The search for exotic phases of matter in quantum systems remains a central goal in condensed matter physics. Recent progress in machine learning has driven the development of powerful new computational methods in quantum chemistry and physics. In particular, the application of neural networks to solve the Schrödinger equation within the variational Monte Carlo (VMC) framework has led to the development of the successful neural network VMC (NNVMC) method. This thesis extends and explores the application of NNVMC to the study of strongly interacting Fermi gases. We first investigated the unitary Fermi gas (UFG)---a system with strong, short-range attractions known to possess a superfluid ground state that is difficult to describe quantitatively. By applying the NNVMC method with the FermiNet Ansatz, we first identified its key limitations before proposing a simple modification. This change led to a substantial improvement in accuracy over the original FermiNet in the UFG, outperforming results from the state-of-the-art simulations while being consistent with experiments. Next, we explored the phases of the 2D spin-imbalanced Fermi gas. The combined power of NNVMC and our modified FermiNet enabled us not only to observe the long-predicted Fulde-Ferrell-Larkin-Ovchinnikov phase but also to discover a previously unknown phase, characterised by the crystallisation of tightly bound Cooper pairs within a uniform sea of unpaired fermions. Lastly, we studied an anisotropic Fermi gas featuring spin-dependent splitting in momentum space --- a prime signature of the recently discovered form of collinear magnetism, termed altermagnetism. We mapped its 2D phase diagram, providing unbiased estimations for the locations of various phases beyond the mean-field approximation. In conclusion, our work has demonstrated that the NNVMC method is a powerful and versatile tool for investigating strongly correlated systems, confirming its potential to uncover novel phenomena in the vast landscape of quantum systems.
The search for exotic phases of matter in quantum systems remains a central goal in condensed matter physics. Recent progress in machine learning has driven the development of powerful new computational methods in quantum chemistry and physics. In particular, the application of neural networks to solve the Schrödinger equation within the variational Monte Carlo (VMC) framework has led to the development of the successful neural network VMC (NNVMC) method. This thesis extends and explores the application of NNVMC to the study of strongly interacting Fermi gases. We first investigated the unitary Fermi gas (UFG)---a system with strong, short-range attractions known to possess a superfluid ground state that is difficult to describe quantitatively. By applying the NNVMC method with the FermiNet Ansatz, we first identified its key limitations before proposing a simple modification. This change led to a substantial improvement in accuracy over the original FermiNet in the UFG, outperforming results from the state-of-the-art simulations while being consistent with experiments. Next, we explored the phases of the 2D spin-imbalanced Fermi gas. The combined power of NNVMC and our modified FermiNet enabled us not only to observe the long-predicted Fulde-Ferrell-Larkin-Ovchinnikov phase but also to discover a previously unknown phase, characterised by the crystallisation of tightly bound Cooper pairs within a uniform sea of unpaired fermions. Lastly, we studied an anisotropic Fermi gas featuring spin-dependent splitting in momentum space --- a prime signature of the recently discovered form of collinear magnetism, termed altermagnetism. We mapped its 2D phase diagram, providing unbiased estimations for the locations of various phases beyond the mean-field approximation. In conclusion, our work has demonstrated that the NNVMC method is a powerful and versatile tool for investigating strongly correlated systems, confirming its potential to uncover novel phenomena in the vast landscape of quantum systems.
Understanding superfluidity remains a major goal of condensed matter physics. Here, we tackle this challenge utilizing the recently developed fermionic neural network (FermiNet) wave function [D. Pfau , .] for variational Monte Carlo calculations. We study the unitary Fermi gas, a system with strong, short-range, two-body interactions known to possess a superfluid ground state but difficult to describe quantitatively. We demonstrate key limitations of the FermiNet in studying the unitary Fermi gas and propose a simple modification based on the idea of an antisymmetric geminal power singlet (AGPs) wave function. The new AGPs FermiNet outperforms the original FermiNet significantly in paired systems, giving results which are more accurate than fixed-node diffusion Monte Carlo and are consistent with experiment. We prove mathematically that the new , which differs from the original only by the method of antisymmetrization, is a strict generalization of the original FermiNet architecture, despite the use of fewer parameters. Our approach shares several advantages with the original FermiNet: The use of a neural network removes the need for an underlying basis set; sand the flexibility of the network yields extremely accurate results within a variational quantum Monte Carlo framework that provides access to unbiased estimates of arbitrary ground-state expectation values. We discuss how the method can be extended to study other superfluid. Published by the American Physical Society 2024
We study the zero-temperature phase diagram of the 2D spin-imbalanced Fermi gas with short-ranged attractive interactions using the recently developed neural network variational Monte Carlo method with the AGPs FermiNet Ansatz. The Fulde-Ferrell-Larkin-Ovchinnikov phase is observed in the weakly interacting BCS limit and a polarised superfluid is seen in the strongly interacting BEC limit. When the interactions are strong, the minority-spin momentum density is reduced almost to zero in the momentum-space region occupied by the unpaired majority-spin electrons. When the interactions are very strong, phase separation occurs, with regions containing bosonic pairs and unpaired regions occupied by the remaining majority-spin particles. In addition, we observe translational symmetry breaking at intermediate interaction strengths, where the system forms an exotic crystal of Cooper pairs in a Fermi fluid of unpaired majority-spin particles. We provide a possible explanation for the formation of the crystalline phase, explain the origins of the k-space momentum-density hole when the pairs are tightly bound, and discuss how our approach opens new directions for future work.
In this thesis we study the quantum many-body problem with tools from machine learning. In particular, we utilize Neural Quantum States (NQS) as an ansatz in Variational Monte Carlo (VMC) simulations. First, we develop a method to train NQS more reliably in an effort to reduce the total computational cost of finding the ground state of strongly-correlated systems. Then, we develop an expressive NQS ansatz to study two-dimensional materials and perform ab-initio phase discovery. In particular, we consider the two-dimensional electron gas, and provide state-of-the-art energies and characterization of the phase diagram. We also find evidence of a novel state between the Fermi liquid and Wigner crystal which we call a Nematic Spin Correlated Liquid (NSCL). Lastly, we turn our attention to a moir\'e system with a honeycomb potential. With minor modifications to the ansatz, we find a highly correlated state we call an interaction-driven molecular Wigner crystal. With the increasing interest in NQS as an accurate method to study the many-body problem and recent advances in experimentally studying two-dimensional materials, we expect the technical developments and physical results presented in this thesis to be valuable to theorists and experimentalists.
Interacting Fermi gases are one of the chief paradigms of condensed matter physics. They have been studied since the beginning of the development of quantum mechanics, but continue to produce surprises today. Recent experimental developments in the field of ultracold atomic gases, as well as conventional solid state materials, have produced new and exotic forms of Fermi gases, the theoretical understanding of which is still in its infancy. This Thesis aims to provide updated tools and additional insights into some of these systems, through the application of both numerical and analytical techniques. The first Part of this Thesis is concerned with the development of improved numerical tools for the study of interacting Fermi gases. These tools take the form of accurate model potentials for the dipolar and contact interactions, as found in various ultracold atomic gas experiments, and a new form of Jastrow correlation factor that interpolates between the radial symmetry of the inter-electron Coulomb potential at short inter-particle distances, and the symmetry of the numerical simulation cell at large separation. These methods are designed primarily for use in quantum Monte Carlo numerical calculations, and provide high accuracy along with considerable acceleration of simulations. The second Part shifts focus to an analytical analysis of spin-imbalanced Fermi gases with an attractive contact interaction. The spin-imbalanced Fermi gas is shown to be unstable to the formation of multi-particle instabilities, generalisations of a Cooper pair containing more than two fermions, and then a theory of superconductivity is built from these instabilities. This multi-particle superconductivity is shown to be energetically favourable over conventional superconducting phases in spin-imbalanced Fermi gases, and its unusual experimental consequences are discussed.
Strongly coupled quantum matter displays a rich phenomenology including phase transitions and often unexpected collective behavior. Remarkable advances in experiments with ultracold Fermi gases allow us to gain deep insight into these intriguing systems. Their theoretical description, however, is often challenging as exact analytic solutions are available only in a few special cases, and approximate techniques such as mean-field or perturbation theory are of limited use. Numerical treatment with Monte Carlo (MC) methods has led to profound success in this regard. Unfortunately, for many systems - and especially for asymmetric quantum gases - the infamous sign problem slows progress due to an exponentially scaling of the computational effort with inceasing system size. In this thesis, we set out to explore the rich physics of two-component Fermi gases in the presence of finite spin polarization and/or mass imbalance. To surmount an arising sign problem, we learn from methodological advances made in the field of quantum chromodynamics and further develop these lattice approaches in the context of nonrelativistic Fermi gases. An extensive overview of the numerical methods is presented, including several toy problems to detail the capabilities and shortcomings of the developed approaches. With these tools in hand, we perform extensive benchmarks of the hybrid Monte Carlo method with imaginary asymmetries (iHMC) and the complex Langevin (CL) method, which is based on a complex version of stochastic quantization. Both approaches are shown to yield excellent results for the ground-state energy equation of state of mass-imbalanced Fermi gases in one spatial dimension. Due to its great versatility, the CL method is subsequently employed to study pairing in one-dimensional Fermi gases, for which suitable two-body correlations are computed, revealing unexpected pairing patterns for spin- and mass-imbalanced systems. Another major system of interest in this thesis is the paradigmatic unitary Fermi gas which is investigated at finite temperature and spin polarization. A precise determination of the density equation of state in the normal phase enables us to explore a broad range of thermodynamic properties. We infer valuable information on the finite-temperature phase diagram, such as a flat phase boundary of the normal-to-superfluid transition near the balanced limit and indications for the absence of an extensive pseudogap phase above this transition. The presented results provide experimentally testable ab initio predictions for a range of previously inaccessible thermodynamic quantities.
Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles.Exact wave functions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially scaling algorithms.The key challenge for many of these algorithms is the choice of wave function approximation, or Ansatz, which must trade off between efficiency and accuracy.Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wave-function Ansatz for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics.Here we introduce a novel deep learning architecture, the Fermionic neural network, as a powerful wave-function Ansatz for many-electron systems.The Fermionic neural network is able to achieve accuracy beyond other variational quantum Monte Carlo Ansatz on a variety of atoms and small molecules.Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry.This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab initio quantum chemistry methods, opening the possibility of accurate direct optimization of wave functions for previously intractable many-electron systems.