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New Journal of Physics

Learning pairing symmetries in disordered superconductors using spin-polarized local density of states

Liang Chen, Chenxi Wang, Rong-Sheng Han, Ye-Qi Zhang

Published Mar 23, 2020• 1 citation
Publisher

Abstract

Abstract We construct an artificial neural network to study the pairing symmetries in disordered superconductors. For Hamiltonians on square lattice with s-wave, d-wave, and nematic pairing potentials, we use the spin-polarized local density of states near a magnetic impurity in the clean system to train the neural network. We find that, when the depth of the artificial neural network is sufficient large, it will have the power to predict the pairing symmetries in disordered superconductors. In a large parameter regime of the potential disorder, the artificial neural network predicts the correct pairing symmetries with relatively high confidences.

DOI
10.1088/1367-2630/ab8261

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Abstract We construct an artificial neural network to study the pairing symmetries in disordered superconductors. For Hamiltonians on square lattice with s-wave, d-wave, and nematic pairing potentials, we use the spin-polarized local density of states near a magnetic impurity in the clean system to train the neural network. We find that, when the depth of the artificial neural network is sufficient large, it will have the power to predict the pairing symmetries in disordered superconductors. In a large parameter regime of the potential disorder, the artificial neural network predicts the correct pairing symmetries with relatively high confidences.
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.
Understanding the phases of a model usually requires knowledge of their characteristic features, which are nonlocal in topologically ordered systems. Here, the authors reframe the phase classification problem in disordered topological superconductors as a data-driven task, motivated by the recent surge of interest in the application of machine-learning techniques including deep learning. It is demonstrated that an artificial neural network learns to extract the essence of the clean system and successfully distinguishes phases even under disorder by statistical recovery of translational symmetry.
The applicability of artificial neural networks (ANNs) is typically limited to the models they are trained with and little is known about their generalizability, which is a pressing issue in the practical application of trained ANNs to unseen problems. Here, by using the task of identifying phase transitions in spin models, we establish a systematic generalizability such that simple ANNs trained with the two-dimensional ferromagnetic Ising model can be applied to the ferromagnetic q q -state Potts model in different dimensions for q \geq 2 q ≥ 2 . The same scheme can be applied to the highly nontrivial antiferromagnetic q q -state Potts model. We demonstrate that similar results can be obtained by reducing the exponentially large state space spanned by the training data to one that comprises only three representative configurations artificially constructed through symmetry considerations. We expect our findings to simplify and accelerate the development of machine learning-assisted tasks in spin-model related disciplines in physics and materials science.
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.
Topological invariants allow one to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wave functions under twisted boundary conditions. However, those procedures do not allow one to calculate a topological invariant by evaluating the system locally, and thus require information about the wave functions in the whole system. Here we show that artificial neural networks can be trained to identify the topological order by evaluating a local projection of the density matrix. We demonstrate this for two different models, a one-dimensional topological superconductor and a two-dimensional quantum anomalous Hall state, both with spatially modulated parameters. Our neural network correctly identifies the different topological domains in real space, predicting the location of in-gap states. By combining a neural network with a calculation of the electronic states that uses the kernel polynomial method, we show that the local evaluation of the invariant can be carried out by evaluating a local quantity, in particular for systems without translational symmetry consisting of tens of thousands of atoms. Our results show that supervised learning is an efficient methodology to characterize the local topology of a system.
The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ans\"atz, we study the frustrated spin-1/$2{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ans\"atz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.