Matching Lagrangian and Hamiltonian Simulations in (2+1)-dimensional U(1) Gauge Theory

TL;DR

The paper tackles the problem of reconciling Lagrangian (path integral) and Hamiltonian lattice formulations for a compact $U(1)$ gauge theory in $(2+1)$ dimensions by taking the temporal continuum limit using anisotropic lattices and nonperturbatively determining the renormalised anisotropy $\xi_{\mathrm{ren}}$ and Sommer scale $r_0/a_s$. The authors perform extensive Lagrangian MC simulations at multiple anisotropies, extract the static potentials, and match to Hamiltonian data obtained from exact diagonalization on small volumes, carefully controlling finite-volume and truncation effects with multiple analysis strategies (cA and cB). They demonstrate that, after temporal continuum extrapolation, Lagrangian results agree with Hamiltonian results within $<2\sigma$ across the matching points, validating the proposed cross-formulation matching approach. The work provides a practical pathway to leverage the strengths of both formulations and sets the stage for extensions to non-Abelian theories and larger systems, with potential applications in quantum simulation and higher-dimensional lattice gauge theories.

Abstract

At finite lattice spacing, Lagrangian and Hamiltonian predictions differ due to discretization effects. In the Hamiltonian limit, i.e. at vanishing temporal lattice spacing $a_t$, the path integral approach in the Lagrangian formalism reproduces the results of the Hamiltonian theory. In this work, we numerically calculate the Hamiltonian limit of a U$(1)$ gauge theory in $(2+1)$ dimensions. This is achieved by Monte Carlo simulations in the Lagrangian formalism with lattices that are anisotropic in the time direction. For each ensemble, we determine the ratio between the temporal and spatial scale with the static quark potential and extrapolate to $a_t \to 0$. Our results are compared with the data from Hamiltonian simulations at small volumes, showing agreement within $<2σ$. These results can be used to match the two formalisms.

Figures (11)

• Figure 1: Spatial and temporal plaquette (left panel) and renormalised anisotropy (right panel) as functions of $\beta$ for $\xi_\text{input}=0.8$ and $\beta_\text{iso}=1.7$. $\xi_\text{ren}$ is determined from the analysis chain N0ET, see \ref{['tab:analysischains']}. The solid lines represent fits to the data, and the shaded regions the corresponding bootstrap errors. The fits are linear in $\beta$ for the plaquette, and a constant for $\xi_\text{ren}$.
• Figure 2: Thermalisation of the plaquette for the input anisotropies $\xi_\text{input} \in\{0.18, 1/5, 1\}$. All data were generated by the heatbath-overrelaxation algorithm, and the ensembles have the lattice sizes $L^2\times T = 16^2 \times 88, 80, 16\approx16^2 \times 16/\xi_\text{input}$. We show the difference between the measurement and the mean value of the plaquette.
• Figure 3: Integrated autocorrelation time of the plaquette for different input anisotropies. $\beta=1.7$ is kept constant, the red squares correspond to points with 100 sweeps between measurements, the black circles to 50 sweeps between measurements. The dashed line shows the ideal case $\tau_\text{int}=0.5$. The inset is a close-up of the lower right region of the larger figure. All simulations were done with the Metropolis-algorithm.
• Figure 4: The matching points for analysis chain N0ET. The upper points correspond to the matching points for $\beta_\mathrm{iso}=1.7$, the lower ones to $\beta_\mathrm{iso}=1.65$.
• Figure 5: $\xi_\text{ren}$ at the matching points for analysis chain N0ET. The red squares correspond to the matching points for $\beta_\mathrm{iso}=1.7$, the black circles to $\beta_\mathrm{iso}=1.65$. The diagonal line shows the line of $\xi_\text{ren}=\xi_\text{input}$.