Lattice gauge theory for Haldane conjecture and central-branch Wilson fermion

TL;DR

This paper develops a lattice gauge theory model using central-branch Wilson fermions to simulate the Haldane conjecture, avoiding fine-tuning and sign problems, and explores its symmetry and anomaly properties relevant to condensed matter physics.

Contribution

It introduces a novel lattice formulation with central-branch Wilson fermions that naturally yields massless fermions and is suitable for simulating the Haldane conjecture without fine-tuning.

Findings

The model has positive semi-definite Dirac determinant, enabling sign-problem-free simulations.

It exhibits a mixed 't Hooft anomaly linking symmetries and lattice transformations.

The formulation provides new insights into the parity-broken phase of 2D Wilson fermions.

Abstract

We develop the $(1+1)$d lattice $U(1)$ gauge theory in order to define $2$-flavor massless Schwinger model, and discuss its connection with Haldane conjecture. We propose to use the central-branch Wilson fermion, which is defined by relating the mass, $m$, and the Wilson parameter, $r$, as $m+2r=0$. 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, $U(1)_{\bar{V}}$, prohibits the additive mass renormalization. Moreover, we show that Dirac determinant is positive semi-definite and this formulation is free from the sign problem, so the Monte Carlo simulation of the path integral is possible. By identifying the symmetry at low energy, we show that this lattice model has the mixed 't Hooft anomaly between $U(1)_{\bar{V}}$, 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 Haldane conjecture. Furthermore, it gives new and strict understanding on parity-broken phase (Aoki phase) of $2$d Wilson fermion.