This paper shows how interactions can open a mass gap for chiral fermions on a strictly one-dimensional lattice while preserving the symmetry that normally forbids mass terms. The authors study the anomaly-free “3–4–5–0” model — a minimal set of four chiral fermion types with charges 3, 4, 5 and 0 — and demonstrate symmetric mass generation, meaning a gap appears without spontaneous symmetry breaking.

Two obstacles stood in the way. First, any local lattice discretization that preserves chirality usually produces extra, unwanted mirror fermions, a problem known as fermion doubling. Second, the natural six-fermion interaction that can gap the 3–4–5–0 system is perturbatively irrelevant at weak coupling: its scaling dimension is 5, so it fades away under renormalization group (RG) flow instead of opening a gap.

To avoid doubling the authors use a nonlocal “tangent-fermion” lattice based on Stacey’s long-range hopping. The hopping between lattice sites is tnm = 2 i t0 (−1)^{n−m}, giving a dispersion E(k)=2 t0 tan(k/2). That dispersion has a single chiral branch in the Brillouin zone, so there is no mirror partner. Although the hopping is all-to-all, the problem can be reformulated as a local generalized eigenvalue problem with nearby couplings, which keeps numerical methods such as tensor networks and density-matrix renormalization group (DMRG) efficient.

To make the six-fermion gapping interaction visible at weak coupling the authors add a Hubbard-type density–density interaction. This four-fermion term does not open a gap by itself, but it renormalizes the scaling dimension of the six-fermion term from 5 down to 5K, where K is an effective Luttinger parameter. When K<2/5 the six-fermion term becomes relevant in RG language and can open a gap. The paper also explains how to avoid rapid oscillations in densities (Friedel oscillations) by choosing fillings so that certain conserved combinations N(p) vanish, which helps the gapping interaction act coherently.