A Convolutional Neural Network (CNN) is designed to study correlation between the temperature and the spin configuration of the 2 dimensional Ising model. Our CNN is able to find the characteristic feature of the phase transition without prior knowledge. Also a novel order parameter on the basis of the CNN is introduced to identify the location of the critical temperature; the result is found to be consistent with the exact value.Studies of phase transition are connected to various areas among theoretical/experimental physics. 1-7) Calculating order parameters is one of the conventional ways to define phases and phase transitions. However, some phases like topological phases 8) do not have any clear order parameters. Even if there are certain theoretical order parameters like entanglement entropy, 9,10) they are difficult to measure in experiments.Machine learning (ML) techniques are useful to resolve this undesirable situation. In fact, ML techniques have been already applied to various problems in theoretical physics: finding approximate potential surface, 11) a study of transition in glassy liquids, 12) solving mean-field equations 14) and quantum many-body systems, 13, 15) a study of topological phases. 16) Especially, ML techniques based on convolutional neural network (CNN) have been developing since the recent groundbreaking record 17) in ImageNet Large Scale Visual Recognition Challenge 2012 (ILSVRC2012), 18) and it is applied to investigate phases of matters with great successes on classifications of phases in 2D systems [19][20][21] and 3D systems. 22,23) It is even possible to draw phase diagrams. 23) In these previous works, however, one needs some informations of the answers for the problems a priori. For example, to classify phases of a system, the training process requires the values of critical temperatures or the location of phase boundaries. This fact prevents applications of the ML techniques to unknown systems so far.The learning process without any answers is called unsupervised learning. Indeed, there are known results on detecting the phase transitions based on typical unsupervised learning architectures called autoencoder which is equivalent to principal component analysis 32) and its variant called variational autoencoder. ?) These architectures encode informations of given samples to lower dimensional vectors, and it is pointed out that such encoding process is similar to encoding physical state informations to order parameters of the systems. However, it is not evident whether the latent variables provide the critical temperature.We propose a novel but simple prescription to estimate the critical temperature of the system via neural network (NN) based on ML techniques without a priori knowledge of the order parameter. Throughout this letter, we focus on the fer- * akinori.tanaka@riken.jp † akio.tomiya@mail.ccnu.edu.cn m m Fig. 1. '(Color online)' Plots of weight matrix components in convolutional neural network (left) and fully connected neural network (right). Horizontal axis correspon...

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PACS 71.10.Fd -Lattice fermion models (Hubbard model, etc.) PACS 71.10.-w -Theories and models of many-electron systems PACS 05.30.Fk -Fermion systems and electron gas Abstract. -We introduce and study two classes of Hubbard models with magnetic flux or with spin-orbit coupling, which have a flat lowest band separated from other bands by a nonzero gap. We study the Chern number of the flat bands, and find that it is zero for the first class but can be nontrivial in the second. We also prove that the introduction of on-site Coulomb repulsion leads to ferromagnetism in both the classes.Introduction. -Motivated by the recent discovery of quantum spin Hall effect in band insulators with strong spin-orbit coupling (SOC) [1][2][3], the topological classification of non-interacting electron systems has attracted a renewed interest [4][5][6]. The states of the systems are characterized by the topological numbers linked to the presence or absence of gapless edge modes carrying electronic or spin current. In integer quantum Hall systems [7], the first Chern number is directly connected to the quantized Hall conductance [8], which is one of the most famous examples of time reversal breaking insulators. On the other hand, in the recently found time reversal invariant insulators, the states are classified by the Z 2 topological number. Since a topological number remains invariant as long as the energy gap does not collapse, adiabatic transformation from the original model to a flat-band model, where all the bands are dispersionless, provides a useful tool for the classification [4][5][6].

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 φ 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 Sm0.6Sr0.4MnO3. 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. FIG. 1: The AdS/CFT and the DL. Top: a typical view of the AdS/CFT correspondence. The CFT at a finite temperature lives at a boundary of asymptotically AdS spacetime with a black hole horizon at the other end. Bottom: a typical neural network of a deep learning.horizon condition. Therefore, a successful machine learning results in a concrete metric of a holographic modeling of the system measured by the experiment [67]. We call this implementation of the holographic model into the deep neural network as AdS/DL correspondence.We check that the holographic DL modeling nicely arXiv:1802.08313v1 [hep-th]

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.

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The Hubbard model on the kagome lattice has highly degenerate ground states (the flat lowest band) in the corresponding single-electron problem and exhibits the so-called flat-band ferromagnetism in the many-electron ground states as was found by Mielke [J. Phys. A 24, L73 (1991)]]. Here we study the model obtained by adding extra hopping terms to the above model. The lowest single-electron band becomes dispersive, and there is no band gap between the lowest band and the other band. We prove that, at half filling of the lowest band, the ground states of this perturbed model remain saturated ferromagnetic if the lowest band is nearly flat.