Ensemble averaging in JT gravity from entanglement in Matrix Quantum Mechanics

Abstract

We consider the generalization of a matrix integral with arbitrary spectral curve ${\rho }_0(E)$ to a 0+1D theory of matrix quantum mechanics (MQM). Using recent techniques for 1D quantum systems at large-$N$, we formulate a hydrodynamical effective theory for the eigenvalues. The result is a simple 2D free boson BCFT on a curved background, describing the quantum fluctuations of the eigenvalues around ${\rho }_0(E)$, which is now the large-$N$ limit of the quantum expectation value of the eigenvalue density operator $\hat{\rho }(E)$. The average over the ensemble of random matrices becomes a quantum expectation value. Equal-time density correlations reproduce the results (including non-perturbative corrections) of random matrix theory. This suggests an interpretation of JT gravity as dual to a $\text{one-time-point}$ reduction of MQM. As an application, we compute the Rényi entropy associated to a bipartition of the eigenvalues. We match a previous result by Hartnoll and Mazenc for the $c=1$ matrix model dual to two-dimensional string theory and extend it to arbitrary ${\rho }_0(E)$. The hydrodynamical theory provides a clear picture of the emergence of spacetime in two dimensional string theory. The entropy is naturally finite and displays a large amount of short range entanglement, proportional to the microcanonical entropy. We also compute the reduced density matrix for a subset of $n<N$ eigenvalues.

arXiv comment29 pages + refs. No figures. v2 Added refs. and minor changes to discussion

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