Here, the authors use gauge-invariant neural network quantum states (NNQS) to study \ensuremath{\mathbb{Z}}${}_{N}$ lattice gauge theories. They compute ground states and map out phase transitions, identifying a continuous transition in \ensuremath{\mathbb{Z}}${}_{2}$ with Ising critical exponents and a weakly first-order transition in \ensuremath{\mathbb{Z}}${}_{3}$. Their work demonstrates how deep learning can overcome challenges in traditional lattice simulations, providing a novel approach for exploring complex quantum phases and topological order in condensed matter systems.
Here, the authors use gauge-invariant neural network quantum states (NNQS) to study \ensuremath{\mathbb{Z}}N lattice gauge theories. They compute ground states and map out phase transitions, identifying a continuous transition in \ensuremath{\mathbb{Z}}2 with Ising critical exponents and a weakly first-order transition in \ensuremath{\mathbb{Z}}3. Their work demonstrates how deep learning can overcome challenges in traditional lattice simulations, providing a novel approach for exploring complex quantum phases and topological order in condensed matter systems.
Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with Z_{d} gauge group and non-Abelian Kitaev D(G) models on different geometries. Focusing on the special case of Z_{2} gauge group on a periodically identified square lattice, the equivariant architecture is analytically shown to contain the loop-gas solution as a special case. Gauge equivariant neural-network quantum states are used in combination with variational quantum Monte Carlo to obtain compact descriptions of the ground state wave function for the Z_{2} theory away from the exactly solvable limit, and to demonstrate the confining or deconfining phase transition of the Wilson loop order parameter.
Symmetries such as gauge invariance and anyonic symmetry play a crucial role in quantum many-body physics.We develop a general approach to constructing gauge-invariant or anyonic-symmetric autoregressive neural networks, including a wide range of architectures such as transformer and recurrent neural network, for quantum lattice models.These networks can be efficiently sampled and explicitly obey gauge symmetries or anyonic constraint.We prove that our methods can provide exact representation for the ground and excited states of the two-and three-dimensional toric codes, and the X-cube fracton model.We variationally optimize our symmetry-incorporated autoregressive neural networks for ground states as well as real-time dynamics for a variety of models.We simulate the dynamics and the ground states of the quantum link model of U(1) lattice gauge theory, obtain the phase diagram for the two-dimensional Z 2 gauge theory, determine the phase transition and the central charge of the SU(2) 3 anyonic chain, and also compute the ground-state energy of the SU(2) invariant Heisenberg spin chain.Our approach provides powerful tools for exploring condensed-matter physics, high-energy physics, and quantum information science.
The identification and characterization of phase transitions is one of the oldest and most central problems in condensed matter physics, traditionally approached through order parameters, free-energy expansions, renormalization-group analysis, and direct numerical simulation. Over the past decade, machine learning (ML) has emerged as a complementary and, in some regimes, transformative methodology for this task. Supervised classifiers can learn to distinguish phases directly from raw configurations without an a priori order parameter; unsupervised and self-supervised techniques such as principal component analysis, autoencoders, and learning-by-confusion can locate transition points with minimal or no labeled data; and generative and variational architectures, including neural-network quantum states, now approximate ground-state wavefunctions of strongly correlated systems well beyond the reach of exact diagonalization. This paper reviews the theoretical foundations of phase transitions relevant to machine-learning applications, surveys the principal families of algorithms used to detect and classify phases, and examines their application across classical spin systems, percolation, topological and quantum phases, strongly correlated electron systems, and experimentally measured spectroscopic data. Representative case studies are discussed, including convolutional and graph-neural-network classifiers of lattice configurations, explainable-ML detection of thermodynamic transitions in photoemission spectra, and deep-learning-guided discovery of high-temperature superconductors. The review closes with a critical discussion of persistent challenges interpretability, data scarcity in correlated-electron experiments, generalization across Hamiltonians, and the reliability of learned order parameters near criticality and outlines directions for future research, including physics-informed architectures, uncertainty quantification, and closed-loop integration with autonomous experimentation.
Gauge theories and topological phases play a fundamental role in different areas of physics. The first ones are at the basis of the Standard Model in the field of particle physics, describing the electroweak and strong interactions through a non-Abelian gauge theory. In condensed matter and statistical physics, gauge theories arise as low-energy effective descriptions of strongly correlated phenomena, such as quantum spin liquids and quantum Hall effect. In this realm there is a strong connection with topological phases and order, as emergent gauge fermions and bosons often describe collective excitations of new exotic states of matter of spin models. The discretization on a lattice is a possible way of dealing with the strongly coupled nature of these theories, due to the fact that this formulation at finite volume provides natural regularizing cut-offs, i.e. the lattice size and spacing. This allows for the investigation of different non-perturbative properties both numerically and analytically. Despite the success of these methods, there are various aspects which remain intractable due to the sign or complex action problems, like the out of equilibrium real time evolution or the analysis of quantum chromodynamics with finite chemical potential. In this respect, quantum simulators of many-body systems to simulate high energy physics arise as promising alternatives to face these problems in the near future. They are quantum systems that can be controlled and used to simulate more complicated systems, whose properties could not be analysed with classical computational, experimental or theoretical tools. In the last decades, there has been a huge development in the fields of quantum optics and atomic physics, allowing for the realization highly precise and controllable platforms by means of trapped ions, superconducting circuits, Rydberg atoms and ultracold atoms in optical lattices. This work of Thesis is part of these fields and has different purposes, all of them connected with the study of gauge theories and topological phases. Firstly, we want to develop a reformulation of lattice gauge theories in terms of gauge invariant fields, in a way to deal solely with physical variables directly in the action. Among several possible advantages, a crucial point is that this can be particularly helpful for the construction of consistent approximation schemes, such as mean-field theories, in order to understand and capture some of the physical features of the theory and make the mean-field approximation consistent with the Elitzur theorem, stating that a local gauge symmetry can not be spontaneously broken. As a second point, we face the problem of simulating higher dimensional gauge theories with ultracold atoms. One of the challenges in more than one dimension is indeed the realization of plaquette interaction terms in the Hamiltonian of lattice gauge theories: these can be engineered through four correlated hoppings in perturbation theory, or by means of constrained hoppings in the dual formulation. In this respect, our main target is to set up an ultracold atomic platform generating the plaquette term in two dimensions using only two correlated hoppings, and protecting gauge invariance through angular momentum conservation. The last point we address in the Thesis is related to the analysis of particles moving in static background gauge potentials, i.e. when the considered gauge field has no dynamical term in the action of the system. Indeed, the dynamics of quantum particles in the presence of static gauge fields gives rise to intriguing physical phenomena. In particular, using ultracold atom setups, the realization of artificial gauge potentials can be used to investigate the physics of topological semimetals, such as Weyl or Dirac type. In the last ten years, a lot of attention has been paid to their characterization, due to the appearence of clear theoretical predictions and very well-controlled experimental techniques. We investigate here the relation between topological phase transitions and van Hove singularities, i.e. the discontinuities in the energy derivative of the density of states, in three-dimensional gapless systems. In such materials, topological phase transitions can be defined by changes of the topological invariants of the Fermi sheets, happening at specific singular points. Moreover, Fermi surface singularities result in the presence of the so-called van Hove points. We then present a general argument to relate topological phase transitions and van Hove singularities, and show observable consequences that are related to the transport properties of the system. We exemplify our argument in Weyl systems by analyzing the three-dimensional Hofstadter model for various commensurate fluxes, which offers the opportunity to consider different kinds of Weyl metals and to understand the features of their density of states.
The Neural-Network Quantum States (NNQS) method is rapidly emerging as a powerful tool for investigating \nquantum many-body physics. By combining variational Monte Carlo techniques with neural network-based \nvariational functions, this approach leverages the remarkable advancements in deep learning achieved in \nrecent years. While substantial progress has been made in simulating magnetic systems on lattices, simple \nmolecules, and even continuous systems, the ab initio simulation of complex strongly correlated electron \nsystems continues to pose significant challenges. With the help of the generalized atomic limit (GAL) – a \nrecently developed model describing a system of quantum dots coupled to a superconducting lead – we \nattempt to show the efficiency and accuracy of NNQS, mainly the Restricted Boltzmann Machine, by comparing \nthe acquired results to other available methods such as exact diagonalization. The simultaneous study of the \nsystem’s properties such as the energy spectrum and quantum phase transitions could bring advancements \nin electronics, sensors or the design of high-quality qubits used in quantum computers.
Gauge Theory plays a crucial role in many areas in science, including high energy physics, condensed matter physics and quantum information science. In quantum simulations of lattice gauge theory, an important step is to construct a wave function that obeys gauge symmetry. In this paper, we have developed gauge equivariant neural network wave function techniques for simulating continuous-variable quantum lattice gauge theories in the Hamiltonian formulation. We have applied the gauge equivariant neural network approach to find the ground state of 2+1-dimensional lattice gauge theory with U(1) gauge group using variational Monte Carlo. We have benchmarked our approach against the state-of-the-art complex Gaussian wave functions, demonstrating improved performance in the strong coupling regime and comparable results in the weak coupling regime.