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Ab-Initio Phase Discovery with Neural Quantum States

Smith, Conor

Published May 17, 2025• 0 citations
OpenAlex

Abstract

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.

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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.
This thesis investigates emergent phenomena in strongly correlated quantum matter, with a central focus on developing first-principles deep learning methods for quantum many-body systems. A common theme throughout this work is the search for microscopic mechanisms that generate unexpected collective behavior, together with computational frameworks capable of describing such behavior in regimes where conventional approaches become inadequate. The first part of the thesis studies a doped Mott insulator in a magnetic field and uncovers a new superconducting state, which we call \emph{magnonic superconductivity}. In this state, magnons mediate pairing, and the superconducting order is carried by composite objects binding electron pairs to spin excitations. This mechanism gives rise to a partially spin-polarized superconducting phase beyond the conventional paradigm of electron pairing and further suggests the possibility of higher-charge bound states, including charge-6e6e6e superconductivity. The main thrust of the thesis is the use of deep neural networks as variational, first-principles representations of many-body wave functions. In contrast to the usual data-driven role of machine learning, the neural-network ans\"atze developed here do not learn from labeled datasets; instead, they are optimized directly from the microscopic Hamiltonian through variational Monte Carlo. Within this framework, we first develop self-attention-based neural-network wave functions for correlated electrons in solids and show that they accurately describe moir\'e systems with minimal human bias, capture interaction-driven phase transitions, and exhibit favorable scaling of representational complexity with system size. We then introduce a general physics-informed initialization strategy that reconstructs complex many-body wave functions from probability density and probability current. This approach enables highly accurate pretraining of neural wave functions, achieves near-exact overlaps for topological and paired quantum states, and allows efficient simulations of fractional quantum Hall systems with realistic interactions at unprecedented sizes. Finally, we introduce QERNEL, an efficient foundational neural variational ansatz designed to learn a family of many-body ground states within a single conditional model. By combining efficient context conditioning with modern permutation-equivariant building blocks, QERNEL achieves substantially improved expressivity at low computational cost. Applied to interacting electrons in semiconductor moir\'e heterobilayers, it captures distinct many-body phases and resolves the sharp phase transition between them in systems as large as 150 electrons. These results show that context-aware neural wave functions can move beyond Hamiltonian-by-Hamiltonian solution toward genuinely foundational models of quantum matter. Taken together, the results of this thesis show that first-principles deep learning can serve not merely as a numerical tool, but as a general framework for quantum many-body physics: one that bridges microscopic Hamiltonians, expressive wave-function ansätze, and scalable computation, while opening new paths to the discovery and characterization of strongly correlated quantum matter.
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.
Neural network quantum state (NNQS) has emerged as a promising candidate for quantum many-body problems, but its practical applications are often hindered by the high cost of sampling and local energy calculation. We develop a high-performance NNQS method for \textit{ab initio} electronic structure calculations. The major innovations include: (1) A transformer based architecture as the quantum wave function ansatz; (2) A data-centric parallelization scheme for the variational Monte Carlo (VMC) algorithm which preserves data locality and well adapts for different computing architectures; (3) A parallel batch sampling strategy which reduces the sampling cost and achieves good load balance; (4) A parallel local energy evaluation scheme which is both memory and computationally efficient; (5) Study of real chemical systems demonstrates both the superior accuracy of our method compared to state-of-the-art and the strong and weak scalability for large molecular systems with up to 120120120 spin orbitals.
In this work, we use neural quantum states (NQS) to describe the high-dimensional wave functions of electron-phonon coupled systems. We demonstrate that NQS can accurately and systematically learn the underlying physics of such problems through a variational Monte Carlo optimization of the energy with minimal incorporation of physical information, even in highly challenging cases. We assess the ability of our approach across various lattice model examples featuring different types of couplings. The flexibility of our NQS formulation is demonstrated via application to ab initio models parametrized by density functional perturbation theory consisting of electron or hole bands coupled linearly to dispersive phonons. We compute accurate real-frequency spectral properties of electron-phonon via a response formalism based on NQS. Our work establishes a general framework for exploring diverse ground-state and dynamical phenomena arising in electron-phonon systems, including the nonperturbative interplay of correlated electronic and electron-phonon effects in systems ranging from simple lattice models to realistic models of materials parametrized by ab initio calculations.
Moiré materials provide an ideal platform for exploring quantum phases of matter. However, solving the many-electron problem in moiré systems is challenging due to strong correlation effects. We introduce a powerful variational representation of quantum states, many-body neural Bloch wavefunction, to solve many-electron problems in moiré materials accurately and efficiently. Applying our method to the semiconductor heterobilayer WSe2/WS2 , we obtain a generalized Wigner crystal at filling factor n = 1/3, a Mott insulator n = 1, and a correlated insulator with local magnetic moments and antiferromagnetic spin correlation at n = 2. Our neural network approach improves the simulation accuracy of strongly interacting moiré materials and paves the way for discovery of new quantum phases with variational learning principle in a unified framework.
Abstract Moiré systems have emerged as an exciting new tunable platform for engineering and probing quantum matter. A large number of exotic states have been observed, stimulating intense efforts in experiment, theory, and simulation. Utilizing a neural-network-based quantum Monte Carlo approach, we discover a new ground state of the two-dimensional electron gas in a honeycomb moiré potential at a filling factor of ν m = 1/4 (one electron every four moiré minima). In this state, two opposite-spin electrons pair to form singlet-like valence bond states which restore local C 6 symmetry in hexagonal molecules each spanning 6 moiré minima. These molecules of pairs then form a molecular Wigner crystal, leaving one quarter of the moiré minima mostly depleted. The formation of such a paired Wigner crystal, absent any confining potential or attractive interaction to facilitate "pre-assembling" the molecule, provides a fascinating case of collective phenomena in strongly interacting quantum many-body systems and opportunities to engineer exotic properties in a system within current experimental setups. This discovery signals a new level of predictive power in quantum many-body computations enabled by neural networks and machine learning.