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arXiv:2008.12331 · Crystals

Neural Network Solver for Small Quantum Clusters

Nicholas Walker, Samuel Kellar, Yi Zhang, Ka-Ming Tam, Juana Moreno

Published Sep 6, 2022• 5 citations
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Abstract

Machine learning approaches have recently been applied to the study of various problems in physics. Most of these studies are focused on interpreting the data generated by conventional numerical methods or the data on an existing experimental database. An interesting question is whether it is possible to use a machine learning approach, in particular a neural network, for solving the many-body problem. In this paper, we present a neural network solver for the single impurity Anderson model, the paradigm of an interacting quantum problem in small clusters. We demonstrate that the neural-network-based solver provides quantitative accurate results for the spectral function as compared to the exact diagonalization method. This opens the possibility of utilizing the neural network approach as an impurity solver for other many-body numerical approaches, such as the dynamical mean field theory.

DOI
10.3390/cryst12091269

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Machine learning approaches have recently been applied to the study of various problems in physics. Most of these studies are focused on interpreting the data generated by conventional numerical methods or the data on an existing experimental database. An interesting question is whether it is possible to use a machine learning approach, in particular a neural network, for solving the many-body problem. In this paper, we present a neural network solver for the single impurity Anderson model, the paradigm of an interacting quantum problem in small clusters. We demonstrate that the neural-network-based solver provides quantitative accurate results for the spectral function as compared to the exact diagonalization method. This opens the possibility of utilizing the neural network approach as an impurity solver for other many-body numerical approaches, such as the dynamical mean field theory.
With the evolution of numerical methods, we are now aiming at not only qualitative understanding but also quantitative prediction and design of quantum many-body phenomena. As a novel numerical approach, machine learning techniques have been introduced in 2017 to analyze quantum many-body problems. Since then, proposed various novel approaches have opened a new era, in which challenging and fundamental problems in physics can be solved by machine learning methods. Especially, quantitative and accurate estimates of material-dependent physical properties of strongly correlated matter have now become realized by combining first-principles calculations with highly accurate quantum many-body solvers developed with the help of machine learning methods. Thus developed quantitative description of electron correlations will constitute a key element of materials science in the next generation.
Quantum impurity solvers are the computational bottleneck of quantum embedding approaches to correlated materials, such as dynamical mean-field theory (DMFT). We show that neural networks trained on synthetic, material-agnostic data learn the impurity mapping from hybridization functions and local interactions to Green's functions with quantitative accuracy for both model systems and real materials, providing fast solvers for single- and multi-orbital models. Benchmarks against numerically controlled quantum Monte Carlo show that the method reproduces the Mott transition, multi-orbital phase diagrams of Hubbard-Kanamori models, and the electronic properties of SrVO3_33​ and SrMnO3_33​. The learned solvers achieve orders-of-magnitude speedup and can initialize controlled calculations, dramatically accelerating DMFT while preserving accuracy.
The Anderson Impurity Model (AIM) is a canonical model of quantum many-body physics. Here we investigate whether machine learning models, both neural networks (NN) and kernel ridge regression (KRR), can accurately predict the AIM spectral function in all of its regimes, from empty orbital, to mixed valence, to Kondo. To tackle this question, we construct two large spectral databases containing approximately 410 000 and 600 000 spectral functions of the single-channel impurity problem. We show that the NN models can accurately predict the AIM spectral function in all of its regimes, with pointwise mean absolute errors down to 0.003 in normalized units. We find that the trained NN models outperform models based on KRR and enjoy a speedup on the order of 105{10}^{5}105 over traditional AIM solvers. The required size of the training set of our model can be significantly reduced using farthest point sampling in the AIM parameter space, which is important for generalizing our method to more complicated multichannel impurity problems of relevance to predicting the properties of real materials.
A neural-network approach to optimize the selection of Slater determinants in configuration interaction calculations for condensed-matter quantum many-body systems is developed. We exemplify our algorithm on the discrete version of the single-impurity Anderson model with up to 299 bath sites. Employing a neural network classifier and active learning, our algorithm enhances computational efficiency by iteratively identifying the most relevant Slater determinants for the ground state wave function. We benchmark our results against established methods and investigate the efficiency of our approach compared to another basis truncation scheme without a neural network. Our algorithm demonstrates a substantial improvement in the efficiency of determinant selection, yielding a more compact and computationally manageable basis without compromising accuracy. Given the straightforward application of our neural-network-supported selection scheme to other model Hamiltonians of quantum many-body clusters, our algorithm can significantly advance selective configuration interaction calculations in the context of correlated condensed matter.
A promising application of machine learning is the creation of low-cost surrogate models to mitigate computational bottlenecks in quantum many-body simulations. Here, we explore whether a neural network (NN) can be trained in the low-data regime, with one to two orders of magnitude fewer training examples than previous works, as an efficient substitute for the impurity solver in dynamical mean-field theory simulations of correlated electron models. We show that the NN solver achieves accuracy comparable to popular continuous-time quantum Monte Carlo (CT-QMC) impurity solvers when interpolating between samples within the training set. While the NN's performance decreases notably when extrapolating to lower temperatures outside the training distribution, its output still provides an excellent initial guess for input to more accurate CT-QMC impurity solvers, thus accelerating the time to solution up to a factor of five. We discuss our results in the context of rapid phase-space exploration.
Machine learning methods are applied to finding the Green's function of the Anderson impurity model, a basic model system of quantum many-body condensed-matter physics. Different methods of parametrizing the Green's function are investigated; a representation in terms of Legendre polynomials is found to be superior due to its limited number of coefficients and its applicability to state of the art methods of solution. The dependence of the errors on the size of the training set is determined. The results indicate that a machine learning approach to dynamical mean-field theory may be feasible.