in the collective field theory formulation. This suggests that the finiteness is a non-perturbative effect. We provide evidence for this expectation by an explicit calculation in the collective field theory of matrix quantum mechanics with vanishing potential. The leading term in the cumulant expansion of the entanglement entropy is calculated using exact eigenstates and eigenvalues of the collective Hamiltonian, yielding a finite result, in precise agreement with the fermion answer. Treating the theory perturbatively, we show that each term in the perturbation expansion is UV divergent. However the series can be resummed, yielding the exact finite result. Our results indicate that the finiteness of the entanglement entropy for higher dimensional string theories is non-perturbative as well, with the scale provided by the Newton constant.