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Deep Quantum Geometry of Matrices

Xizhi Han (韩希之) and Sean A. Hartnoll
Phys. Rev. X 10, 011069 – Published 23 March 2020
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Abstract

We employ machine learning techniques to provide accurate variational wave functions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. The variational quantum Monte Carlo method is implemented with deep generative flows to search for gauge-invariant low-energy states. The ground state (and also long-lived metastable states) of an SU(N) matrix quantum mechanics with three bosonic matrices, and also its supersymmetric “mini-BMN” extension, are studied as a function of coupling and N. Known semiclassical fuzzy sphere states are recovered, and the collapse of these geometries in more strongly quantum regimes is probed using the variational wave function. We then describe a factorization of the quantum mechanical Hilbert space that corresponds to a spatial partition of the emergent geometry. Under this partition, the fuzzy sphere states show a boundary-law entanglement entropy in the large N limit.

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  • Received 11 August 2019
  • Revised 5 January 2020
  • Accepted 12 February 2020

DOI:https://doi.org/10.1103/PhysRevX.10.011069

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Particles & Fields

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Machine Learning Tackles Spacetime

Published 23 March 2020

Neural networks enable an important calculation in a popular approach to unifying quantum mechanics with general relativity.

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Authors & Affiliations

Xizhi Han (韩希之) and Sean A. Hartnoll

  • Department of Physics, Stanford University, Stanford, California 94305-4060, USA

Popular Summary

A cornerstone of modern fundamental physics is the notion that space itself is emergent from more elementary nongeometric entities. Holographic duality gives the best-understood framework for this emergence of space. In holographic dualities, certain quantum-mechanical theories are known to reorganize themselves into theories of classical gravity, with gravitational interactions arising in a spacetime that was not present originally. However, very little is understood about how this emergence actually occurs. This is because the quantum mechanics in question has proved too difficult to solve. In this paper, we harness the power of machine learning to solve the kinds of quantum-mechanical theories that arise in holographic dualities.

Starting in a well-understood semiclassical limit of these theories, we are able to vary a parameter that causes the theory to behave in a strongly quantum-mechanical way. We find that the wave function of the quantum system, described semiclassically by a geometry known as a fuzzy sphere, undergoes a qualitative change in the quantum regime that can be associated with gravitational collapse.

The methods we introduce in this paper will allow for the investigation of a wealth of holographic physics that has previously been inaccessible.

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Vol. 10, Iss. 1 — January - March 2020

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