Abstract
We employ machine learning techniques to provide accurate variational wavefunctions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. Variational quantum Monte Carlo 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 $\mathrm{SU}(N)$ matrix quantum mechanics with three bosonic matrices, as well as 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 wavefunction. 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.
Highlights
A quantitative, first-principles understanding of the emergence of spacetime from nongeometric microscopic degrees of freedom remains among the key challenges in quantum gravity
Holographic duality has provided a firm foundation for attacking this problem; we know that supersymmetric large N matrix theories can lead to emergent geometry [1,2]
We focus on quantum mechanical models with three bosonic large N matrices
Summary
A quantitative, first-principles understanding of the emergence of spacetime from nongeometric microscopic degrees of freedom remains among the key challenges in quantum gravity. The large ν limit is a semiclassical limit in which the classical fuzzy sphere solution accurately describes the quantum state. By using the variational Monte Carlo method with generative flows, we obtain a fully quantum mechanical description of this emergent space. This result, in itself, is excessive given that the physics of the fuzzy sphere is accessible to semiclassical computations. In the bosonic sector of the model, the fuzzy sphere is a metastable state, and it collapses in a first-order large N transition at ν ∼ νc ≈ 4. Beyond the energetics of the fuzzy sphere state, we define a factorization of the microscopic quantum mechanical Hilbert space that leads to a boundary-law entanglement entropy at large ν. In the final section of the paper, we discuss how richer, gravitating states may arise in the opposite small ν limit of the model
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