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Exploring Phase Diagrams with Functional Renormalization and Artificial Neural Networks: From the Hubbard Model to Lattice Gauge Theory

Sebastian J. Wetzel

Published Jan 1, 2018• 0 citations
Publisher

Abstract

This thesis is dedicated to provide physicists with new and improved techniques to \nexamine phase diagrams and phase transitions. \nOne the one hand, an analysis of the effects of different regularization schemes in \nfunctional renormalization group calculations is provided. Building on this knowledge, \nan investigation of the phase diagram of the Hubbard-Model on the square \nlattice is performed using the functional renormalization group. The calculation \nreveals leading instabilities in the d-wave superconducting and different antiferromagnetic \nchannels. In the symmetry broken phases there are a changing Fermi \nsurface geometry, coexistence phases of d-wave superconductivity and antiferromagnetism \nas well as a mutual tendency of superconductivity and antiferromagnetism \nto repel each other. \nOn the other hand, a scheme to discover phase transitions using unsupervised \nartifcial neural networks is developed. Further, a method to interpret artifcial \nneural networks is introduced. These methods are applied to systems ranging from \nthe two dimensional Ising Model to four dimensional SU(2) lattice gauge theory. \nThey find the existence of different phases, calculate phase boundaries and derive \nthe explicit formulas of the quantities by which the neural network distinguishes \nbetween phases. It turns out that these quantities are order parameters and other \nthermodynamic quantities.

DOI
10.11588/heidok.00025528

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This thesis is dedicated to provide physicists with new and improved techniques to \nexamine phase diagrams and phase transitions. \nOne the one hand, an analysis of the effects of different regularization schemes in \nfunctional renormalization group calculations is provided. Building on this knowledge, \nan investigation of the phase diagram of the Hubbard-Model on the square \nlattice is performed using the functional renormalization group. The calculation \nreveals leading instabilities in the d-wave superconducting and different antiferromagnetic \nchannels. In the symmetry broken phases there are a changing Fermi \nsurface geometry, coexistence phases of d-wave superconductivity and antiferromagnetism \nas well as a mutual tendency of superconductivity and antiferromagnetism \nto repel each other. \nOn the other hand, a scheme to discover phase transitions using unsupervised \nartifcial neural networks is developed. Further, a method to interpret artifcial \nneural networks is introduced. These methods are applied to systems ranging from \nthe two dimensional Ising Model to four dimensional SU(2) lattice gauge theory. \nThey find the existence of different phases, calculate phase boundaries and derive \nthe explicit formulas of the quantities by which the neural network distinguishes \nbetween phases. It turns out that these quantities are order parameters and other \nthermodynamic quantities.
The phases with spontaneously broken symmetries corresponding to antiferromagnetic and ddd-wave superconducting order in the two-dimensional $t\ensuremath{-}{t}^{'}$-Hubbard model are investigated by means of the functional renormalization group. The introduction of composite boson fields in the magnetic, charge density, and superconducting channels allows an efficient parametrization of the four-fermion vertex and the study of regimes where either the antiferromagnetic or superconducting order parameter, or both, are nonzero. We compute the phase diagram and the temperature dependence of the order parameter below the critical temperature, where antiferromagnetic and superconducting order show a tendency of mutual exclusion.
We present a solution to the problem of interpreting neural networks classifying phases of matter. We devise a procedure for reconstructing the decision function of an artificial neural network as a simple function of the input, provided the decision function is sufficiently symmetric. In this case one can easily deduce the quantity by which the neural network classifies the input. The method is applied to the Ising model and SU(2) lattice gauge theory. In both systems we deduce the explicit expressions of the order parameters from the decision functions of the neural networks. We assume no prior knowledge about the Hamiltonian or the order parameters except Monte Carlo--sampled configurations.
Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.
We employ the functional renormalization group to investigate the phase diagram of the t−t′t-t't−t′ Hubbard model on the square lattice with finite chemical potential μμμ at zero temperature. A unified scheme to derive flow equations in the symmetric and symmetry broken regimes allows a consistent continuation of the renormalization flow in the symmetry broken regimes. At the transition from the symmetric regime to the symmetry broken regimes, our calculation reveals leading instabilities in the d-wave superconducting and antiferromagnetic channels. Furthermore, we find a first order transition between commensurate and incommensurate antiferromagnetism. In the symmetry broken regimes our flow equations are able to renormalize around a changing Fermi surface geometry. We find a coexistence of d-wave superconductivity and antiferromagnetism at intermediate momentum scales k. However, there is a mutual tendency of superconductivity and antiferromagnetism to repel each other at even smaller scales k, which leads to the eradication of the coexistence phase in the limit of macroscopic scales.
The study of phase transitions plays an important role in understanding qualitative changes in the behaviour of physical systems at criticality. Despite decades of progress, there is still a strong demand for high-precision numerical tools capable of resolving subtle critical phenomena. Motivated by this need, in this thesis, we present two complementary numerical investigations of phase transitions in lattice systems. The first uses GPU-accelerated higher-order tensor renormalization group (HOTRG) techniques to study the two-dimensional generalized XY model, characterizing its ferromagnetic, nematic, and paramagnetic phases and mapping their phase boundaries using thermodynamic observables in the thermodynamic limit. The second develops and benchmarks a configurational temperature estimator, constructed from gradients and Hessians of the Euclidean lattice action, in compact U(1) lattice gauge theories. On one hand, tensor network methods capture rich phase structures when truncation and finite-bond effects are adequately controlled. On the other hand, the configurational temperature estimator provides an independent, low-overhead means of validating thermal sampling across different algorithms and models, and can also be used as a runtime diagnostic to identify sampling pathologies before large-scale production runs.
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.