Grassmann tensor renormalization group for the massive Schwinger model with a $θ$ term using staggered fermions

TL;DR

This work develops a Grassmann tensor renormalization group framework to study the two-flavor Schwinger model with a $2\pi$-periodic $\theta$ term using staggered fermions. It computes the $\theta$-dependence of the free energy and topological charge in the thermodynamic limit across a broad mass range, validating against the analytic large-mass Maxwell limit and prior MC results in the small-mass regime. The results indicate that the lattice $N_f=2$ Schwinger model exhibits finite-\beta phase-structure differences from the continuum limit, with mass-dependent topological features, potential first-order behavior at $\theta=\pi$ at large mass, and an exponentially large correlation length at small mass. The study demonstrates a viable Grassmann TRG pipeline for fermionic gauge theories with a $\theta$ term and outlines clear paths for improving continuum extrapolation and exploring alternative actions.

Abstract

We use the Grassmann tensor renormalization group method to investigate the $N_f=2$ Schwinger model with the staggered fermions in the presence of a $2π$ periodic $θ$ term in a broad range of mass. The method allows us to deal with the massive staggered fermions straightforwardly and to study the $θ$ dependence of the free energy and topological charge in the thermodynamic limit. Our calculation provides consistent results with not only the analytical solution in the large mass limit but also the previous Monte Carlo studies in the small mass regime. Our numerical results also suggest that the $N_f=2$ Schwinger model on a lattice has a different phase structure, than the model in the continuum limit.

Figures (11)

• Figure 1: (A) Schematic picture of the Grassmann tensor network in Eq. \ref{['eq:tn_rep']}. Since the fundamental tensor defined in Eq. \ref{['eq:fundamental_tensor']} depends on $n_{1}$ due to the staggered sign function, two kinds of tensors, white and gray symbols, are necessary to restore the path integral. (B) Structure of the fundamental tensor in Eq. \ref{['eq:fundamental_tensor']}. Red and green symbols show $T^{(g)}$ and $\mathcal{T}^{(f)}$, respectively. Dotted lines represent the square lattice. Each external line represents the auxiliary Grassmann field. Gauge fields are denoted by the diamonds.
• Figure 2: Free energy density as a function of the bond dimension $D$ (left) and the cutoff $K$ (right) in the Gauss-Legendre quadrature rule.
• Figure 3: Free energy density as a function of the $\theta$, in the range of $-1.1\pi\le\theta\le2.2\pi$.
• Figure 4: Free energy density as a function of $\theta/\pi$ with $\sqrt{\beta m_0^2} \geq 0.14$ (left) and $\sqrt{\beta m_0^2} \leq 0.2$ (right). A solid curve shows the analytical solution of the Maxwell theory on a lattice in the left panel while the mass perturbation result for $\sqrt{\beta m_0^2} =0.01$ in the right.
• Figure 5: Free energy density as a function of $\theta/\pi$ at $\sqrt{\beta m_0^2}=0$ with various $\beta$. Note that the plot of $\beta=1/(0.5^2)=4$ is also depicted in Fig. \ref{['fig:mass_f']}.