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Normalizing-flow-based density of states for (1+1)D U(1) lattice gauge theory with a $θ$-term

Simran Singh, Lena Funcke

Abstract

A normalizing-flow-based implementation of the density-of-states approach has recently been used to successfully reconstruct the partition function of (1+1)D scalar lattice field theory. In this preliminary work, we extend this framework to a lattice gauge theory by employing gauge-equivariant normalizing flows to reconstruct the density of states of pure (1+1)D U(1) lattice gauge theory, both with and without a $θ$-term. In the absence of a $θ$-term, we first demonstrate that the normalizing-flow-based reconstruction of the density of states reproduces the known analytic results for this theory. We further show that, in the presence of a $θ$-term, this formulation enables the generation of gauge-field configurations at fixed values of the topological charge.

Normalizing-flow-based density of states for (1+1)D U(1) lattice gauge theory with a $θ$-term

Abstract

A normalizing-flow-based implementation of the density-of-states approach has recently been used to successfully reconstruct the partition function of (1+1)D scalar lattice field theory. In this preliminary work, we extend this framework to a lattice gauge theory by employing gauge-equivariant normalizing flows to reconstruct the density of states of pure (1+1)D U(1) lattice gauge theory, both with and without a -term. In the absence of a -term, we first demonstrate that the normalizing-flow-based reconstruction of the density of states reproduces the known analytic results for this theory. We further show that, in the presence of a -term, this formulation enables the generation of gauge-field configurations at fixed values of the topological charge.
Paper Structure (8 sections, 13 equations, 3 figures)

This paper contains 8 sections, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The density of states
  3. The normalizing-flow-based density of states
  4. U(1) lattice gauge theory in (1+1)D
  5. Results
  6. Results for (1+1)D U(1) gauge theory without a $\theta$-term
  7. Results for (1+1)D U(1) gauge theory with a $\theta$-term
  8. Conclusions and outlook

Figures (3)

  • Figure 1: Results for (1+1)D U(1) gauge theory without a $\theta$-term obtained with the DoS method. Left: Expectation value of the Wilson action from NF-generated configurations at fixed constraint $c$ for an $L=8$ lattice and $P \in\{ 100, 2000\}$. The dashed line indicates perfect constraint satisfaction. Right: Reconstructed DoS compared to the exact result, illustrating that tighter constraints ($P=2000$) yield improved agreement.
  • Figure 2: Results for (1+1)D U(1) gauge theory without a $\theta$-term obtained with the DoS method. Left: Expectation value of the Wilson action from NF-generated configurations at fixed constraint $c$ for $L = 4$ and $L = 8$ lattices with $P=2000$. The dashed line indicates perfect constraint satisfaction. Right: Reconstructed DoS compared to the exact result for both lattice sizes.
  • Figure 3: Results for (1+1)D U(1) gauge theory with a $\theta$-term obtained with the gDoS method. Verification of the topological charge constraint for an $L=8$ lattice. Shown is the expectation value of the topological charge from NF-generated gauge field configurations at fixed constraint, with $\beta=1.0$ and $P = \{5,50\}$. The dashed line indicates perfect constraint satisfaction.