This thesis investigates gauge theories by combining theoretical analysis of non-invertible symmetries with computational simulations using neural network quantum states. We first investigate (3+1)d ZN gauge theories with non-invertible duality symmetries that arise from gauging one-form symmetries. These symmetries impose strict constraints on renormalization group flows, often prohibiting symmetry-preserving gapped phases. Our proof demonstrates that a self-dual theory with ZN one-form symmetry must either remain gapless or break self-duality, except when N = k²ℓ with –1 as a quadratic residue modulo ℓ. Lattice gauge theories provide concrete realizations of these constraints. We then focus on the development of neural network quantum states for studying lattice gauge theories. By implementing gauge-invariant neural networks, we compute ground-state wavefunctions for ZN lattice gauge theories in (2+1)d. For Z2 gauge theory, we characterize the continuous confinement transition and identify critical exponents consistent with the Ising universality class. In the Z3 case we observe a weakly first-order transition between the confined and deconfined phases. For both theories, we compute order and disorder parameters and verify the linear confining potential in the confining phase. These results demonstrate that neural network methods provide a powerful tool for studying quantum field theories computationally. By combining these simulations with symmetry-based analysis, this work uncovers new insights into the nonperturbative dynamics and phase structure of gauge theories.
This thesis investigates gauge theories by combining theoretical analysis of non-invertible symmetries with computational simulations using neural network quantum states. We first investigate (3+1)d ZN gauge theories with non-invertible duality symmetries that arise from gauging one-form symmetries. These symmetries impose strict constraints on renormalization group flows, often prohibiting symmetry-preserving gapped phases. Our proof demonstrates that a self-dual theory with ZN one-form symmetry must either remain gapless or break self-duality, except when N = k²ℓ with –1 as a quadratic residue modulo ℓ. Lattice gauge theories provide concrete realizations of these constraints. We then focus on the development of neural network quantum states for studying lattice gauge theories. By implementing gauge-invariant neural networks, we compute ground-state wavefunctions for ZN lattice gauge theories in (2+1)d. For Z2 gauge theory, we characterize the continuous confinement transition and identify critical exponents consistent with the Ising universality class. In the Z3 case we observe a weakly first-order transition between the confined and deconfined phases. For both theories, we compute order and disorder parameters and verify the linear confining potential in the confining phase. These results demonstrate that neural network methods provide a powerful tool for studying quantum field theories computationally. By combining these simulations with symmetry-based analysis, this work uncovers new insights into the nonperturbative dynamics and phase structure of gauge theories.
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.
Symmetry is foundational in theoretical physics. In this thesis, we show how recent advances in symmetry have reinforced its power to classify, characterize, and constrain quantum systems non-perturbatively. We study symmetries, ranging from spatial symmetries to generalized symmetries, in both quantum field theory and quantum lattice systems. In particular, this thesis develops five sets of results. First, we show that ordinary ordered phases generically host emergent generalized symmetries in the infrared, associated with homotopy defects of the order-parameter space. We use these emergent generalized symmetries to characterize and classify defect-suppressed disorder transitions to phases with topological order and emergent gauge structures. Second, we turn to lattice anomalies and show how the chiral anomaly of a massless Dirac fermion in two spacetime dimensions can be realized on the lattice. In particular, we identify lattice realizations of the vector and axial symmetries and demonstrate how their anomaly is matched by the non-abelian Onsager algebra in the staggered fermion Hamiltonian. Third, we give a microscopic lattice understanding of T-duality in quantum spin chains using non-invertible symmetry. This non-invertible symmetry, together with lattice momentum and winding symmetries, also enforces gaplessness, hence providing a novel way to realize perturbative anomalies in quantum spin-chain models. Fourth, we show that in local lattice models of spinless fermions, fermion-number conservation combined with a non-onsite Majorana translation symmetry enforces a Fermi surface. This symmetry-enforced Fermi surface is a strong form of symmetry-enforced gaplessness. Finally, we introduce a framework of stratified symmetries and stratified anomalies to understand and discover new Lieb–Schultz–Mattis constraints. We apply this framework to modulated symmetries and find an explicit criterion for when local anomaly data imply obstructions to a trivial gapped phase. Together, these results demonstrate the broad applicability and power of symmetry and its generalizations, both for organizing quantum phases and for deriving robust non-perturbative constraints.
Simulating quantum-mechanical systems is a very challenging issue due to the high computational costs. The main reason for this is related to the dimension of the Hilbert space that grows exponentially with the number of degrees of freedom, making an exact implementation impossible even for the most powerful today’s supercomputers. \nOn this ground it appears clear the need of a new simulation method, i.e. quantum simulation. The general idea is very simple: using a controllable quantum system, called quantum simulator, to emulate and to analyze another quantum system that usually results less controllable or accessible. Quantum simulation is currently a growing and multidisciplinary physical area that involves theoretical and experimental research. From the theoretical side, the main focus is on the study of simulation-models that create a mapping between the target physics and the simulator. \nIn the first two parts of this thesis, we explore these ideas by studying the ground-state properties and the real-time dynamics of a class of Zn lattice gauge theories in 1+1 dimensions, in which the gauge fields are coupled to fermionic matter. These models can be considered as quantum simulation-models of lattice 1+1 Quantum Electrodynamics (QED), which is the simplest gauge theory that shows non-trivial phenomena, like confinement, that are also observed in more complicated gauge theories, such as 3+1 Quantum Chromodynamics (QCD). \nIn the third part of this thesis, we unveil an interesting interplay of symmetry and topology by demonstrating the existence of symmetry-protected topological (SPT) phases in the presence of gauge interactions. More specifically, we introduce an alternative discretization of the Schwinger model leading to a new type of topological-QED. We use bosonization and Density-Matrix Renormalization Group (DMRG) techniques to analyze the phase diagram of the model, opening an interesting route to study topological phases of matter in the context of gauge theories.
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.
Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) S_3 S 3 symmetry and the other with non-invertible \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1d symmetry-topological-order (SymTO) of type JK_4\boxtimes \overline{JK}_4 J K 4 ⊠ J K ¯ 4 . The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual S_3 S 3 -symmetric and \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) -symmetric models.