Finiteness of Entanglement Entropy in Collective Field Theory

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.

arXiv comment30 pages, 5 figures

References in corpus (39)

• M Theory As A Matrix Model: A Conjecture
• Strings in flat space and pp waves from $\mathcal{N}=4$ Super Yang Mills
• Entropy and Area
• Geometric and Renormalized Entropy in Conformal Field Theory
• Entanglement entropy and conformal field theory
• On Geometric Entropy
• Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory
• Bipartite Fluctuations as a Probe of Many-Body Entanglement
• Entanglement from Charge Statistics: Exact Relations for Many-Body Systems
• Dynamical structure factor at small q for the XXZ spin-1/2 chain
• Long strings in two dimensional string theory and non-singlets in the matrix model
• Strings on a Cone and Black Hole Entropy
• Random matrices and entanglement entropy of trapped Fermi gases
• Developments in 2D String Theory
• Nonlinear Bosonization of Fermi Surfaces: The Method of Coadjoint Orbits
• Non-relativistic Fermions, Coadjoint Orbits of \winf\ and String Field Theory at $c=1$
• Bose-Fermi duality and entanglement entropies
• Classical Fermi Fluid and Geometric Action for $c=1$
• Matrix Model Maps in AdS/CFT
• Entanglement entropy in two dimensional string theory
• Collective String Field Theory of Matrix Models in the BMN Limit
• Open Strings On The Rindler Horizon
• Entanglement and Complexity of Purification in (1+1)-dimensional free Conformal Field Theories
• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Full counting statistics for interacting trapped fermions
• Derivation of String Field Theory from the Large N BMN Limit
• Geometric Entropy of Nonrelativistic Fermions and Two Dimensional Strings
• Gauge Invariant Target Space Entanglement in D-Brane Holography
• Bulk Entanglement Entropy and Matrices
• Long String Scattering in c = 1 String Theory
• Applications of the Collective Field Theory for the Calogero-Sutherland Model
• Notes on Collective Field Theory of Matrix and Spin Calogero Models
• Target space entanglement in Matrix Models
• Entanglement and Confinement in Coupled Quantum Systems
• Degrees of freedom in two dimensional string theory
• Entanglement in the Quantum Hall Matrix Model
• Target space entanglement in quantum mechanics of fermions and matrices
• Free fermions and $\alpha$-determinantal processes
• Fluctuations around Periodic BPS-Density Waves in the Calogero Model

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