Machine learning phases of matter

Abstract

Neural networks can be used to identify phases and phase transitions in condensed matter systems via supervised machine learning. Readily programmable through modern software libraries, we show that a standard feed-forward neural network can be trained to detect multiple types of order parameter directly from raw state configurations sampled with Monte Carlo. In addition, they can detect highly nontrivial states such as Coulomb phases, and if modified to a convolutional neural network, topological phases with no conventional order parameter. We show that this classification occurs within the neural network without knowledge of the Hamiltonian or even the general locality of interactions. These results demonstrate the power of machine learning as a basic research tool in the field of condensed matter and statistical physics. arXiv:1605.01735v1 [cond-mat.str-el] 5 May 20162 Condensed matter physics is the study of the collective behavior of massively complex assemblies of electrons, nuclei, magnetic moments, atoms or qubits [1]. This complexity is reflected in the size of the classical or quantum state space, which grows exponentially with the number of particles. This exponential growth is reminiscent of the "curse of dimensionality" commonly encountered in machine learning. That is, a target function to be learned requires an amount of training data that grows exponentially in the dimension (e.g. the number of image features). Despite this curse, the machine learning community has developed a number of techniques with remarkable abilities to recognize, classify, and characterize complex sets of data. In light of this success, it is natural to ask whether such techniques could be applied to the arena of condensed-matter physics, particularly in cases where the microscopic Hamiltonian contains strong interactions, where numerical simulations are typically employed in the study of phases and phase transitions [2,3]. We demonstrate that modern machine learning architectures, such as fully-connected and convolutional neural networks [4], can provide a complementary approach to identifying phases and phase transitions in a variety of systems in condensed matter physics. The training of neural networks on data sets obtained by Monte Carlo sampling provides a particularly powerful and simple framework for the supervised learning of phases and phase boundaries in physical models, and can be easily built from readily-available tools such as Theano [5] or TensorFlow [6] libraries.Conventionally, the study of phases in condensed matter systems is performed with the help of tools that have been carefully designed to elucidate the underlying physical structures of various states. Among the most powerful are Monte Carlo simulations, which consist of two steps: a stochastic importance sampling over state space, and the evaluation of estimators for physical quantities calculated from these samples [3]. These estimators are constructed based on a variety of physical impetuses; e.g. the ready availability of an analogous experi...

Artificial neural network in cosmic landscape

“…As one of the most promising area in computer science, machine learning (or artificial intelligence) [29][30][31] is widely used to understand difference areas of physics, including condensed matter phases [39], high energy experiment [40], energy landscape [41][42][43], particle phenomenology [44], tensor networks [45] and cosmic non-Gaussianities [46]. In recent research string theorists also find that machine learning algorithm is efficient to study manifold data in the string landscape [47][48][49][50], which may give us the motivation to think about the learning algorithm landscape from cosmological point of view.…”

Learning phase transitions from dynamics

“…Such a finite-size scaling can indeed be successfully attempted using machine learned data [1,15], and provides a useful and interesting alternative for locating a phase boundary.…”

“…So far, these methods have relied only on static properties of the underlying physical systems, such as raw state configurations sampled from Monte Carlo simulations [1,15] or entanglement spectra obtained using exact diagonalization [3,11,17]. However, dynamics of physical observables are often more accessible experimentally.…”

“…Machine learning is emerging as a novel tool for identifying phases of matter [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. At its core, this problem can be cast as a classification problem in which data obtained from physical systems are assigned a class (i.e., a phase) using machine learning methods.…”

“…At its core, this problem can be cast as a classification problem in which data obtained from physical systems are assigned a class (i.e., a phase) using machine learning methods. This approach has enabled the autonomous detection of order parameters [2,5,6], phase transitions [1,3], and entire phase diagrams [4,7,16,17]. Simultaneous research effort at the interface between machine learning and many-body physics has focused on the use of neural networks for efficient representations of quantum wave functions [18][19][20][21][22][23][24][25][26], drawing a parallel between deep networks and the renormalization group [27][28][29].…”