Domain-wall fermion

The domain-wall fermion is a lattice fermion formulation that preserves chiral symmetry by embedding the theory in five dimensions, with mass defects at the boundaries of the extra dimension. A four-dimensional chiral fermion emerges as a bound state localized on the domain wall, while its mirror image — an anti-chiral mode — is trapped on the opposite wall, separated by a gap that suppresses mixing. This formulation evades the Nielsen-Ninomiya theorem by sacrificing locality in the fifth dimension, but the residual chiral symmetry breaking decays exponentially with the extent of the extra dimension, making it the method of choice for precision calculations in lattice QCD where chiral purity is paramount.

Domain-wall fermions are typically defended as a technical fix for the lattice chiral problem, an expensive workaround that buys correctness at the cost of computational efficiency. The deeper insight is that the fifth dimension is not an artifice but a revelation: chiral fermions in four dimensions are intrinsically non-local at the cutoff scale, and the domain-wall construction makes this non-locality explicit. The extra dimension is where the anomaly lives in bulk, and the four-dimensional theory on the boundary inherits the chiral anomaly through a mechanism that mirrors the Atiyah-Singer index theorem. The domain wall is not a trick; it is a window.