Establishing quantum spin liquid physics in microscopic models is a daunting task due to a variety of competing low-energy states, requiring sophisticated computational approaches to find the true ground state. Here, the authors present such a technique, based on neural networks with space-group symmetry, to obtain significantly improved ground-state wave functions of frustrated Heisenberg models. The authors also use this approach to find excited states and provide a blueprint for testing analytical predictions from spin-liquid theories.
Establishing quantum spin liquid physics in microscopic models is a daunting task due to a variety of competing low-energy states, requiring sophisticated computational approaches to find the true ground state. Here, the authors present such a technique, based on neural networks with space-group symmetry, to obtain significantly improved ground-state wave functions of frustrated Heisenberg models. The authors also use this approach to find excited states and provide a blueprint for testing analytical predictions from spin-liquid theories.
We show that neural quantum states based on very deep (4--16-layered) neural networks can outperform state-of-the-art variational approaches on highly frustrated quantum magnets, including quantum-spin-liquid candidates. We focus on group convolutional neural networks (GCNNs) that allow us to impose space-group symmetries on our ansätze. We achieve state-of-the-art ground-state energies for the J1−J2 Heisenberg models on the square and triangular lattices, in both ordered and spin-liquid phases, and discuss ways to access low-lying excited states in nontrivial symmetry sectors. We also compute spin and dimer correlation functions for the quantum paramagnetic phase on the triangular lattice, which do not indicate either conventional or valence-bond ordering.
Artificial neural networks have been recently introduced as a general ansatz to represent many-body wave functions. In conjunction with variational Monte Carlo calculations, this ansatz has been applied to find Hamiltonian ground states and their energies. Here, we provide extensions of this method to study excited states, a central task in several many-body quantum calculations. First, we give a prescription that allows us to target eigenstates of a (nonlocal) symmetry of the Hamiltonian. Second, we give an algorithm to compute low-lying excited states without symmetries. We demonstrate our approach with both restricted Boltzmann machines and feed-forward neural networks. Results are shown for the one-dimensional spin-1/2 Heisenberg model, and for the one-dimensional Bose-Hubbard model. When comparing to exact results, we obtain good agreement for a large range of excited-states energies. Interestingly, we find that deep networks typically outperform shallow architectures for high-energy states.
Artificial neural networks have been recently introduced as a general ansatz to represent many-body wave functions. In conjunction with variational Monte Carlo calculations, this ansatz has been applied to find Hamiltonian ground states and their energies. Here, we provide extensions of this method to study excited states, a central task in several many-body quantum calculations. First, we give a prescription that allows us to target eigenstates of a (nonlocal) symmetry of the Hamiltonian. Second, we give an algorithm to compute low-lying excited states without symmetries. We demonstrate our approach with both restricted Boltzmann machines and feed-forward neural networks. Results are shown for the one-dimensional spin-1/2 Heisenberg model, and for the one-dimensional Bose-Hubbard model. When comparing to exact results, we obtain good agreement for a large range of excited-states energies. Interestingly, we find that deep networks typically outperform shallow architectures for high-energy states.
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.
Neural networks are a promising tool for simulating quantum many body systems. Recently, it has been shown that neural network-based models describe quantum many body systems more accurately when they are constrained to have the correct symmetry properties. In this paper, we show how to create maximally expressive models for quantum states with specific symmetry properties by drawing on literature from the machine learning community. We implement group equivariant convolutional networks (G-CNN) \cite{cohen2016group}, and demonstrate that performance improvements can be achieved without increasing memory use. We show that G-CNNs achieve very good accuracy for Heisenberg quantum spin models in both ordered and spin liquid regimes, and improve the ground state accuracy on the triangular lattice over other variational Monte-Carlo methods.
This thesis is devoted to the numerical investigation of quantum spin models which describe the low-energy physics of frustrated magnets. At extremely low temperatures, these systems can host the so-called spin liquid phase, an unconventional state of matter characterized by a high degree of quantum entanglement and the absence of magnetic order. The experimental identification of the spin liquid phase in actual materials relies on the detection of its distinctive excitations (named spinons), which possess fractional quantum numbers and can be probed by inelastic neutron scattering experiments. From the theoretical point of view, variational methods have been largely employed to tackle ground state properties of frustrated spin models. In particular, variational Monte Carlo techniques based on Gutzwiller-projected fermionic wave functions have been shown to provide accurate results for several frustrated systems. In this thesis, we pursue an extension of this variational scheme to target dynamical spectra, which are directly measured by inelastic neutron scattering experiments. Specifically, we compute the dynamical structure factor by constructing approximate excited states, which are obtained by applying two-spinon operators to the ground state wave function. Our results prove that this variational method can accurately describe the spectral features of different spin systems. Focusing on prototypical frustrated models on the square and triangular lattices, we observe how the dynamical structure factor reflects the phase transition between a magnetically ordered phase with spin wave excitations, to a spin liquid state with fractional degrees of freedom. In addition to spectral properties, we also explore new directions to improve the accuracy of Gutzwiller-projected states by the application of a neural network correlator, in the form of a restricted Boltzmann machine. While this hybrid variational scheme provides a considerable improvement of the variational energy in the case of unfrustrated spin models, less satisfactory results are obtained for frustrated systems, which call for further refinements of the neural network variational Ansätze.