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It has been proposed that random wide neural networks near Gaussian process are quantum field theories around Gaussian fixed points. In this paper, we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural networks to generate arbitrary paths in the Feynman's path integral. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world., Comment: 24 pages, 6 figures

Understanding the inner workings of neural networks, including transformers, remains one of the most challenging puzzles in machine learning. This study introduces a novel approach by applying the principles of gauge symmetries, a key concept in physics, to neural network architectures. By regarding model functions as physical observables, we find that parametric redundancies of various machine learning models can be interpreted as gauge symmetries. We mathematically formulate the parametric redundancies in neural ODEs, and find that their gauge symmetries are given by spacetime diffeomorphisms, which play a fundamental role in Einstein's theory of gravity. Viewing neural ODEs as a continuum version of feedforward neural networks, we show that the parametric redundancies in feedforward neural networks are indeed lifted to diffeomorphisms in neural ODEs. We further extend our analysis to transformer models, finding natural correspondences with neural ODEs and their gauge symmetries. The concept of gauge symmetries sheds light on the complex behavior of deep learning models through physics and provides us with a unifying perspective for analyzing various machine learning architectures., Comment: 11 pages, 3 figures

According to the AdS/CFT correspondence, certain quantum many-body systems in $d$-dimensions are equivalent to gravitational theories in $(d+1)$-dimensional asymptotically AdS spacetimes. When a massless particle is sent from the AdS boundary to the bulk curved spacetime, it reaches another point of the boundary after a time lag. In the dual quantum system, it should appear as if quasiparticles have been transferred between two separated points. We theoretically demonstrate that this phenomenon, which we call "spacetime-localized response," is actually observed in the dynamics of the one-dimensional transverse-field Ising model near the quantum critical point. This result suggests that, if we can realize a holographic spin system in a laboratory, the experimental probing of the emergent extra-dimension is possible by applying a designed stimulus to a quantum many-body system, which is holographically equivalent to sending a massless particle through the higher-dimensional curved bulk geometry. We also discuss possible experimental realizations using Rydberg atoms in an optical tweezers array., Comment: 10 pages, 6 figures, Ver. 2: matches the PRD published version

We find that simple neural networks with ReLU activation generate polytopes as an approximation of a unit sphere in various dimensions. The species of polytopes are regulated by the network architecture, such as the number of units and layers. For a variety of activation functions, generalization of polytopes is obtained, which we call neural polytopes. They are a smooth analogue of polytopes, exhibiting geometric duality. This finding initiates research of generative discrete geometry to approximate surfaces by machine learning., Comment: 5 pages, 9 figures. v2: References added. Accepted at the 1st Workshop on the Synergy of Scientific and Machine Learning Modeling at International Conference on Machine Learning (ICML), Honolulu, Hawaii, USA. 2023

Photon spheres are the characteristic of general black holes, thus are a suitable touchstone for the emergence of gravitational spacetime in the AdS/CFT correspondence. We provide a spectral analysis of an AdS Schwarzschild black hole near its photon sphere. We find that quasinormal modes near the photon sphere reflect the AdS boundary, resulting in a peculiar spectral pattern. Our large angular momentum analysis owes to an analogue to solvable Schr\"odinger equations such as an inverted harmonic oscillator and the P\"oschl-Teller model, with a Dirichlet boundary condition. Through the AdS/CFT dictionary, it predicts the existence of a peculiar subsector in the large angular momentum spectrum of thermal holographic CFTs on a sphere., Comment: 20 + 19 pages, 9 figures

Recently, Krylov complexity was proposed as a measure of complexity and chaoticity of quantum systems. We consider the stadium billiard as a typical example of the quantum mechanical system obtained by quantizing a classically chaotic system, and numerically evaluate Krylov complexity for operators and states. Despite no exponential growth of the Krylov complexity, we find a clear correlation between variances of Lanczos coefficients and classical Lyapunov exponents, and also a correlation with the statistical distribution of adjacent spacings of the quantum energy levels. This shows that the variances of Lanczos coefficients can be a measure of quantum chaos. The universality of the result is supported by our similar analysis of Sinai billiards. Our work provides a firm bridge between Krylov complexity and classical/quantum chaos., Comment: 41 pages, 35 figures, 8 tables; v2: journal version, appendices added

We propose a method to calculate wave functions and energies not only of the ground state but also of low-lying excited states using a deep neural network and the unsupervised machine learning technique. For systems composed of identical particles, a simple method to perform symmetrization for bosonic systems and antisymmetrization for fermionic systems is also proposed., Comment: 23 pages, 18 figures, 8 tables; Typo of Eq. (25) corrected

We propose a way to discover, in tabletop experiments, spacetime-emergent materials, that is, materials holographically dual to higher-dimensional quantum gravity systems under the AdS/CFT correspondence. The emergence of the holographic spacetime is verified by a mathematical imaging transform of the response function on the material. We consider theories on a 1-dimensional ring-shaped material, and compute the response to a scalar source locally put at a point on the ring. When the theory on the material has a gravity dual, the imaging in the low temperature phase exhibits a distinct difference from the ordinary materials: the spacetime-emergent material can look into the holographically emergent higher-dimensional curved spacetime and provides an image as if a wave had propagated there. Therefore the image is an experimental signature of the spacetime emergence. We also estimate temperature, ring size and source frequency usable in experiments, with an example of a quantum critical material, TlCuCl$_3$., Comment: 23 pages, 9 figures, discussion expanded

We derive an explicit form of the dilaton potential in improved holographic QCD (IHQCD) from the QCD lattice data of the chiral condensate as a function of the quark mass. This establishes a data-driven holographic modeling of QCD -- machine learning holographic QCD. The modeling consists of two steps for solving inverse problems. The first inverse problem is to find the emergent bulk geometry consistent with the lattice QCD simulation data at the boundary. We solve this problem with the refinement of neural ordinary differential equation, a machine learning technique. The second inverse problem is to derive a bulk gravity action with a dilaton potential such that its solution is the emergent bulk geometry. We solve this problem at non-zero temperature, and derive the explicit form of the dilaton potential. The dilaton potential determines the bulk action, the Einstein-dilaton system, thus we derive holographically the bulk system from the QCD chiral condensate data. The usefulness of the model is shown in the example of the prediction of the string breaking distance, whose value is found to be consistent with another lattice QCD data., Comment: 14 pages, 10 figures

It has long been thought that a highly excited string can be regarded as a black hole: the correspondence principle between strings and a black hole, while recent studies found that black holes are characterized by chaos. This suggests that highly excited strings are the source of the black hole chaoticity. We study the chaoticity of a string amplitude where a tachyon is scattered by a highly excited string. Our strategy to extract the chaos in the amplitude is a generalization of the transient chaos analysis for classical scattering. We look for the fractal structure in the plots of incoming/outgoing scattering angles, where the outgoing angle is defined as the maximum pole of the amplitude. Within our strategy, we could not identify any fractal structure in the scattering data. We also discuss other possible setups and strategies to extract the chaos, hoping that our present work serves as a step toward the formulation of chaos in string scattering amplitudes., Comment: 26 pages, 6 figures

We perform imaging of a fundamental string from string scattering amplitudes, and show that its image is a double slit., Comment: 19 pages, 10 figures

Predictability is ensured by causality while lost in chaos. To reconcile these two popular notions, we study chaos in geodesic motions in generic curved spacetimes with external potentials, where causality is controlled by a scalar potential. We develop a reparametrization-independent method to analytically estimate the Lyapunov exponent $\lambda$ of a particle motion. We show that causality gives the universal upper bound $\lambda\propto E\ (E\rightarrow\infty)$, which coincides with the chaos energy bound proposed by Murata, Tanahashi, Watanabe, and one of the authors (K.H.). We also find that the chaos bound discovered by Maldacena, Shenker, and Stanford can be violated in particular potentials, even with causality. Our estimates, although waiting for numerical confirmation, reveal the hidden nature of physical theories: causality bounds chaos., Comment: 20 pages. v2: typos corrected, references added

The International Linear Collider (ILC) is on the table now as a new global energy-frontier accelerator laboratory taking data in the 2030s. The ILC addresses key questions for our current understanding of particle physics. It is based on a proven accelerator technology. Its experiments will challenge the Standard Model of particle physics and will provide a new window to look beyond it. This document brings the story of the ILC up to date, emphasizing its strong physics motivation, its readiness for construction, and the opportunity it presents to the US and the global particle physics community., Comment: 356 pages, Large pdf file (40 MB) submitted to Snowmass 2021; v2 references to Snowmass contributions added, additional authors; v3 references added, some updates, additional authors

We conjecture a chaos energy bound, an upper bound on the energy dependence of the Lyapunov exponent for any classical/quantum Hamiltonian mechanics and field theories. The conjecture states that the Lyapunov exponent $\lambda(E)$ grows no faster than linearly in the total energy $E$ in the high energy limit. In other words, the exponent $c$ in $\lambda(E) \propto E^c \,(E\to\infty)$ satisfies $c\leq 1$. This chaos energy bound stems from thermodynamic consistency of out-of-time-order correlators (OTOC's) and applies to any classical/quantum system with finite $N$ / large $N$ ($N$ is the number of degrees of freedom) under plausible physical conditions on the Hamiltonians. To the best of our knowledge the chaos energy bound is satisfied by any classically chaotic Hamiltonian system known, and is consistent with the cerebrated chaos bound by Maldacena, Shenker and Stanford which is for quantum cases at large $N$. We provide arguments supporting the conjecture for generic classically chaotic billiards and multi-particle systems. The existence of the chaos energy bound may put a fundamental constraint on physical systems and the universe., Comment: 3 pages, plus 7 pages of supplemental material; v2: minor revision with a reference and a footnote added

We derive an explicit form of the dilaton potential in improved holographic QCD (IHQCD) from the experimental data of the $\rho$ meson spectrum. For this purpose we make use of the emergent bulk geometry obtained by deep learning from the hadronic data in arXiv:2005.02636. Requiring that the geometry is a solution of an IHQCD derives the corresponding dilaton potential backwards. This determines the bulk action in a data-driven way, which enables us at the same time to ensure that the deep learning proposal is a consistent gravity. Furthermore, we find that the resulting potential satisfies the requirements normally imposed in IHQCD, and that the holographic Wilson loop for the derived model exhibits quark confinement., Comment: 7pages, 5figures. v2: footnote 6 added

We provide a formula to reconstruct bulk spacetime metrics inside black holes by the time dependence of complexity in the dual quantum field theory, based on the complexity=volume (CV) conjecture in the holographic duality., Comment: 22 pages, 2 figures

Saturation of the nuclear binding energy is one of the most important properties of atomic nuclei. We derive the saturation in holographic QCD, by building a shell-model-like mean-field nuclear potential from the nuclear density profile obtained in a holographic multi-baryon effective theory. The numerically estimated binding energy is close to the experimental value., Comment: 7 pages, 1 fugure; v2: a footnote added; v3: published version

We provide formulas for holographically building a bulk metric from given expectation values of rectangular Wilson loops. As a corollary, we prove that any confining quark potential necessarily leads to the existence of a bulk IR bottom., Comment: 12 pages; v2: references added, typos corrected

We provide a detailed examination of a thermal out-of-time-order correlator (OTOC) growing exponentially in time in systems without chaos. The system is a one-dimensional quantum mechanics with a potential whose part is an inverted harmonic oscillator. We numerically observe the exponential growth of the OTOC when the temperature is higher than a certain threshold. The Lyapunov exponent is found to be of the order of the classical Lyapunov exponent generated at the hilltop, and it remains non-vanishing even at high temperature. We adopt various shape of the potential and find these features universal. The study confirms that the exponential growth of the thermal OTOC does not necessarily mean chaos when the potential includes a local maximum. We also provide a bound for the Lyapunov exponent of the thermal OTOC in generic quantum mechanics in one dimension, which is of the same form as the chaos bound obtained by Maldacena, Shenker and Stanford., Comment: 24 pages, 14 figures; v2: Appendix B (studying other operator orderings of OTOC) added, a reference added

The neural ordinary differential equation (Neural ODE) is a novel machine learning architecture whose weights are smooth functions of the continuous depth. We apply the Neural ODE to holographic QCD by regarding the weight functions as a bulk metric, and train the machine with lattice QCD data of chiral condensate at finite temperature. The machine finds consistent bulk geometry at various values of temperature and discovers the emergent black hole horizon in the holographic bulk automatically. The holographic Wilson loops calculated with the emergent machine-learned bulk spacetime have consistent temperature dependence of confinement and Debye-screening behavior. In machine learning models with physically interpretable weights, the Neural ODE frees us from discretization artifact leading to difficult ingenuity of hyperparameters, and improves numerical accuracy to make the model more trustworthy., Comment: 10 pages, 6 figures

We propose a deep learning method to build an AdS/QCD model from the data of hadron spectra. A major problem of generic AdS/QCD models is that a large ambiguity is allowed for the bulk gravity metric with which QCD observables are holographically calculated. We adopt the experimentally measured spectra of $\rho$ and $a_2$ mesons as training data, and perform a supervised machine learning which determines concretely a bulk metric and a dilaton profile of an AdS/QCD model. Our deep learning (DL) architecture is based on the AdS/DL correspondence (arXiv:1802.08313) where the deep neural network is identified with the emergent bulk spacetime., Comment: 13 pages, 9 figures, v2: Figure display problem resolved, the content unchanged

We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole., Comment: 20 pages, 2 figures; v2: typos corrected

Exponential growth of thermal out-of-time-order correlator (OTOC) is an indicator of a possible gravity dual, and a simple toy quantum model showing the growth is being looked for. We consider a system of two harmonic oscillators coupled nonlinearly with each other, and numerically observe that the thermal OTOC grows exponentially in time. The system is well-known to be classically chaotic, and is a reduction of Yang-Mills-Higgs theory. The exponential growth is certified because the growth exponent (quantum Lyapunov exponent) of the thermal OTOC is well matched with the classical Lyapunov exponent, including their energy/temperature dependence. Even in the presence of the exponential growth in the OTOC, the energy level spacings are not sufficient to judge a Wigner distribution, hence the OTOC is a better indicator of quantum chaos., Comment: 27 pages, 14 figures. v2: analysis of level statistics improved, references and comments added

Higher-order topological insulators (HOTIs) or multipole insulators, hosting peculiar corner states, were discovered [arXiv:1611.07987, arXiv:1708.03636]. It was independently discovered [arXiv:1702.00624] that continuum 5-dimensional (5D) Weyl semimetals generically host the corner states, and so do 4D class A and 3D class AIII topological insulators. In this paper we further confirm that the 5D Weyl semimetals, upon dimensional reduction, lead to universal higher-order topology. First we explain a discrete symmetry protecting the 5D Weyl semimetals, and describe dimensional reductions of the 5D Weyl semimetals to the popular HOTIs in the continuum limit. We calculate the topological charge carried by edge states of the 5D Weyl semimetal, for the most generic boundary condition. The topological charge is a Dirac monopole, which can also be seen from that edge Hamiltonians are always of the form of a 3D Weyl semimetal. This edge topology leads to the edge-of-edge states, or the corner states, generically, suggesting that the 5D Weyl semimetal is thought of as a physical structural origin of corner states in HOTIs. In addition, we explicitly calculate a nested Wilson loop of the 5D Weyl semimetal and find that the topological structure is identical to that of a Wilson loop of a Dirac monopole., Comment: 11 pages, 2 figures; v2: 12 pages, 2 figures, version to be published in Phys.Rev.B

Type II Weyl semimetals are dictated by bulk excitations with tilted light cones, resembling the inside of black holes. We obtain generic boundary conditions for surface boundaries of the type II Weyl semimetals near Weyl nodes, and show that for a certain boundary condition edge states can escape out of the "black hole" event horizon. This means that for realization of the material "black hole" by the Type II Weyl semimetals a careful choice of the boundary condition is necessary., Comment: 8 pages, 4 figures; v2: references added

Clarifying conditions for the existence of a gravitational picture for a given quantum field theory (QFT) is one of the fundamental problems in the AdS/CFT correspondence. We propose a direct way to demonstrate the existence of the dual black holes: imaging an Einstein ring. We consider a response function of the thermal QFT on a two-dimensional sphere under a time-periodic localized source. The dual gravity picture, if it exists, is a black hole in an asymptotic global AdS$_4$ and a bulk probe field with a localized source on the AdS boundary. The response function corresponds to the asymptotic data of the bulk field propagating in the black hole spacetime. We find a formula that converts the response function to the image of the dual black hole: The view of the sky of the AdS bulk from a point on the boundary. Using the formula, we demonstrate that, for a thermal state dual to the Schwarzschild-AdS$_4$ spacetime, the Einstein ring is constructed from the response function. The evaluated Einstein radius is found to be determined by the total energy of the dual QFT. Our theoretical proposal opens a door to gravitational phenomena on strongly correlated materials., Comment: 5+2 pages, 5 figures, accepted for publication in Physical Review Letters. arXiv admin note: substantial text overlap with arXiv:1811.12617

We study $1/N$ corrections to a Wilson loop in holographic duality. Extending the AdS/CFT correspondence beyond the large $N$ limit is an important but a subtle issue, as it needs quantum gravity corrections in the gravity side. To find a physical property of the quantum corrected geometry of near-horizon black 0-branes previously obtained by Hyakutake, we evaluate a Euclidean string worldsheet hanging down in the geometry, which corresponds to a rectangular Wilson loop in the $SU(N)$ quantum mechanics with 16 supercharges at a finite temperature with finite $N$. We find that the potential energy defined by the Wilson loop increases due to the $1/N$ correction, therefore the quantum gravity correction weakens the gravitational attraction., Comment: 15 pages, 9 figures

We provide a deep Boltzmann machine (DBM) for the AdS/CFT correspondence. Under the philosophy that the bulk spacetime is a neural network, we give a dictionary between those, and obtain a restricted DBM as a discretized bulk scalar field theory in curved geometries. The probability distribution as training data is the generating functional of the boundary quantum field theory, and it trains neural network weights which are the metric of the bulk geometry. The deepest layer implements black hole horizons, and an employed regularization for the weights is an Einstein action. A large $N_c$ limit in holography reduces the DBM to a folded feed-forward architecture. We also neurally implement holographic renormalization into an autoencoder. The DBM for the AdS/CFT may serve as a platform for studying mechanisms of spacetime emergence in holography., Comment: 13 pages, 5 figures

It is challenging to quantify chaos of QCD, because non-perturbative QCD accompanies non-local observables. By using holography, we find that QCD strings at large $N_c$ and strong coupling limit exhibit chaos, and measure their Lyapunov exponent at zero temperature. A pair of a quark and an antiquark separated by $L_q$ in the large $N_c$ QCD is dual to a Nambu-Goto string hanging from the spatial boundary of the D4-soliton geometry. We numerically solve the motion of the string after putting a pulse force on its boundaries. The chaos is observed for the amplitude of the force larger than a certain lower bound. The bound increases as $L_q$ grows, and its dependence is well approximated by a hypothesis that the chaos originates in the endpoints of the QCD string., Comment: 12 pages, 13 figures

A new method to study nuclear physics via holographic QCD is proposed. Multiple baryons in the Sakai-Sugimoto background are described by a matrix model which is a low energy effective theory of D-branes of the baryon vertices. We study the quantum mechanics of the matrix model and calculate the eigenstates of the Hamiltonian. The obtained states are found to coincide with known nuclear and baryonic states, and have appropriate statistics and charges. Calculated spectra of the baryon/nucleus for small baryon numbers show good agreement with experimental data. For hyperons, the Gell-Mann--Okubo formula is approximately derived. Baryon resonances up to spin $5/2$ and isospin $5/2$ and dibaryon spectra are obtained and compared with experimental data. The model partially explains even the magic numbers of light nuclei, $N=2,8$ and $20$., Comment: 65 pages; v2: 69 pages, discussions on the magic number are clarified; v3: 75 pages, published version

Thermal states in some quantum field theories (QFTs) correspond to black holes in asymptotically AdS spacetime in the AdS/CFT correspondence. We propose a direct procedure to construct holographic images of the black hole in the bulk from a given response function of the QFT on the boundary. The response function with respect to an external source corresponds to the asymptotic data of the bulk field generated by the source on the AdS boundary. According to the wave optics, we can obtain the images from the bulk field propagating in the bulk spacetime. For a thermal state on two-dimensional sphere dual to Schwarzschild-AdS$_4$ black hole, we demonstrate that the holographic images gravitationally lensed by the black hole can be constructed from the response function. In particular, the Einstein rings on the image can be clearly observed and their radius depends on the total energy of the QFT thermal state. These results are consistent with the size of the photon sphere of the black hole calculated in geometrical optics. This implies that, if there exists a dual gravitational picture for a given quantum system, we would be able to probe existence of the dual black hole by the Einstein rings constructed from observables of the quantum system., Comment: 22 pages, 12 figures

We apply the relation between deep learning (DL) and the AdS/CFT correspondence to a holographic model of QCD. Using a lattice QCD data of the chiral condensate at a finite temperature as our training data, the deep learning procedure holographically determines an emergent bulk metric as neural network weights. The emergent bulk metric is found to have both a black hole horizon and a finite-height IR wall, so shares both the confining and deconfining phases, signaling the cross-over thermal phase transition of QCD. In fact, a quark antiquark potential holographically calculated by the emergent bulk metric turns out to possess both the linear confining part and the Debye screening part, as is often observed in lattice QCD. From this we argue the discrepancy between the chiral symmetry breaking and the quark confinement in the holographic QCD. The DL method is shown to provide a novel data-driven holographic modeling of QCD, and sheds light on the mechanism of emergence of the bulk geometries in the AdS/CFT correspondence., Comment: 15 pages, 14 figures

In this paper, we argue that holographic complexity should be a basis-dependent quantity. Computational complexity of a state is defined as a minimum number of gates required to obtain that state from the reference state. Due to this minimality, it satisfies the triangle inequality, and can be regarded as a (discrete version of) distance in the Hilbert space. However, we show a no-go theorem that any basis-independent distance cannot reproduce the behavior of the holographic complexity. Therefore, if holographic complexity is dual to a distance in the Hilbert space, it should be basis-dependent, i.e., it is not invariant under a change of the basis of the Hilbert space., Comment: 7 pages

Measuring chaos of QCD-like theories is a challenge for formulating a novel characterization of quantum gauge theories. We define a chaos phase diagram of QCD allowing us to locate chaos in the parameter space of energy of homogeneous meson condensates and the QCD parameters such as pion/quark mass. We draw the chaos phase diagrams obtained in two ways: first, by using a linear sigma model, varying parameters of the potential, and second, by using the D4/D6 holographic QCD, varying the number of colors $N_c$ and the 't Hooft coupling constant $\lambda$. A scaling law drastically simplifies our analyses, and we discovered that the chaos originates in the maximum of the potential, and larger $N_c$ or larger $\lambda$ diminishes the chaos., Comment: 25 pages, 7 figures

To find the origin of chaos near black hole horizon in string-theoretic AdS/CFT correspondence, we perform a chaos analysis of a suspended string in AdS black hole backgrounds. It has a definite CFT interpretation: chaos of Wilson loops, or in other words, sensitive time-evolution of a quark antiquark force in thermal gauge theories. Our nonlinear numerical simulation of the suspended Nambu-Goto string shows chaos, which would be absent in pure AdS background. The calculated Lyapunov exponent $\lambda$ satisfies the universal bound $\lambda \leq 2\pi T_{\rm H}$ for the Hawking temperature $T_{\rm H}$. We also analyze a toy model of a rectangular string probing the horizon and show that it contains a universal saddle characterized by the surface gravity $2\pi T_{\rm H}$. Our work demonstrates that the black hole horizon is the origin of the chaos, and suggests a close interplay between chaos and quark deconfinement., Comment: 12 pages, 7 figures

We present a deep neural network representation of the AdS/CFT correspondence, and demonstrate the emergence of the bulk metric function via the learning process for given data sets of response in boundary quantum field theories. The emergent radial direction of the bulk is identified with the depth of the layers, and the network itself is interpreted as a bulk geometry. Our network provides a data-driven holographic modeling of strongly coupled systems. With a scalar $\phi^4$ theory with unknown mass and coupling, in unknown curved spacetime with a black hole horizon, we demonstrate our deep learning (DL) framework can determine them which fit given response data. First, we show that, from boundary data generated by the AdS Schwarzschild spacetime, our network can reproduce the metric. Second, we demonstrate that our network with experimental data as an input can determine the bulk metric, the mass and the quadratic coupling of the holographic model. As an example we use the experimental data of magnetic response of a strongly correlated material Sm$_{0.6}$Sr$_{0.4}$MnO$_3$. This AdS/DL correspondence not only enables gravity modeling of strongly correlated systems, but also sheds light on a hidden mechanism of the emerging space in both AdS and DL., Comment: 7 pages, 6 figures; Supplemental material: 8 pages, 5 figures

Quantum complexity is conjectured to probe inside of black hole horizons (or wormhole) via gauge gravity correspondence. In order to have a better understanding of this correspondence, we study time evolutions of complexities for generic Abelian pure gauge theories. For this purpose, we discretize $U(1)$ gauge group as $\mathbf{Z}_N$ and also continuum spacetime as lattice spacetime, and this enables us to define a universal gate set for these gauge theories, and evaluate time evolutions of the complexities explicitly. We find that for a generic class of diagonal Hamiltonians to achieve a large complexity $\sim \exp(\mbox{entropy})$, which is one of the conjectured criteria necessary to have a dual black hole, the Abelian gauge theory needs to be maximally nonlocal., Comment: 53 pages, 13 figures. v2: minor improvements. v3: added clarifications

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time., Comment: 30 pages, 13 figures

We study an exotic state which is localized only at an intersection of edges of a topological material. This "edge-of-edge" state is shown to exist generically. We construct explicitly generic edge-of-edge states in 5-dimensional Weyl semimetals and their dimensional reductions, such as 4-dimensional topological insulators of class A and 3-dimensional chiral topological insulators of class AIII. The existence of the edge-of-edge state is due to a topological charge of the edge states. The notion of the Berry connection is generalized to include the space of all possible boundary conditions, where Chern-Simons forms are shown to be nontrivial., Comment: 11 pages, 1 figure, references added

Floquet states can be realized in quantum systems driven by continuous time-periodic perturbations. It is known that a state known as the Floquet Weyl semimetal can be realized when free Dirac fermions are placed in a rotating electric field. What will happen if strong interaction is introduced to this system? Will the interaction wash out the characteristic features of Weyl semimetals such as the Hall response? Is there a steady state and what is its thermodynamic behavior? We answer these questions using AdS/CFT correspondence in the $\mathcal{N}=2$ supersymmetric massless QCD in a rotating electric field in the large $N_c$ limit realizing the first example of a "holographic Floquet state". In this limit, gluons not only mediate interaction, but also act as an energy reservoir and stabilize the nonequilibrium steady state (NESS). We obtain the electric current induced by a rotating electric field: In the high frequency region, the Ohm's law is satisfied, while we recover the DC nonlinear conductivity at low frequency, which was obtained holographically in a previous work. The thermodynamic properties of the NESS, e.g., fluctuation-dissipation relation, is characterized by the effective Hawking temperature that is defined from the effective horizon giving a holographic meaning to the "periodic thermodynamic" concept. In addition to the strong (pump) rotating electric field, we apply an additional weak (probe) electric field in the spirit of the pump-probe experiments done in condensed matter experiments. Weak DC and AC probe analysis in the background rotating electric field shows Hall currents as a linear response, therefore the Hall response of Floquet Weyl semimetals survives at the strong coupling limit. We also find frequency mixed response currents, i.e., a heterodyning effect, characteristic to periodically driven Floquet systems., Comment: 30 pages, 12 figures

Motion of a particle near a horizon of a spherically symmetric black hole is shown to possess a universal Lyapunov exponent of a chaos provided by its surface gravity. To probe the horizon, we introduce electromagnetic or scalar force to the particle so that it does not fall into the horizon. There appears an unstable maximum of the total potential where the evaluated maximal Lyapunov exponent is found to be independent of the external forces and the particle mass. The Lyapunov exponent is universally given by the surface gravity of the black hole. Unless there are other sources of a chaos, the Lyapunov exponent is subject to an inequality $\lambda \leq 2\pi T_{\rm BH}/\hbar$, which is identical to the bound recently discovered by Maldacena, Shenker and Stanford., Comment: 12 pages, 4 figures; v2: references added, numerical plots in Fig. 4 corrected

We find that generic boundary conditions of Weyl semimetal is dictated by only a single real parameter, in the continuum limit. We determine how the energy dispersions (the Fermi arcs) and the wave functions of edge states depend on this parameter. Lattice models are found to be consistent with our generic observation. Furthermore, the enhanced parameter space of the boundary condition is shown to support a novel topological number., Comment: 17 pages, 11 figures; v3: section of lattice model modified, figures added, references added, typos corrected