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arXiv:2301.07906 · Physical review. B./Physical review. B

Machine learning the spectral function of a hole in a quantum antiferromagnet

Jackson Lee, Matthew R. Carbone, Wei‐Guo Yin

Published May 16, 2023• 7 citations
arXivHTMLPDF

Abstract

Understanding charge motion in a background of interacting quantum spins is a fundamental problem in quantum many-body physics. The most extensively studied model for this problem is the so-called $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}{t}^{\ensuremath{'}\ensuremath{'}}\text{\ensuremath{-}}J$ model, where the determination of the parameter ${t}^{\ensuremath{'}}$ in the context of cuprate superconductors is challenging. Here we present a theoretical study of the spectral functions of a mobile hole in the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}{t}^{\ensuremath{'}\ensuremath{'}}\text{\ensuremath{-}}J$ model using two machine-learning techniques: $K$-nearest neighbor regression (KNN) and a feed-forward neural network (FFNN). We employ the self-consistent Born approximation to generate a dataset of about $1.3\ifmmode\times\else\texttimes\fi{}{10}^{5}$ spectral functions. We show that, for the forward problem, both methods allow for the accurate and efficient prediction of spectral functions, allowing, e.g., rapid searches through parameter space. Furthermore, we find that for the inverse problem (inferring Hamiltonian parameters from spectra), the FFNN can, but the KNN cannot, accurately predict the model parameters using merely the density of states. Our results suggest that it may be possible to use deep-learning methods to predict materials parameters from experimentally measured spectral functions.

DOI
10.1103/physrevb.107.205132

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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.
In the past decades, methods to solve the ground state given a quantum many-body Hamiltonian have been well established. In this work, we consider an inverse problem and demonstrate that convolutional neural network (CNN) can predict the physical parameters of interacting Hamiltonians, such as coupling strengths and magnetic fields, providing the quantum many-body wave-functions as the ground states. We propose QubismNet that consists of two main parts: the Qubism map that visualizes the ground states (or the purified reduced density matrices) as images, and a CNN that maps the images to the target physical parameters. QubismNet exhibits impressive powers of learning and generalization on several quantum spin models. While the training samples are restricted to the states from certain ranges of the parameters, QubismNet can accurately predict the parameters of the states beyond such training regions. For instance, our results show that QubismNet can predict the magnetic fields near the critical point by learning from the states away from the critical vicinity. Our work provides a data-driven way to infer the Hamiltonians that give the designed ground states, and therefore would benefit the existing and future generations of quantum technologies such as Hamiltonian-based quantum simulations.
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Spectral density functions quantify how environmental modes couple to quantum systems and govern their open dynamics. Inferring such frequency-dependent functions from time-domain measurements is an ill-conditioned inverse problem. Here, we use exactly solvable spin-boson models with pure-dephasing and amplitude-damping channels to reconstruct spectral density functions from noisy simulated data. First, we introduce a parameter estimation approach based on machine learning regressors to infer Lorentzian and Ohmic-like spectral density parameters, quantifying robustness to noise. Second, we show that a cosine transform inversion yields a physics-consistent spectral prior estimation, which is refined by a constrained neural network enforcing positivity and correct asymptotic behaviour. Our neural network framework robustly reconstructs structured spectral densities by filtering simulated noisy signals and learning general functional dependencies.
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.
One-dimensional quantum spin chains provide a versatile platform for design of artificial quantum matter that enables breakthroughs in spintronics and quantum computing. These spin systems can be engineered at the atomic scale with scanning tunneling microscopy and their local excitations can be measured with electrically driven spin resonance. When engineering these system, a major open challenge is to extract parameters of the many-body spin Hamiltonian, since the extraction must be done indirectly. This thesis presents a machine learning technique that allows to infer the underlying Hamiltonian parameters via dynamical spin excitations. The methodology is based on the exploitation of many-body reconstructions around engineered quantum impurities adjacent to S=1/2 spin chains. These systems can be realized by hydrogenated titanium atoms in magnesium oxide films, featuring S=1/2 impurities next to the chain. The presented supervised methodology exploits the local dynamical correlator functions to provide the full many-body Hamiltonian as a result. Two neural networks using different quantum impurity placements are implemented. The networks allow to predict long range Heisenberg exchange interactions J2 and J3, anisotropic exchange interaction JZ, as well as antisymmetric Dzyaloshinskii-Moriya exchange interaction JDMI accurately for spin chain with a length of 21 spins from computational data. It is demonstrated that the approach is robust to noise in the spin excitations, offering an experimentally realistic strategy for Hamiltonian learning. It is also shown that different impurity placements offer better estimates for the parameters. With further research, the networks could be used to predict the parameters from experimental data. The introduced technique demonstrates the applicability of machine learning methodology to study nano-scale quantum systems and puts forward confined excitations in quantum magnets as a powerful strategy to understand the nature of quantum spin many-body models.