DMRG study of the higher-charge Schwinger model and its 't Hooft anomaly

TL;DR

This study uses DMRG to numerically analyze the charge-$q$ Schwinger model, exploring its symmetries, anomalies, and boundary effects, providing insights into its nonperturbative properties.

Contribution

It applies DMRG to the lattice Hamiltonian of the charge-$q$ Schwinger model, revealing boundary effects and confirming the role of Wilson loops in chiral transformations.

Findings

Local operators reduce boundary effects in calculations.

Wilson loops generate discrete chiral transformations.

Boundary effects are significant in extensive quantities.

Abstract

The charge-$q$ Schwinger model is the $(1+1)$-dimensional quantum electrodynamics (QED) with a charge-$q$ Dirac fermion. It has the $\mathbb{Z}_q$ $1$-form symmetry and also enjoys the $\mathbb{Z}_q$ chiral symmetry in the chiral limit, and there is a mixed 't Hooft anomaly between those symmetries. We numerically study the charge-$q$ Schwinger model in the lattice Hamiltonian formulation using the density-matrix renormalization group (DMRG). When applying DMRG, we map the Schwinger model to a spin chain with nonlocal interaction via Jordan-Wigner transformation, and we take the open boundary condition instead of the periodic one to make the Hilbert space finite-dimensional. When computing the energy density or chiral condensate, we find that using local operators significantly reduces the boundary effect compared with the computation of corresponding extensive quantities divided by the volume. To discuss the consequence of the 't Hooft anomaly, we carefully treat the renormalization of the chiral condensates, and then we confirm that Wilson loops generate the discrete chiral transformations in the continuum limit.