Abstract
Abstract We develop a $(1+1)$D lattice $U(1)$ gauge theory in order to define the two-flavor massless Schwinger model, and discuss its connection with the Haldane conjecture. We propose to use the central-branch Wilson fermion, which is defined by relating the mass, $m$, and the Wilson parameter, $r$, by $m+2r=0$. This setup gives two massless Dirac fermions in the continuum limit, and it turns out that no fine-tuning of $m$ is required because the extra $U(1)$ symmetry at the central branch, $U(1)_{\overline{V}}$, prohibits additive mass renormalization. Moreover, we show that the Dirac determinant is positive semi-definite and this formulation is free from the sign problem, so a Monte Carlo simulation of the path integral is possible. By identifying the symmetry at low energy, we show that this lattice model has a mixed ’t Hooft anomaly between $U(1)_{\overline{V}}$, lattice translation, and lattice rotation. We discuss its relation to the anomaly of half-integer anti-ferromagnetic spin chains, so our lattice gauge theory is suitable for numerical simulation of the Haldane conjecture. Furthermore, it gives a new and strict understanding on the parity-broken phase (Aoki phase) of the $2$D Wilson fermion.
Highlights
Most reliable techniques to study non-perturbative physics of asymptotically-free QFTs, including Yang-Mills theory and Quantum Chromodynamics (QCD)
We identify its symmetry at low energy scale and find that this lattice model has the same ’t Hooft anomaly with that of half-integer spin chain
We emphasize that all the symmetries relevant for ’t Hooft anomaly are exact symmetries at the lattice level, and this means that we find the Z2 ’t Hooft anomaly between U (1) × U (1) symmetry at the central branch, lattice translation, and lattice rotation symmetries
Summary
We first give a brief review on Wilson fermion and central-branch Wilson fermion. After that, using this knowledge, we discuss the symmetry of the Dirac spectrum for the central-branch Wilson fermion. It is notable that this extra symmetry prohibits the on-site mass term ψnψn, and eventually prohibits additive mass renormalization as the chiral symmetry in staggered fermion does [50,51,52]. This formulation is regarded as another realization of lattice fermions with the remnant of chiral symmetry, which means we do not need fine-tuning of the mass parameter. A, where 4d two-flavor centralbranch fermion is discussed
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