Deep Quantum Geometry of Matrices

Han, Xizhi; Hartnoll, Sean A.

doi:10.1103/PhysRevX.10.011069

Title: Deep Quantum Geometry of Matrices

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.

Authors:: Han, Xizhi; Hartnoll, Sean A.

Publication Date:: 2020-03-23

Research Org.:: Stanford Univ., CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1605980

Alternate Identifier(s):: OSTI ID: 1803256

Grant/Contract Number:: de-sc0018134; SC0018134

Resource Type:: Published Article

Journal Name:: Physical Review. X

Additional Journal Information:: Journal Name: Physical Review. X Journal Volume: 10 Journal Issue: 1; Journal ID: ISSN 2160-3308

Publisher:: American Physical Society

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics

Citation Formats


                    Han, Xizhi, and Hartnoll, Sean A. Deep Quantum Geometry of Matrices.  United States: N. p., 2020. 
Web.  doi:10.1103/PhysRevX.10.011069.

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                    Han, Xizhi, & Hartnoll, Sean A. Deep Quantum Geometry of Matrices.  United States.  https://doi.org/10.1103/PhysRevX.10.011069

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                    Han, Xizhi, and Hartnoll, Sean A. Mon .  
"Deep Quantum Geometry of Matrices".  United States.  https://doi.org/10.1103/PhysRevX.10.011069.

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@article{osti_1605980,

  title        = {Deep Quantum Geometry of Matrices},

  author       = {Han, Xizhi and Hartnoll, Sean A.},

  abstractNote = {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.},

  doi          = {10.1103/PhysRevX.10.011069},

  journal      = {Physical Review. X},

  number       = 1,

  volume       = 10,

  place        = {United States},

  year         = {2020},

  month        = {3}

}

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Impact of supersymmetry on the nonperturbative dynamics of fuzzy spheres journal, September 2005

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A gauge invariant UV–IR mixing and the corresponding phase transition for fields on the fuzzy sphere journal, January 2005

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On the architecture of spacetime geometry journal, October 2014

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Degrees of freedom in two dimensional string theory journal, February 1996

Entanglement entropy on fuzzy spaces journal, August 2006

Geometry of the gauge algebra in noncommutative Yang-Mills theory journal, August 2001

Black hole thermodynamics from simulations of lattice Yang-Mills theory journal, August 2008

Noncommutative gauge theory on fuzzy sphere from matrix model journal, June 2001

Brane dynamics in background fluxes and non-commutative geometry journal, May 2000

A practical introduction to tensor networks: Matrix product states and projected entangled pair states journal, October 2014

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Geometry in Transition: A Model of Emergent Geometry journal, May 2008

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The Ryu–Takayanagi Formula from Quantum Error Correction journal, May 2017

The fuzzy sphere journal, January 1992

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Neural-Network Quantum States, String-Bond States, and Chiral Topological States journal, January 2018

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journal, April 2002

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journal, November 1989

Degrees of freedom in two dimensional string theory
journal, February 1996

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A practical introduction to tensor networks: Matrix product states and projected entangled pair states
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journal, January 2008

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