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arXiv:1506.08858

Machine learning for many-body physics: efficient solution of dynamical mean-field theory

Louis-François Arsenault, O. Anatole von Lilienfeld, Andrew J. Millis

Published Jun 29, 2015• 29 citations
arXivHTMLPDF

Abstract

Machine learning methods for solving the equations of dynamical mean-field theory are developed. The method is demonstrated on the three dimensional Hubbard model. The key technical issues are defining a mapping of an input function to an output function, and distinguishing metallic from insulating solutions. Both metallic and Mott insulator solutions can be predicted. The validity of the machine learning scheme is assessed by comparing predictions of full correlation functions, of quasi-particle weight and particle density to values directly computed. The results indicate that with modest further development, machine learning approach may be an attractive computational efficient option for real materials predictions for strongly correlated systems.

DOI
10.48550/arxiv.1506.08858

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Machine learning methods for solving the equations of dynamical mean-field theory are developed. The method is demonstrated on the three dimensional Hubbard model. The key technical issues are defining a mapping of an input function to an output function, and distinguishing metallic from insulating solutions. Both metallic and Mott insulator solutions can be predicted. The validity of the machine learning scheme is assessed by comparing predictions of full correlation functions, of quasi-particle weight and particle density to values directly computed. The results indicate that with modest further development, machine learning approach may be an attractive computational efficient option for real materials predictions for strongly correlated systems.
Machine learning opens new avenues for modeling correlated materials. Quantum embedding approaches, such as dynamical mean-field theory (DMFT), provide corrections to first-principles calculations for strongly correlated materials, which are poorly described at lower levels of theory. Such embedding approaches are computationally demanding on classical computing architectures, and hence remain restricted to small systems, which limits the scope of applicability without exceptional computational resources. Here we outline a data-driven machine-learning process for solving the Anderson impurity model (AIM)---the central component of DMFT calculations. The key advance is the use of an ensemble error-correction approach to generate fast and accurate solutions of AIM. An example calculation of the Mott transition using DMFT in the single band Hubbard model is given as an example of the technique, and is validated against the most accurate available method. This approach is called data-driven dynamical mean-field theory (d3MFT{\mathbf{d}}^{3}\mathbf{MFT}d3MFT).
Modeling many-body quantum systems is widely regarded as one of the most promising applications for near-term noisy quantum computers. However, in the near term, system size limitation will remain a severe barrier for applications in materials science or strongly correlated systems. A promising avenue of research is to combine many-body physics with machine learning for the classification of distinct phases. I present a workflow that synergizes quantum computing, many-body theory, and quantum machine learning (QML) for studying strongly correlated systems. In particular, it can capture a putative quantum phase transition of the stereotypical strongly correlated system, the Hubbard model. Following the recent proposal of the hybrid classical-quantum algorithm for the two-site dynamical mean-field theory (DMFT), I present a modification that allows the self-consistent solution of the single bath site DMFT. The modified algorithm can be generalized for multiple bath sites. This approach is used to generate a database of zero-temperature wavefunctions of the Hubbard model within the DMFT approximation. I then use a QML algorithm to distinguish between the metallic phase and the Mott insulator phase to capture the metal-to-Mott insulator phase transition. I train a recently proposed quantum convolutional neural network(QCNN) and then utilize the QCNN as a quantum classifier to capture the phase transition region. This work provides a recipe for application to other phase transitions in strongly correlated systems and represents an exciting application of small-scale quantum devices realizable with near-term technology.
Modeling many-body quantum systems is widely regarded as one of the most promising applications for near-term noisy quantum computers. However, in the near term, system size limitation will remain a severe barrier for applications in materials science or strongly correlated systems. A promising avenue of research is to combine many-body physics with machine learning for the classification of distinct phases. We present a workflow that synergizes quantum computing, many-body theory, and quantum machine learning (QML) for studying strongly correlated systems. In particular, it can capture a putative quantum phase transition of the stereotypical strongly correlated system, the Hubbard model. Following the recent proposal of the hybrid quantum-classical algorithm for the two-site dynamical mean-field theory (DMFT), we present a modification that allows the self-consistent solution of the single bath site DMFT. The modified algorithm can be generalized for multiple bath sites. This approach is used to generate a database of zero-temperature wavefunctions of the Hubbard model within the DMFT approximation. We then use a QML algorithm to distinguish between the metallic phase and the Mott insulator phase to capture the metal-to-Mott insulator phase transition. We train a recently proposed quantum convolutional neural network (QCNN) and then utilize the QCNN as a quantum classifier to capture the phase transition region. This work provides a recipe for application to other phase transitions in strongly correlated systems and represents an exciting application of small-scale quantum devices realizable with near-term technology.
Learning from data has led to a paradigm shift in computational materials science. In particular, it has been shown that neural networks can learn the potential energy surface and interatomic forces through examples, thus bypassing the computationally expensive density functional theory calculations. Combining many-body techniques with a deep-learning approach, we demonstrate that a fully connected neural network is able to learn the complex collective behavior of electrons in strongly correlated systems. Specifically, we consider the Anderson-Hubbard (AH) model, which is a canonical system for studying the interplay between electron correlation and strong localization. The ground states of the AH model on a square lattice are obtained using the real-space Gutzwiller method. The obtained solutions are used to train a multitask multilayer neural network, which subsequently can accurately predict quantities such as the local probability of double occupation and the quasiparticle weight, given the disorder potential in the neighborhood as the input.
We present SCALINN—strongly correlated approach with language-inspired neural network—as a method for solving the Anderson impurity model and reducing the computational cost of dynamical mean-field theory calculations. Inspired by the success of generative Transformer networks in natural language processing, SCALINN utilizes an in-house modified Transformer network in order to learn correlated Matsubara Green's functions, which act as solutions to the impurity model. This is achieved by providing the network with low-cost Matsubara Green's functions, thereby overcoming the computational cost of high accuracy solutions. Across different temperatures and interaction strengths, the performance of SCALINN is demonstrated in both physical observables (spectral function, Matsubara Green's functions, quasiparticle weight), and the mean squared error cost values of the neural network, showcasing the network's ability to accelerate Green's function based calculations of correlated materials by predicting DMFT solver solutions, which—given a sufficiently comprehensive training database—can be applied across materials and regimes, enabling DFT + DMFT calculations without prior knowledge of the material properties.
Abstract We demonstrate that a machine learning framework based on kernel ridge regression can encode and predict the self-energy of one-dimensional Hubbard models using only mean-field features such as static and dynamic Hartree–Fock quantities and first-order GW calculations. This approach is applicable across a wide range of on-site Coulomb interaction strengths U / t , ranging from weakly interacting systems ( U / t ≪ 1 ) to strong correlations ( U / t > 8 ). The predicted self-energy is transformed via Dyson’s equation and analytic continuation to obtain the real-frequency Green’s function, which allows access to the spectral function and density of states. This method can be used for nearest-neighbor interactions t and long-range hopping terms t ′, t ′′, and t ′ ′ ′ .