(2020) Featured in Physics

PHYSICAL REVIEW X 10, 011069 (2020) Featured in Physics

Xizhi Han (韩希之) and Sean A. Hartnoll Department of Physics, Stanford University, Stanford, California 94305-4060, USA

(Received 11 August 2019; revised manuscript received 5 January 2020; accepted 12 February 2020; published 23 March 2020)

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 SUðNÞ 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 Subject Areas: Computational Physics, String Theory

I. INTRODUCTION [8,9]. In contrast, we are faced with models where there is an emergent locality that is not manifest in the micro- A quantitative, first-principles understanding of the scopic interactions. This locality cannot be used apriori; emergence of spacetime from nongeometric microscopic it must be uncovered. Facing a similar challenge of degrees of freedom remains among the key challenges in extracting the most relevant variables in high-dimensional quantum gravity. Holographic duality has provided a data, deep learning has demonstrated remarkable success firm foundation for attacking this problem; we now [10–12],intasksrangingfromimageclassification[13] to know that supersymmetric large N matrix theories can game playing [14].Thesesuccesses,andothers,have lead to emergent geometry [1,2]. What remains is the motivated tackling many-body physics problems with the technical challenge of solving these strongly quantum machine learning toolbox [15]. For example, there has mechanical systems and extracting the emergent space- been much interest and progress in applications of time dynamics from their quantum states. Recent years restricted Boltzmann machines to characterize states of have seen significant progress in numerical studies of spin systems [16–19]. large N matrix quantum mechanics at nonzero temper- In this work, we solve for low-energy states of quantum ature. Using Monte Carlo simulations, quantitatively mechanical Hamiltonians with both bosons and fermions, correct features of emergent black hole geometries have using generative flows (normalizing flows [20–22] and been obtained; see, e.g., Refs. [3–5]. To grapple with masked autoregressive flows [23–25], in particular) and the questions such as the emergence of local spacetime variational quantum Monte Carlo method. Compared with physics, and its associated short-distance entanglement spin systems, the problem we are trying to solve contains [6,7], new and inherently quantum mechanical tools are continuous degrees of freedom and gauge symmetry, and needed. there is no explicit spatial locality. Recent works have Variational wave functions can capture essential aspects applied generative models to physics problems [26–28] and of low-energy physics. However, the design of accurate have aimed to understand holographic geometry, broadly many-body wave-function ansatze has typically required conceived, with machine learning [29–31]. We use gen- significant physical insight. For example, the power of erative flows to characterize emergent geometry in large N tensor network states, such as matrix product states, hinges multimatrix quantum mechanics. As we have noted above, upon an understanding of entanglement in local systems such models form the microscopic basis of established holographic dualities. We focus on quantum mechanical models with three Published by the American Physical Society under the terms of bosonic large N matrices. These are among the simplest the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to models with the core structure that is common to holo- the author(s) and the published article’s title, journal citation, graphic theories. The bosonic part of the Hamiltonian takes and DOI. the form

2160-3308=20=10(1)=011069(15) 011069-1 Published by the American Physical Society XIZHI HAN and SEAN A. HARTNOLL PHYS. REV. X 10, 011069 (2020) 1 1 1 i i i j i j 2 i i number R. The first will be purely bosonic states, with H ¼ tr Π Π − ½X ;X ½X ;X þ ν X X 2 B 2 4 2 R ¼ 0. The second will have an R ¼ N − N. In this latter sector, the fuzzy sphere state is supersymmetric at large þ iνϵijkXiXjXk : ð1:1Þ positive ν, so we refer to it as the “supersymmetric sector.” In the bosonic sector of the model, the fuzzy sphere is a metastable state, and it collapses in a first-order large N Here, the Xi are N-by-N traceless Hermitian matrices, transition at ν ∼ ν ≈ 4. See Figs. 2 and 3 below. In the with i 1,2,3.TheΠi are conjugate momenta, and ν is c ¼ supersymmetric sector of the model, where the fuzzy a mass deformation parameter. The potential energy sphere is stable, the collapse is found to be more gradual. 1 νϵijk k in Eq. (1.1) is a total square: VðXÞ¼4 tr½ð X þ See Figs. 6 and 7. In Fig. 8, we start to explore the small ν i j 2 i½X ;X Þ . The supersymmetric extension of this model limit of the supersymmetric sector. [32], discussed below, can be thought of as a simplified Beyond the energetics of the fuzzy sphere state, we version of the BMN matrix quantum mechanics [33]. define a factorization of the microscopic quantum We refer to the supersymmetric model as mini-BMN, mechanical Hilbert space that leads to a boundary-law following Ref. [34]. For the low-energy physics we are entanglement entropy at large ν.SeeEq.(5.14) below. exploring, the large N planar diagram expansion in this This factorization both captures the emergent local model is controlled by the dimensionless coupling 3 dynamics of fields on the fuzzy sphere and also reveals λ ≡ N=ν . Here, λ can be understood as the usual a microscopic cutoff to this dynamics at a scale set by N. dimensionful ’t Hooft coupling of a large N quantum The nature of the emergent fields and their cutoffs can be mechanics at an energy scale set by the mass term usefully discussed in string theory realizations of the (cf. Ref. [35]). model. In string-theoretic constructions, fuzzy spheres The mass deformation in the Hamiltonian (1.1) inhibits arise from the polarization of D branes in background the spatial spread of wave functions—which will be helpful fields [41–44]. A matrix quantum mechanics theory such for numerics—and leads to minima of the potential at as Eq. (1.1) describes N “D0 branes”—see Ref. [32] and the discussion section below for a more precise charac- ½Xi;Xj¼iνϵijkXk: ð1:2Þ terization of the string theory embedding of mini-BMN theory—and the maximal fuzzy sphere corresponds to a In particular, one can have Xi ¼ νJi, with the Ji being, for configuration in which the D0 branes polarize into a single example, the N-dimensional irreducible representation of spherical D2 brane. There is no gravity associated with the suð2Þ algebra. This set of matrices defines a “fuzzy this emergent space; the emergent fields describe the low- sphere” [36]. There are two important features of this energy worldvolume dynamics of the D2 brane. In this solution. First, in the large N limit, the noncommutative case, the emergent fields are a Maxwell field and a single algebra generated by the Xi approaches the commutative scalar field corresponding to transverse fluctuations of the algebra of functions on a smooth two-dimensional sphere brane. In the final section of the paper, we discuss how [37,38]. Second, the large ν limit is a semiclassical limit in richer, gravitating states may arise in the opposite small ν which the classical fuzzy sphere solution accurately limit of the model. describes the quantum state. In this semiclassical limit, the low-energy excitations above the fuzzy sphere state are II. THE MINI-BMN MODEL obtained from classical harmonic perturbations of the matrices about the fuzzy sphere [39]. See also Ref. [40] The mini-BMN Hamiltonian is [32] for an analogous study of the large-mass BMN theory. At 3 3 large N and ν, these excitations describe fields propagating † k k † 2 H ¼ HB þ tr λ σ ½X ;λþ νλ λ − νðN − 1Þ: ð2:1Þ on an emergent spatial geometry. 2 2 By using the variational Monte Carlo method with σk generative flows, we obtain a fully quantum mechanical The bosonic part HB is given in Eq. (1.1). The are Pauli λ description of this emergent space. This result, in itself, is matrices. The are matrices of two-component SO(3) excessive given that the physics of the fuzzy sphere is spinors. It can be useful to write the matrices in terms of su 1 2 … 2 − 1 accessible to semiclassical computations. Our variational the ðNÞ generators TA, with A ¼ ; ; ;N , which wave functions will quantitatively reproduce the semi- obey ½TA;TB¼ifABCTC and are Hermitian and ortho- classical results in the large ν limit, thereby providing a normal (with respect to the Killing form). In other words, i i A λα λα A solid starting point for extending the variational method X ¼ XAT and ¼ AT . The ijk and ABC indices are across the entire N and ν phase diagram. Exploring the freely raised and lowered. Lower αβ indices are for spinors parameter space, we find that the fuzzy sphere collapses transforming in the 2 representation of SO(3), while upper upon moving into the small ν quantum regime. We consider indices are for 2¯. We will not raise or lower spinor indices. two different “sectors” of the model, with different fermion The full Hamiltonian can then be written as

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2 1 ∂ 1 2 1 seen to be completely general (but not unique) if we have H ¼ − þ ðf Xi Xj Þ þ ν2ðXi Þ2 2 ∂ i 2 4 ABC B C 2 A the number of free fermion states D sufficiently large. ð XAÞ 1 For purely bosonic models, jMðXÞi is simply the phase − ν ϵijk i j k λα† k σkβλ of the wave function. 2 fABC XAXBXC þ ifABC A XB α Cβ

3 α† 3 2 νλ λ α − ν − 1 B. Gauge invariance and gauge fixing þ 2 A A 2 ðN Þ; ð2:2Þ The generators (2.5) correspond to the following action λα† ≡ λ † λα† λ δ δα ∈ where A ð AαÞ and f A ; Bβg¼ AB β are complex of an element U G ¼ SUðNÞ on the wave function: fermion creation and annihilation operators. This −1 −1 −1 Hamiltonian is seen to have four supercharges, ðUψÞðXÞ¼fðU XUÞjðUMU ÞðU XUÞi: ð2:7Þ ∂ In other words, the group acts by matrix conjugation. The i i j k iβ Qα ¼ −i þ iνX − f ϵ X X σα λ β; ∂Xi A 2 ABC ijk B C A wave function is required to be invariant under the group A action; i.e., Uψ ψ for any U ∈ G. ¯ α † ¼ Q ¼ðQαÞ ; ð2:3Þ Gauge invariance allows us to evaluate the wave function using a representative for each orbit of the gauge group. Let which obey X˜ be the representative in the gauge orbit of X. Gauge ¯ α invariance of the wave function implies that there must fQα; Q g¼4H: ð2:4Þ exist functions f˜ and M˜ such that States that are invariant under all supercharges therefore f X f˜ X˜ ; have vanishing energy. ð Þ¼ ð Þ Matrix quantum mechanics theories arising from micro- jMðXÞi ¼ jUM˜ ðX˜ ÞU−1i; where X ¼ UXU˜ −1: ð2:8Þ scopic string theory constructions are typically gauged. This means that physical states must be invariant under the The functions f˜ and M˜ take gauge representatives as inputs SUðNÞ symmetry. In particular, physical states are annihi- or may be thought of as gauge-invariant functions. The lated by the generators wave function we use will be in the form (2.8). The functions f˜ and M˜ will be parametrized by neural networks, ∂ i α† G −if X λ λ α : 2:5 as we describe in Sec. III. A ¼ ABC B ∂ i þ B C ð Þ XC We proceed to describe the gauge fixing we use to select the representative for each orbit, as well as the measure factor associated with this choice. The SUðNÞ gauge A. Representation of the fermion wave function representative X˜ will be such that ˜ −1 The mini-BMN wave function can be represented as a (1) Xi ¼ UXiU for i ¼ 1, 2, 3 and some unitary function from bosonic matrix coordinates to fermionic states matrix U. ˜ 1 ˜ 1 ≤ ˜ 1 ≤ … ≤ ˜ 1 ψðXÞ¼fðXÞjMðXÞi. Here, X denotes the three bosonic (2) X is diagonal and X11 X22 XNN. ≥ 0 ˜ 2 traceless Hermitian matrices. The function fðXÞ is (3) Xiðiþ1Þ is purely imaginary, with the imaginary part the norm of the wave function at X, while jMðXÞi is a positive for i ¼ 1; 2; …;N− 1. normalized state of matrix fermions. A fermionic state with The third condition is needed to fix the Uð1ÞN−1 residual definite fermion number R is parametrized by a complex gauge freedom after diagonalizing X1. The representative X˜ ra tensor MAα such that is well defined except on a subspace of measure zero where the matrices are degenerate. Then, X˜ can be represented as a 2 XD YR X2 NX−1 2ðN2−1Þ α† vector in R with a positivity constraint on some jMi ≡ Mra λ j0i; ð2:6Þ Aα A components. The change of variables from X to X˜ leads to a r¼1 a¼1 α¼1 A¼1 measure factor given by the volume of the gauge orbit: where j0i is the state with all fermionic modes unoccupied. 2 2 d3ðN −1ÞX ¼ ΔðX˜ Þd2ðN −1ÞX;˜ ð2:9Þ The definitionP(2.6) is parsed as follows: For any fixed r ηra† ra λα† and a, ¼ αA MAα A is the creation operator for the with matrix fermionic modes, where A runs over some ortho- su α 1 YN NY−1 normal basis of the ðNÞ QLie algebra and ¼ , 2 for two ˜ ˜ 1 ˜ 1 ˜ 2 † Δ ∝ − ηra 0 ðXÞ jXii Xjjj jXiðiþ1Þj: ð2:10Þ fermionic matrices. Then, a j i is a state of multiple i≠j¼1 i¼1 free fermions created by η†. The final summation over r in Eq. (2.6) is a decomposition of a general fermionic state Keeping track of this measure (apart from an overall into a sum of free fermion states. Such a representation is prefactor) will be important for proper sampling in the

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Monte Carlo algorithm. The derivation of (2.10) is shown concerning the training are given in the SM, Sec. D [45]. in Sec. A of the Supplemental Material (SM) [45]. Benchmarks are presented at the end of this section.

III. ARCHITECTURE DESIGN FOR MATRIX A. Parametrizing and sampling the QUANTUM MECHANICS gauge-invariant wave function We first describe how gauge invariance is incorporated In this work, we propose a variational Monte Carlo into the variational Monte Carlo algorithm. As already method with importance sampling to approximate the discussed, an important step is to sample according to ground state of matrix quantum mechanics theories, 2 X ∼ jψðXÞj . From Eq. (2.8), for a gauge-invariant wave leading to an upper bound on the ground-state energy. ψ X 2 f˜ X˜ 2 X˜ The importance sampling is implemented with generative function, j ð Þj ¼j ð Þj . However, in sampling ,we Δ ˜ flows. The basic workflow is sketched as follows: must keep track of the measure factor ðXÞ in Eq. (2.10). We do this as follows: (1) Start with a wave function ψ θ with variational ˜ ˜ Δ ˜ ˜ ˜ 2 parameters θ. In our case, θ will characterize neural (1) Sample X according to pðXÞ¼ ðXÞjfðXÞj . (2) Generate Haar random elements U ∈ SUðNÞ. networks. ˜ −1 (2) Write the expectation value of the Hamiltonian to be (3) Output samples X ¼ UXU . minimized as The correctness of this procedure is shown in the SM, Z Sec. A [45]. 2 Conversely, at the evaluation stage, ψ X can be com- Eθ ¼hψ θjHjψ θi¼ dXjψ θðXÞj HX½ψ θ ð Þ puted in the following steps for gauge-invariant wave E 2 ψ functions (2.8): ¼ X∼jψ θj ½HX½ θ: ð3:1Þ (1) Gauge fix X¼UXU˜ −1 as discussed in the last section. In the mini-BMN case, X denotes three traceless (2) Compute M˜ ðX˜ Þ and f˜ðX˜ Þ. Details of the structure of ψ ˜ ˜ Hermitian matrices (indices omitted), and HX½ θ is M and f will be discussed below. E ˜ ˜ ˜ ˜ −1 the energy density at X. Notationally, X∼pðXÞ is the (3) Return ψðXÞ¼fðXÞjUMðXÞU i according to expectation value, with the random variable X drawn Eq. (2.8). from the probability distribution pðXÞ. We now describe the implementation of M˜ and f˜ as (3) Generate random samples according to the wave- neural networks. The basic building block, a multilayer 2 function probabilities X ∼ pθðXÞ¼jψ θðXÞj , and fully connected (also called dense) neural network, is an evaluate their energy densities HX½ψ θ. The varia- elemental architecture capable of parametrizing compli- tional energy (3.1) can then be estimated as the cated functions efficiently [12]. The neural network defines average of energy densities of the samples. a function F∶x ↦ y mapping an input vector x to an (4) Update the parameters θ (via stochastic gradient output vector y via a sequence of affine and nonlinear descent) to minimize Eθ: transformations:

θ 1 θ − α∇θ Eθ ; 3:2 m m−1 1 tþ ¼ t t t ð Þ F ¼ Aθ ∘ tanh ∘ Aθ ∘ tanh ∘ … ∘ tanh ∘ Aθ: ð3:4Þ

where t ¼ 1; 2; … denotes the steps of training and 1 1 1 Here, A ðxÞ¼M x þ b is an affine transformation, where the parameter α > 0 sets the learning rate. The θ θ θ the weights M1 and the biases b1 are trainable parameters. gradient of energy is estimated from Monte Carlo θ θ The hyperbolic tangent nonlinearity then acts elementwise samples: 1 on AθðxÞ. We experimented with different activation ∇ E ∇ ψ θEθ ¼ X∼pθ ½ θHX½ θ functions; the final result is not sensitive to this choice. i i Similar mappings are applied m times, allowing Mθ and bθ þ E ∼ ½∇θ( ln pθðXÞ)ðH ½ψ θ − EθÞ: X pθ X to be different for different layers i, to produce the output ð3:3Þ vector y. The mapping F∶x ↦ y is nonlinear and capable of approximating any square integrable function if the The method is applicable even if the probabilities are number of layers and the dimensions of the affine trans- available only up to an unknown normalization formations are sufficiently large [46]. factor. The function M˜ ðX˜ Þ is implemented as such a multilayer (5) Repeat steps 3 and 4 until Eθ converges. Observ- fully connected neural network, mapping from vectorized ables of physical interest are evaluated with respect 2 2 X˜ to M˜ in Eq. (2.6), i.e., R2ðN −1Þ → RDR2ðN −1Þ. The to the optimal parameters after training. ˜ ˜ In the following, we discuss details of parametrizing and implementation of fðXÞ is more interesting, as both ˜ ˜ sampling from gauge-invariant wave functions with fer- evaluating fðXÞ and sampling from the distribution ˜ ˜ ˜ ˜ 2 mions. Technicalities concerning the evaluation of HX½ψ θ pðXÞ¼ΔðXÞjfðXÞj are necessary for the Monte Carlo are spelled out in the SM, Sec. B [45]. More details algorithm. Generative flows are powerful tools to efficiently

011069-4 DEEP QUANTUM GEOMETRY OF MATRICES PHYS. REV. X 10, 011069 (2020) parametrize and sample from complicatedpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi probability distributions. The function f˜ðX˜ Þ¼ pðX˜ Þ=ΔðX˜ Þ,sowe can focus on sampling and evaluating pðX˜ Þ, which will be implemented by generative flows. Two generative flow architectures are implemented for comparison: a normalizing flow and a masked autoregressive flow. The normalizing flow starts with a product of simple univariate probability distributions pðxÞ¼p1ðx1Þ… pMðxMÞ, where the pi can be different. Values of x sampled from this distribution are passed through an invertible multilayer dense network as in Eq. (3.4). The probability distribution of the output y is then

−1 Dy −1 −1 qðyÞ¼pðxÞ det ¼ p(F ðyÞ)j det DFj : ð3:5Þ Dx The masked autoregressive flow generates samples progressively. It requires an ordering of the components of the input, say, x1;x2; …;xM. Each component is drawn from a parametrized distribution pi(xi; Fiðx1; …;xi−1Þ), where the parameter depends only on previous components. Thus, x1 is sampled independently, and for other components, the dependence Fi is given by Eq. (3.4). The overall probability is the product FIG. 1. Benchmarking the architecture: Variational ground- YM state energies for the mini-BMN model with N ¼ 2 and fermion ( ; … ) qðxÞ¼ pi xi Fiðx1; ;xi−1Þ : ð3:6Þ numbers R ¼ 0 and R ¼ 2 (shown as dots) compared to the exact i¼1 ground-state energy in the j ¼ 0 sector, obtained in Ref. [34] (shown as the dashed curve). Uncertainties are at or below the When piðxiÞ are chosen as normal distributions, both flows are able to represent any multivariate normal dis- scale of the markers; in particular, the variational energies slightly tribution exactly. Features of the wave function (such as below the dashed line are within the numerical error of the line. NF stands for normalizing flows and MAF for masked autore- polynomial or exponential tails) can be probed by exper- gressive flows. As described in the main text, the numbers in the imenting with different base distributions piðxiÞ. Choices brackets are, first, the number of layers in the neural networks of the base distributions and performances of the two flows and, second, the number of generalized normal distributions in are assessed in the following benchmark subsection and each base distribution. also in the SM, Sec. D [45]. We use both types of flow in the numerical results of Sec. IV. Hamiltonian (2.2) via ν → −ν, λ → λ†, λ† → λ, and X → −X). The masked autoregressive flow yields better B. Benchmarking the architecture (lower) variational energies. These energies are seen to be In Ref. [34], the Schrödinger equation for the N ¼ 2 close to the j ¼ 0 results obtained in Ref. [34]. The mini-BMN model was solved numerically. Comparison variational results seem to be asymptotically accurate as with the results in that paper allows us to benchmark our jνj → ∞, while remaining a reasonably good approxima- architecture before moving to larger values of N.In tion at small ν. Small ν is an intrinsically more difficult Ref. [34], the Schrödinger equation is solved in sectors regime, as the potential develops flat directions (visualized with a fixed fermion number in Ref. [34]), and hence the wave function is more X complicated, possibly with long tails. In the supersym- λα†λ 0 R ¼ A Aα; ½R; H¼ ; ð3:7Þ metric R ¼ 2 sector, where quantum mechanical effects at Aα small ν are expected to be strongest, further significant and total SO(3) angular momentum j ¼ 0; 1=2. We do not improvement at the smallest values of ν is seen with deeper constrain j, but we do fix the number of fermions in the autoregressive networks and more flexible base distribu- variational wave function. tions, as we describe shortly. Analogous improvements in The variational energies obtained from our machine these regimes will also be seen at larger N in Sec. IV C and learning architecture with R ¼ 0 and R ¼ 2 are shown the SM, Sec. D [45]. as a function of ν in Fig. 1. We take negative ν to compare In Fig. 1, the base distributions piðxiÞ, introduced in the with the results given in Ref. [34], which uses an opposite previous subsection, are chosen to be a mixture of s sign convention. (There is a particle-hole symmetry of the generalized normal distributions:

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FIG. 2. Expectation value of the radius in the zero fermion FIG. 3. Variational energies in the zero fermion sector of the sector of the mini-BMN model, for different N and ν. The dashed mini-BMN model, for different N and ν. The dashed lines are − 3 ν 2 − 1 Δ Δ lines are the semiclassical values (4.4). Solid dots are initialized semiclassical values: E ¼ 2 ðN Þþ Ejbos, with Ejbos near the fuzzy sphere configuration, and the open markers are given in Eq. (4.8). As in Fig. 2, solid dots are initialized near the initialized near zero. We have used normalizing and autoregres- fuzzy sphere configuration, and the open markers are initialized sive flows, respectively, as these produce more accurate varia- near zero. tional wave functions in the two different regimes. e.g., Figs. 2 and 3. The wave function in this semiclassical Xs Xs βi i r − −μi αi β regime is almost Gaussian, and indeed, the NF(1, 1) and p x ki e ðjxi rj= rÞ r ; ki 1: 3:8 ið iÞ¼ r 2αi Γ 1 βi r ¼ ð Þ MAF(1, 1) flows give similar energies when initialized near 1 r ð = rÞ 1 r¼ r¼ fuzzy sphere configurations. The NF architecture, in fact, i gives slightly lower energies in this regime, so we have used Here, the kr are positive weights for each generalized normal distribution in the mixture. In Eq. (3.8), the ki , αi , normalizing flows in Figs. 2 and 3 for the fuzzy sphere. r r The numerics above and below are performed with βi , and μi are learnable (i.e., variational) parameters. For r r D ¼ 4 in Eq. (2.6), so the fermionic wave function autoregressive flows, these parameters further depend on jMðXÞi is a sum of four free fermion states for each value x , with 1 ≤ j

s αi s The architecture described above gives a variational X βi r X ð rÞ αi −1 −βi p x >0 ki x r e rxi ; ki 1; 3:9 wave function for low-energy states of the mini-BMN ið i Þ¼ r Γ αi ð iÞ r ¼ ð Þ r¼1 ð rÞ r¼1 model. With the wave function in hand, we can evaluate observables. We start with the purely bosonic sector of the i i i where again the kr, αr,andβr depend on xj,with1 ≤ j

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The minima of the classical potential occur at

½Xi;Xj¼iνϵijkXk: ð4:2Þ

These are supersymmetric solutions of the classical theory, annihilated by the supercharges (2.3) in the classical limit; therefore, they have vanishing energy. The solution of Eq. (4.2) is

Xi ¼ νJi; ð4:3Þ

FIG. 4. Probability distribution, from the variational wave where the Ji are representations of the suð2Þ algebra, function, for the radius in the fuzzy sphere phase for N ¼ 8 ½Ji;Jj¼iϵijkJk. Here, we are interested in maximal, and different ν. The horizontal axis is rescaled by the semi- N-dimensional irreducible representations. (Reducible rep- classical value of the radius r0, given in Eq. (4.4) below. The resentations can also be studied, corresponding to multiple width of the distribution in units of the classical radius becomes polarized D branes.) smaller as ν is increased. The suð2Þ Casimir operator suggests a notion of “radius” given by ν ν ν , as the collapsed state has lower energy. For < c, the X3 2 2 1 ν ðN − 1Þ fuzzy sphere is no longer even metastable. We gain a r2 ¼ trðXiÞ2 ¼ : ð4:4Þ N 4 semiclassical understanding of this transition in Sec. IV B. i¼1 Figures 2 and 3 show that the radius and energy of the fuzzy sphere state are accurately described by semi- Indeed, the algebra generated by the Xi matrices tends classical formulas (derived in the following section) towards the algebra of functions on a sphere as N → ∞ ν ν 3 for all > c. In particular, E=N and r=N are rapidly [37,38]. At finite N, a basis for this space of matrices is ˆ converging towards their large N values. Figure 4 further provided by the matrix spherical harmonics Yjm. These shows that the probability distribution for the radius r harmonics obey becomes strongly peaked about its semiclassical expect- 3 ation value at large ν. X i i ˆ 1 ˆ 3 ˆ ˆ Analogous behavior to that shown in Figs. 2 and 3 has ½J ;½J ;Yjm¼jðjþ ÞYjm; ½J ;Yjm¼mYjm: ð4:5Þ 1 previously been seen in classical Monte Carlo simula- i¼ ˆ tions of a thermal analogue of our quantum transition We construct the Yjm explicitly in the SM, Sec. C [45]. The – 0 ≤ ≤ − 1 [47 49]. These papers study the thermal partition func- j index is restricted to j jmax ¼ N . The space of tion of models similar to Eq. (1.1) in the classical limit, matrices therefore defines a regularized or “fuzzy” 2 i.e., without the Π kinetic energy term. The fuzzy sphere [36]. geometry emerges in a first-order phase transition as a Matrix spherical harmonics are useful for parametrizing low-temperature phase in these models. We see that, in fluctuations about the classical state (4.3). Writing our quantum mechanical context, the geometric phase is X i ν i i ˆ associated with the presence of a specific boundary-law X ¼ J þ yjmYjm; ð4:6Þ entanglement. jm the classical equations of motion can be perturbed about B. Semiclassical analysis of the fuzzy sphere the fuzzy sphere background to give linear equations for the i The results above describe the emergence of a (meta- parameters yjm. The solutions of these equations define ν ν stable) geometric fuzzy sphere state at > c. In this the classical normal modes. We find the normal modes in section, we recall that in the ν → ∞ limit, the fluctuations the SM, Sec. C [45], proceeding as in Refs. [39,40]. The ν ν of the geometry are classical fields. For finite > c, the normal mode frequencies are found to be νω, with background geometry is well defined at large N,but fluctuations will be described by an interacting (noncom- ω2 ¼ 0 multiplicity N2 − 1; mutative) quantum field theory. ω2 j2 multiplicity 2 j − 1 1; In the large ν limit, the wave function can be described ¼ ð Þþ semiclassically [39,40]. We now briefly review this limit, ω2 ¼ðj þ 1Þ2 multiplicity 2ðj þ 1Þþ1: ð4:7Þ with details given in the SM, Sec. C [45]. These results 1 ≤ ≤ − 1 provide a further useful check on the numerics and will Recall that j jmax ¼ N . The three different sets guide our discussion of entanglement in Sec. V. of frequencies in Eq. (4.7) correspond to the group theoretic

011069-7 XIZHI HAN and SEAN A. HARTNOLL PHYS. REV. X 10, 011069 (2020) suð2Þ decomposition j ⊗ 1 ¼ðj − 1Þ ⊕ j ⊕ ðj þ 1Þ. Here, j is the “orbital” angular momentum, and the 1 is due to the vector nature of the Xi. We will give a field theoretic interpretation of these modes shortly. The modes give the following semiclassical contribution to the energy of the fuzzy sphere state:

ν X 4 3 5 − 9 Δ j j ω N þ N ν Ejbos ¼ 2 j j¼ 6 j j: ð4:8Þ

This energy is shown in Fig. 3. The scaling as N3 arises because there are N2 oscillators, with maximal frequency of Γ order N. This semiclassical contribution will be canceled FIG. 5. One-loop effective potential ðrÞ for the radius of the 0 → ∞ out in the supersymmetric sector studied in Sec. IV C bosonic (R ¼ ) fuzzy sphere as N . The fuzzy sphere is ν ν1-loop ≈ 3 03 below. only metastable when > c;N¼∞ . ; see the SM [45]. The normal modes (4.7) can be understood by mapping rffiffiffiffiffiffiffiffi the matrix quantum mechanics Hamiltonian onto a non- 4π commutative gauge theory. The analogous mapping for the δai ¼ −iLiy − ðn × ∇y · ∇Þai: ð4:11Þ Nν3 classical model has been discussed in Ref. [50]. We carry out this map in the SM, Sec. C [45]. The original Here, n is the normal vector and yðθ; ϕÞ a local field on the Hamiltonian (1.1) becomes the following noncommutative 2 sphere. The first term in Eq. (4.11) is the usual U(1) U(1) gauge theory on a unit spatial S [setting the sphere transformation. The second term describes a coordinate radius to 1 in the field theory description will connect easily transformation with infinitesimal displacement n × ∇y. to the quantized modes in Eq. (4.7)]: Indeed, it is known that noncommutative gauge theories Z mix internal and spacetime symmetries, which, in this case, 1 1 H ¼ ν dΩ ðπiÞ2 þ ðfijÞ2 þ const: ð4:9Þ are area-preserving diffeomorphisms of the sphere [51,52]. 2 4 The emergent U(1) noncommutative gauge theory thereby realizes the large N limit of the microscopic SUðNÞ gauge ⋆ The noncommutative star product is defined in the SM symmetry, as area-preserving diffeomorphisms [37,38]. [45] and The fluctuation modes about the fuzzy sphere back- rffiffiffiffiffiffiffiffi ground allow a one-loop quantum effective potential for the 4π ij i j j i ijk k i j radius to be computed in the SM, Sec. C [45]. The potential f ≡ iðL a − L a Þþϵ a þ i ½a ;a ⋆; ð4:10Þ Nν3 at N → ∞ is shown in Fig. 5. At large ν, the effective potential shows a metastable minimum at r ∼ Nν=2.For where the derivatives generate rotations on the sphere Li ¼ 1-loop ν < ν ∞, this minimum ceases to exist. The large N, − ϵ j∂ ≡ ⋆ − ⋆ c;N¼ i ijkx k and ½f; g⋆ f g g f. In Eqs. (4.9) and one-loop analysis therefore qualitatively reproduces the (4.10), the vector potential ai can be decomposed into behavior seen in Figs. 2 and 3. The quantitative disagree- two components tangential to the sphere, which become the ment is mainly due to finite N corrections. The transition is two-dimensional gauge field, and a component transverse only sharp as N → ∞. to the sphere, which becomes a scalar field. This decomposition is described in the SM, Sec. C [45]. The normal C. Numerical results, supersymmetric sector modes (4.7) are coupled fluctuations of the gauge field and We now consider states with fermion number the transverse scalar field. The zero modes in Eq. (4.7) are R N2 − N. The fuzzy sphere background is now super- pure gauge modes, given in Eq. (4.11) below. In Eq. (4.10), ¼ symmetric at large positive ν [32]. The contribution of the the effective coupling controlling quantum field theoretic fermions to the ground-state energy is seen in the SM, interactions is seen to be 1= Nν 3=2. The extra 1=N arises ð Þ Sec. C [45], to cancel the bosonic contribution (4.8) at one i j → ∞ because the commutator ½a ;a ⋆ vanishes as N ; see loop: the SM [45]. Corrections to the Gaussian fuzzy sphere state are therefore controlled by a different coupling than that of 3 3 − ν 2 − 1 Δ Δ 0 the ‘t Hooft expansion (recall λ ¼ N=ν ). 2 ðN Þþ Ejfer þ Ejbos ¼ : ð4:12Þ The SUðNÞ gauge symmetry generators (2.5) are realized in an interesting way in the noncommutative field In Fig. 6, the variational upper bound on the energy of the theory description. We see in the SM [45] that upon fuzzy sphere state remains close to zero for all values of ν. mapping to noncommutative fields, the gauge transforma- Figure 7 shows the radius as a function of ν. Probing tions become the smallest values of ν requires a more powerful

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FIG. 6. Variational energies in the SUSY sector of the mini- FIG. 8. Distribution of radius for different N and small ν. Bands BMN model, for different N and ν. Solid dots are initialized near show the standardpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi deviationP of the quantum mechanical distri- the fuzzy sphere configuration, and the open markers are 1 2 bution of r ¼ ð =NÞ trXi , not to be confused with numeri- initialized near zero. We use normalizing and autoregressive cal uncertainty of the average. Recall that the numbers in the flows, respectively, as these produce more accurate variational brackets are, first, the number of layers in the neural networks wave functions in the two different regimes. and, second, the number of generalized normal distributions in each base distribution. wave-function ansatz than those of Figs. 6 and 7. We will consider that regime shortly. around and starts to increase. The variance in the distri- In contrast to the states with zero fermion number in bution of the radius is also seen to increase towards small ν, Fig. 3, here the fuzzy sphere is seen to be the stable ground revealing the quantum mechanical nature of this regime. state at large ν. However, the fuzzy sphere appears to merge These behaviors (nonmonotonicity of radius and increasing ν with the collapsed state below a value of that decreases variance) are expected—and proven for N ¼ 2—because with N, which is physically plausible: While the classical the flat directions of the classical potential at ν ¼ 0 show 2 2 2 fuzzy sphere radius r ∼ ν N decreases at small ν, that the extent of the wave function is set by purely quantum fluctuations of the collapsed state are expected quantum mechanical effects in this limit. to grow in space as ν → 0. This is because the flat The small ν regime here is furthermore an opportunity to directions in the classical potential of the ν ¼ 0 theory, test the versatility of our variational ansatz away from given by commuting matrices, are not lifted in the presence semiclassical regimes. In the SM, Sec. D [45], we see that of supersymmetry [53]. Eventually, the fuzzy sphere should for small ν, MAFs achieve much lower energies than NFs. be subsumed into these quantum fluctuations. This Increasing the number of distributions in the mixture and smoother large N evolution towards small ν (relative to the number D of free fermions states in Eq. (2.6) further the bosonic sector) is mirrored in the thermal behavior of lowers the energy. These facts mirror the behavior we found classical supersymmetric models [54,55]. in our N ¼ 2 benchmarking in Sec. III B at small ν, Indeed, exploring the small ν region with more precision, increasing our confidence in the ability of the network we observe a physically expected feature. In Fig. 8, we see to capture this regime for large N also. The error in a that as ν decreases towards zero, the radius not only ceases variational ansatz is, as always, not controlled, and there- to follow the semiclassical decreasing behavior but turns fore, further exploration of this regime is warranted before very strong conclusions can be drawn. We plan to revisit this regime in future work, to search for the possible presence of emergent “throat” geometries, as we discuss in Sec. VI below.

V. ENTANGLEMENT ON THE FUZZY SPHERE In this section, we show that the large ν fuzzy sphere state discussed above contains boundary-law entanglement. To compute the entanglement, one must first define a factorization of the Hilbert space. For our emergent space at finite N and ν, the geometry is both fuzzy and fluctuating, FIG. 7. Expectation value of radius in the SUSY sector of the and hence lacks a canonical spatial partition. The fuzziness mini-BMN model, for different N and ν. Solid dots are initialized of the sphere is captured by a toy model of a free field on a near the fuzzy sphere configuration, and the open markers are sphere with an angular-momentum cutoff. Recall from initialized near zero. The dashed lines are the semiclassical Sec. IV that the noncommutative nature of the fuzzy sphere − 1 values (4.4). amounts to an angular-momentum cutoff jmax ¼ N .

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≤ We will start, then, by defining a partition of the space of with j jmax. However, we can still do our best to ∞ χ functions with such a cutoff. approximate the projector PA of multiplication by A,as jmax defined in the previous paragraph, with a projector PA A. Free field with an angular-momentum cutoff ≤ jmax that lives in the subspace with j jmax. Formally, let Q Consider a free massive complex scalar field φðθ; ϕÞ on a be the space of functions on the sphere spanned by θ ϕ ≤ unit two-sphere with the following Hamiltonian: Yjmð ; Þ, with j jmax. Define the orthogonal projector j j ∞ max ∶ jmax → jmax max − Z PA Q Q to minimize the distance kPA PA k. 2 2 2 2 jmax H ¼ dΩ½jπj þj∇φj þ μ jφj : ð5:1Þ The projector PA annihilates all functions in the orthogo- 2 S nal complement of Qjmax , when viewed as an operator acting on Q∞. It is convenient to choose · to be the Here, π is the field conjugate to φ. We impose a cutoff k k ≤ Frobenius norm, and in the SM, Sec. E [45], an explicit j jmax on the angular momentum, rending the quantum jmax mechanical problem well defined. The fields can therefore formula for PA is obtained. jmax be decomposed into a sum of spherical harmonic modes: The projector PA then defines a factorization of the 2 2 j 2 j jmax max ⊗ max Hilbert space L ðQ Þ¼L ðimPA Þ L ðker PA Þ for jXmj≤j any region A, and entanglement can be evaluated in the φðθ; ϕÞ¼ ajmYjmðθ; ϕÞ: ð5:2Þ usual way. In particular, the second Rényi entropy of a pure 0≤ ≤ j jmax state jψi on a region A is Z The wave functional of the quantum field φðθ; ϕÞ is then a 0 0 0 S2 ρ − ln dx dx ¯ dx dx ¯ ψ x x ¯ ψ x x ¯ mapping from coefficients ajm to complex amplitudes. The ð AÞ¼ A A A A ð A þ AÞ ð A þ AÞ ground-state wave functional of the Hamiltonian (5.1) is 0 0 0 × ψðx þ x ¯ Þψ ðxA þ x ¯ Þ P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ZA A A − jðjþ1Þþμ2 ja j2 ψ ∝ jm jm ðajmÞ e : ð5:3Þ ¼ − ln dxdx0ψðxÞψ ðPx0 þðI − PÞxÞψðx0Þ

To calculate entanglement for quantum states, a factori- × ψ ðPx þðI − PÞx0Þ; ð5:4Þ zation of the Hilbert space H ¼ H1 ⊗ H2 is prescribed. To − motivate the construction of such a factorization in the where xA ¼ Px and xA¯ ¼ðI PÞx are integrated over imP j fuzzy sphere case, we now review the general framework of max ¯ and ker P, for P ¼ PA , and xA and xA can be more defining entanglement in (factorizable) quantum field ≤ compactly combined into a field x with j jmax. Note that theories. the various x’s in Eq. (5.4) denote functions on the sphere. In quantum mechanics, a quantum state is a function jmax The projector PA is found to have two important from the configuration space Q to complex numbers, and geometric features: the Hilbert space of all quantum states is commonly that of (1) The trace of the projector, which counts the number H 2 square integrable functions ¼ L ðQÞ. In quantum field of modes in a region, is proportional to the size of the theories, the space Q is furthermore a linear space of jmax ∝ 2 region. Specifically, at large jmax,trPA jmaxjAj functions on some geometric manifold M, and thus an as is seen numerically in Fig. 9 and understood orthogonal decomposition Q ¼ Q1 ⊕ Q2 induces a fac- 2 2 analytically in the SM, Sec. E [45]. torization of H ¼ L ðQ1Þ ⊗ L ðQ2Þ, which can be exploited to define entanglement. To define entanglement, it then suffices to find an orthogonal decomposition of the space of fields on the fuzzy sphere. Without an angular-momentum cutoff, i.e., → ∞ with jmax , there is a natural choice for any region A on the sphere, which sets Q1 to be all functions supported on ¯ A, and Q2 all functions supported on A, the complement of A. Any function f on M can be uniquely written as a sum of f1 ∈ Q1 and f2 ∈ Q2, where f1 ¼fχA and f2 ¼ fð1 − χAÞ. Here, χA is the function on the sphere that is 1 on A and 0 otherwise. Note that the map of multiplication χ ↦ χ ⊕ → by A, f f A, acts as the projection Q1 Q2 Q1. FIG. 9. Trace of the projector versus fractional area of the Conversely, given any orthogonal projection operator region (a spherical cap with polar angle θA), with different P∶Q → Q, we can decompose Q ¼ imP ⊕ ker P. angular-momentum cutoffs j . A linear proportionality is χ max When the cutoff jmax is finite, multiplication by A will observed at large jmax. The discreteness in the plot arises because generally take the function out of the subspace of functions the finite jmax space of functions cannot resolve all angles.

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and does not need to approximate a spatial region in other geometries. The partition is even less meaningful in nongeometric regions of the Hilbert space. The variational wave function we have constructed can be used to compute entanglement for any given factorization of the Hilbert space, but it is unclear if preferred factorizations exist away from geometric limits. In this work, we focus on the entanglement in the ν → ∞ limit where the fields are infinitesimal and hence do not backreact on the spherical geometry. In this limit, the factorization is precisely—up to issues of gauge invariance—that of the free-field case FIG. 10. The second R´enyi entropy for a complex scalar free field (with mass μ ¼ 1) versus the polar angle θ of a spherical discussed in the previous subsection. A The matrices corresponding to the infinitesimal fields on cap. The entropy with different cutoffs jmax is shown. At large 0 03 2π θ the fuzzy sphere are, cf. Eq. (4.6), jmax, the curve approaches the boundary law . × sin A, shown as a dashed line. Discreteness in the plot is again due to the i i − ν i finite jmax space of functions. A ¼ X J ; ð5:5Þ

which should be thought of as living in the tangent space at (2) The second R´enyi entropy defined by the projector Xi ¼ νJi. At large ν, the wave function is strongly follows a boundary law. At large jmax, with the mass μ 1 ≈ 0 03 ∂ supported on the classical configuration, and hence in this fixed to ¼ , the entropy S2 . jmaxj Aj as is seen numerically in Fig. 10 and understood analyti- limit, the infinitesimal description is accurate. Gauge cally in the SM, Sec. E [45]. transformations then act as This boundary entanglement law in Fig. 10 is of course i → i ϵ ν i … precisely the expected entanglement in the ground state of a A A þ i ½Y; J þ ; ð5:6Þ local quantum field [6,7]. As the cutoff jmax is removed, the ϵ entanglement grows unboundedly. where is infinitesimal and Y is an arbitrary Hermitian ϵ i The partition we have just defined can now be adapted to matrix. The ½Y;A term is omitted in Eq. (5.6) as it is of the fluctuations about the large ν fuzzy sphere state in the higher order. Gauge invariance of the state is manifested as matrix quantum mechanics model. We do this in the ψ ν i i ψ ν i i ϵ ν i following subsection. Intuitively, we would like to replace ð J þ A Þ¼ ð J þ A þ i ½Y; J Þ: ð5:7Þ the jðj þ 1Þþμ2 spectrum of the free field in the wave Physical states are wave functions on gauge orbits ½Ai, function (5.3) with the matrix mechanics modes (4.7). i Recall that the matrix modes are cut off at angular the set of infinitesimal matrices differing from A by a momentum j ¼ N − 1. gauge transformation (5.6). Similarly to the discussion of max free fields above, a partition of the space of gauge orbits is specified by a projector P. We now explain how this B. Fuzzy sphere in the mini-BMN model projector is constructed. Given a projector P0 acting on Now, we address two additional subtleties that arise infinitesimal matrices Ai, a projector acting on gauge orbits when adapting the free field ideas above to the mini-BMN can be defined as fuzzy sphere. First, the mini-BMN theory is an SUðNÞ gauge theory. It is known that entanglement in gauge Pð½AiÞ ¼ ½P0ðAiÞ: ð5:8Þ theories may depend upon the choice of gauge-invariant algebras associated with spatial regions [56]. Different However, for P to be well defined, P0 must preserve gauge prescriptions correspond to different boundary or gauge directions: conditions [57]. However, for a fuzzy geometry, the boundaries of regions and gauge edge modes are not P0ðAi þ iϵ½Y;νJiÞ ¼ P0ðAiÞþiϵ½Y0; νJi; ð5:9Þ sharply defined. To introduce the fewest additional degrees of freedom, we choose to factorize the physical Hilbert for any Ai, Y and some Y0 dependent on Y. Let V be the space, instead of an extended one [58,59], to evaluate subspace of gauge directions: entanglement in the mini-BMN model. This method is similar to the “balanced center” procedure in Ref. [56], V ¼fi½Y;Ji∶Yis Hermitiang: ð5:10Þ where edge modes are absent [60]. Second, the emergent fields include fluctuations of the Then, Eq. (5.9) is equivalent to the requirement that geometry itself. The factorization that we have discussed in P0ðVÞ ⊂ V. The strategy for finding the projector P is to 0 0 the previous subsection is tailored to a region on the sphere solve for the projector P that minimizes kP − χAk subject

011069-11 XIZHI HAN and SEAN A. HARTNOLL PHYS. REV. X 10, 011069 (2020) to the constraint that Eq. (5.9) is satisfied. Then, P is defined via P0 as in Eq. (5.8). 0 The problem of minimizing kP − χAk for orthogonal projectors P0 such that P0ðVÞ ⊂ V is exactly solvable as follows. The condition that P0ðVÞ ⊂ V is equivalent to 0 ⊕ imposing that P ¼ PV PV⊥ , where PV is some projector in the subspace V and PV⊥ in its orthogonal complement 0 V⊥. Here, kP − χAk is minimized if and only if − χ − χ kPV AjV k and kPV⊥ AjV⊥ k are both minimized. Via the correspondence between matrix spherical harmon- ˆ ics Yjm and spherical harmonic functions Yjmðθ; ϕÞ—in the SM, Sec. C [45]—both of these minimizations become the FIG. 11. The second Rényi entropy for a spherical cap on the matrix theory fuzzy sphere versus the polar angle θ of the cap. same problem as in the free field case, with a detailed A Solid curves are exact values at ν ¼ ∞, and dots are numerical solution in the SM, Sec. E [45]. values from variational wave functions at ν ¼ 10 for different N. The second Rényi entropy, in terms of gauge orbits, is The wave functions are NF(1, 1) in the zero fermion sector as evaluated similarly to Eq. (5.4): shown in Figs. 2 and 3. Z ρ − 0 Δ Δ 0 S2ð AÞ¼ ln d½Ad½A ð½AÞ ð½A Þ S2ðρAÞ ≈ 0.03Nj∂Aj: ð5:14Þ × ψ A ψ P A0 I − P A invð½ Þ invð ½ þð Þ½ Þ Here, j∂Aj¼2π sin θA is again the circumference of the ψ 0 ψ − 0 spherical cap A [in units where the sphere has radius one, × invð½A Þ ðP½AþðI PÞ½A Þ; ð5:11Þ inv consistent with the field theoretic description in Eq. (4.9)]. where Δ are measure factors for gauge orbits and The result (5.14) is the same as that of the toy model in ψ ð½AÞ ¼ ψðνJ þ AÞ. Recall that ψ is gauge invariant Fig. 10, with jmax now set by the microscopic matrix inv − 1 according to Eq. (5.7). The formula (5.11) as displayed dynamics to be N . (A simpler instance of entanglement does not involve any gauge choice. However, there are revealing the inherent graininess of a spacetime built from some gauges where evaluating Eq. (5.11) is particularly matrices is two-dimensional string theory [62,63]). This convenient. The gauge we choose for this purpose, which is regulated boundary-law entanglement underpins the emergent locality on the fuzzy sphere at large N and ν. Recall different from that in Sec. II B, is that A ∈ V⊥; i.e., the fields are perpendicular to gauge directions. In this gauge, from the discussion around Eq. (4.9) that there are only two measure factors are trivial, and the projector is simply P , emergent fields on the sphere: a Maxwell field and a scalar V⊥ field. The perpendicular gauge choice we have made trans- which minimizes kP − χ j k: V⊥ A V⊥ lates into the Coulomb gauge for the emergent Maxwell Z field, cf. the discussion around Eq. (4.11) above. The factor ρ − 0ψ ψ 0 − S2ð AÞ¼ ln dAdA ⊥ðAÞ ⊥ðPV⊥ A þðI PV⊥ ÞAÞ of N in Eq. (5.14) is due to the microscopic cutoff at a V⊥ ∼ scale Lfuzz Lsph=N. ψ 0 ψ − 0 × ⊥ðA Þ ⊥ðPV⊥ A þðI PV⊥ ÞA Þ; ð5:12Þ Previous work on the entanglement of a free field on a fuzzy sphere involved similar wave functions but a different where ψ ⊥ðAÞ is defined as ψðνJ þ AÞ for A ∈ V⊥ [61]. factorization of the Hilbert space, which was inspired The bosonic fuzzy sphere wave function can be written in instead by coherent states [64–67]. Those results did not ν→∞ the limit as follows. AsP in Eq.P(4.6), the perturbations always produce boundary-law entanglement. Here, we see can be decomposed as Ai ¼ δx yi Yˆ , where the that the UV/IR mixing in noncommutative field theories a a jm jma jm ν yi diagonalize the potential energy at quadratic order in A does not preclude a partition of the large N and large jma P Hilbert space with a boundary-law entanglement. so that V ν2=2 ω2 δx 2 (see SM, Sec. C [45]). ¼ð Þ a að aÞ þ We can also evaluate the entropy (5.12) using the large ν The wave function is then, analogously to Eq. (5.3), variational wave functions, without assuming the asymp- P ν totic form (5.13). The results are shown as dots in Fig. 11. −j j jω jðδx Þ2 ψ ⊥ðAÞ ∝ e 2 a a a : ð5:13Þ However, we stress that only the ν → ∞ limit has a clear physical meaning, where fluctuations are infinitesimal. The The frequencies are given by Eq. (4.7), excluding the pure variational results are close to the exact values in Fig. 11, gauge zero modes. Using this wave function, the Rényi showing that the neural network ansatz captures the entropy (5.12) can be computed exactly and is shown as a entanglement structure of these matrix wave functions. solid line in Fig. 11.AsN → ∞, these curves approach a The results in this section are for the bosonic fuzzy boundary law sphere. The projection we introduced in order to partition

011069-12 DEEP QUANTUM GEOMETRY OF MATRICES PHYS. REV. X 10, 011069 (2020) the space of matrices can be extended in a similar, but more the usual gauge fields and transverse scalar fields of string involved, way to factorize the fermionic Hilbert space. theoretic D-branes. For N=ν3 ≫ 1, it is possible that the strongly interacting, collapsed D-particles will develop a geometric “throat,” in VI. DISCUSSION the spirit of the canonical holographic correspondence [1]. We have seen that neural network variational wave It is not well understood when such a throat would be functions capture, in detail, the physics of a semiclassical captured by the mini-BMN matrix quantum mechanics. spherical geometry that emerges in the mini-BMN model The variational wave functions that we have developed here (2.1) at large ν. Away from the semiclassical limit, the provide a new window into this problem. In particular, we spherical geometry either abruptly or gradually collapses hope to investigate the small ν collapsed state in more detail towards a new state. In Fig. 8, we saw that in the “super- in the future, with the objective of revealing any entangle- symmetric” sector, this new state was characterized by an ment associated with emergent local dynamics in the throat increase in both the expectation value and quantum spacetime. If the emergent dynamics includes gravity, there mechanical variance of the radius as ν → 0. To understand are two potentially interesting complications. First, the the physics of this process, and to start thinking about the entanglement of bulk fields may be entwined with entan- nature of the collapsed state as ν → 0, it is helpful to glement due to the “stringy” degrees of freedom that seem consider the string theoretic embedding of the model. to be manifested in the Bekenstein-Hawking entropy of The mini-BMN model can be realized in string theory as black holes as well as in the Ryu-Takayanagi formula – the description of N D-particles in an AdS4 spacetime. [68 71]. Second, and perhaps relatedly, it may become Let us review some aspects of this realization [32]. The crucial to understand the edge-mode contribution to the parameter entanglement, which we have avoided in our discussion here [72,73]. 1 3 More generally, the methods we have developed will be ∼ LAdS 3 gs : ð6:1Þ applicable to a wide range of quantum problems of interest ν Ls in the holographic correspondence. The benefit of the

Here, LAdS is the AdS radius, Ls is the string length, and gs variational neural network approach is direct access to is the string coupling. The proportionality in Eq. (6.1) properties of the zero-temperature quantum mechanical depends on the volume, in units of the string length, of state. Optimizing the numerical methods and variational internal cycles wrapped by the branes in the compactifi- ansatz further, and with more computational power, it cation down to AdS4. In particular, the mass of a single D- should not be difficult to work with larger values of N. 1 In addition to understanding the emergence of spacetime particle is proportional to =gs times the wrapped internal volume. The strength of the gravitational backreaction of N from first principles, it should also be possible to study, for example, the microstates and dynamics of quantum coincident D-particles is then controlled by GN × N=gs. ∼ 2 black holes. Here, GN gs is the four-dimensional Newton constant, where we have suppressed a factor of the volume of the The code used in the paper is available online [74]. compactification manifold. Therefore, if we keep the AdS radius fixed in string units, gravitational backreation ACKNOWLEDGMENTS ∼ ν3 ≳ 1 becomes important when gsN N= . Up to factors of It is a pleasure to thank Frederik Denef and Xiaoliang the volume of compactification cycles, this is equivalent to Qi for helpful discussions; Aitor Lewkowycz, Raghu the statement that the dimensionless ’t Hooft coupling Mahajan, and Edward Mazenc for comments on the draft λ ¼ N=ν3, introduced below Eq. (1.1), becomes large. of this article; and Tarek Anous for sharing his code with us. For N=ν3 ≲ 1, then, the D-particles can be treated as light We also thank Zhaoheng Guo and Yang Song for collabo- probes on the background AdS spacetime. The fuzzy ration on a related project. S. A. H. is partially funded by sphere configuration describes a polarization of the D- DOE Award No. de-sc0018134. Computational work was particles into spherical “dual giant gravitons.” From the performed on the Sherlock Cluster at Stanford University, string theory perspective, this polarization is driven by with the TensorFlow code for the project available online. Ω ∼ 1 the fourform flux =LAdS supporting the background AdS4 spacetime. Together with the discussion in the previous paragraph on the strength of the gravitational interaction, we can write the heuristic relation N=ν3 ∼ [1] J. M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, Int. J. Theor. Phys. 38, gravity=flux. At large ν, the flux wins out, and semiclassical ν 1113 (1999). fuzzy spheres can exist; however, at small , gravitational [2] J. Polchinski, Introduction to Gauge/Gravity Duality,in forces cause the spheres to collapse. The entanglement and Proceedings, Theoretical Advanced Study Institute in emergent locality that we have described in this paper is that Elementary Particle Physics (TASI 2010). String Theory of the polarized spheres, whose excitations are described by and Its Applications: From meV to the Planck Scale:

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