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arXiv:1902.04079 · Physical review. B./Physical review. B

Multifaceted machine learning of competing orders in disordered interacting systems

Michael Matty, Yi Zhang, Zlatko Papić, Eun-Ah Kim

Published Oct 28, 2019• 12 citations
arXivHTMLPDF

Abstract

There are growing efforts to use artificial neural networks (ANNs) for phase recognition. The authors target the problem of competition between topological order and broken symmetry states in the presence of disorder. The lack of traditional diagnostics for this problem makes any progress using ANNs particularly significant. The authors successfully obtain a phase diagram consisting of fractional quantum Hall, charge density wave, and localized phases using multifaceted inputs that inform defining traits of these phases to the ANN.

DOI
10.1103/physrevb.100.155141

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There are growing efforts to use artificial neural networks (ANNs) for phase recognition. The authors target the problem of competition between topological order and broken symmetry states in the presence of disorder. The lack of traditional diagnostics for this problem makes any progress using ANNs particularly significant. The authors successfully obtain a phase diagram consisting of fractional quantum Hall, charge density wave, and localized phases using multifaceted inputs that inform defining traits of these phases to the ANN.
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Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge.The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network.When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of nonlocal properties.Here, we introduce quantum loop topography (QLT): a procedure of constructing a multidimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps.The loop configuration is guided by the characteristic response for defining the phase, which is Hall conductivity for the cases at hand.Feeding QLT to a fully connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish the Chern insulator and the fractional Chern insulator from trivial insulators with high fidelity.In addition to establishing the first case of obtaining a phase diagram with a topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.
Fractional quantum Hall systems host strongly correlated topological phases with potential applications in topological quantum computation, yet identifying their microscopic ground states remains challenging because competing phases are often nearly degenerate and Landau-level mixing is difficult to treat. Here, we introduce a symmetry-adapted neural-network wavefunction that describes Abelian and non-Abelian states while incorporating all Landau levels. The method accurately captures composite-fermion liquids and the emergence of Wigner crystallization at low filling. When the first Landau level is partially filled, finite-size overlaps and entanglement spectra provide evidence consistent with 2ˉk1k+1\bar{2}^{k}1^{k+1}2ˉk1k+1-type parton phases at ν=4/3ν=4/3ν=4/3 and 7/57/57/5, while the ν=3/2ν=3/2ν=3/2 state exhibits anti-Pfaffian-like entanglement structure. These results demonstrate that neural-network wavefunctions offer a flexible framework for studying competing topological orders in fractional quantum Hall systems.
The identification of topological phases in disordered systems is a significant field in condensed matter physics, where disorder breaks translational symmetry and invalidates conventional topological invariants defined in momentum space. Currently, machine learning offers promising alternatives in the identification of topological phases. In this work, by regarding the population dynamics as input data, we used feedforward neural networks (FNNs), vanilla recurrent neural networks (RNNs), and long short-term memory (LSTM) networks to identify the topology in the Su–Schrieffer–Heeger (SSH) model and the disordered SSH model, respectively. We also compared the identification capabilities of different neural networks using different input data. Our results show that FNN has the lowest training cost and a relatively high prediction accuracy. However, when increasing the time length and reducing the number of time points, vanilla RNN has higher prediction accuracy. Furthermore, we develop an interactive web-based tool, enabling real-time topological phase prediction based on user-specified parameters. This study not only lays the foundation for researchers to identify topology by using population dynamics as the input data of neural networks but also provides an accessible platform to support data-driven exploration of complex quantum phases.
Neural-network-based machine learning is emerging as a powerful tool for obtaining phase diagrams when traditional regression schemes using local equilibrium order parameters are not available, as in many-body localized (MBL) or topological phases. Nevertheless, instances of machine learning offering new insights have been rare up to now. Here we show that a single feed-forward neural network can decode the defining structures of two distinct MBL phases and a thermalizing phase, using entanglement spectra obtained from individual eigenstates. For this, we introduce a simplicial geometry-based method for extracting multipartite phase boundaries. We find that this method outperforms conventional metrics for identifying MBL phase transitions, revealing a sharper phase boundary and shedding new insight on the topology of the phase diagram. Furthermore, the phase diagram we acquire from a single disorder configuration confirms that the machine-learning-based approach we establish here can enable speedy exploration of large phase spaces that can assist with the discovery of new MBL phases. To our knowledge, this Letter represents the first example of a standard machine learning approach revealing new information on phase transitions.
Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge. The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network. When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of nonlocal properties. Here, we introduce quantum loop topography (QLT): a procedure of constructing a multidimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps. The loop configuration is guided by the characteristic response for defining the phase, which is Hall conductivity for the cases at hand. Feeding QLT to a fully connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish the Chern insulator and the fractional Chern insulator from trivial insulators with high fidelity. In addition to establishing the first case of obtaining a phase diagram with a topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.