Finite entanglement entropy and spectral dimension in quantum gravity

Abstract

What are the conditions on a field theoretic model leading to a finite entanglement entropy density? We prove two very general results: 1) Ultraviolet finiteness of a theory does not guarantee finiteness of the entropy density; 2) If the spectral dimension of the spatial boundary across which the entropy is calculated is non-negative at all scales, then the entanglement entropy cannot be finite. These conclusions, which we verify in several examples, negatively affect all quantum-gravity models, since their spectral dimension is always positive. Possible ways out are considered, including abandoning the definition of the entanglement entropy in terms of the boundary return probability or admitting an analytic continuation (not a regularization) of the usual definition. In the second case, one can get a finite entanglement entropy density in multi-fractional theories and causal dynamical triangulations.

15 pages, 2 figures. v2: references added, cosmetic changes, figure improved; v3: extension of the theorem to positive values of the spectral dimension, new discussion on an analytic continuation of the definition of entanglement entropy, new figure added; v4: minor typos corrected, matches published version

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Cited by in corpus (5)

• Finite Entanglement Entropy in Asymptotically Safe Quantum Gravity
• Signal propagation on $\kappa$-Minkowski spacetime and non-local two-point functions
• Spectral dimensions and dimension spectra of quantum spacetimes
• Vacuum energy from noncommutative models
• Accelerated horizons and Planck-scale kinematics