Weyl Fermions on a Finite Lattice

The phenomenon of unpaired Weyl fermions appearing on the sole $2n$-dimensional boundary of a ($2n+1$)-dimensional manifold with massive Dirac fermions was recently analyzed in D. B. Kaplan [preceding Letter, Chiral gauge theory at the boundary between topological phases, Phys. Rev. Lett. 132, 141603 (2024).]. In this Letter, we show that similar unpaired Weyl edge states can be seen on a finite lattice. In particular, we consider the discretized Hamiltonian for a Wilson fermion in ($2+1$) dimensions with a $1+1$ dimensional boundary and continuous time. We demonstrate that the low lying boundary spectrum is indeed Weyl-like: it has a linear dispersion relation and definite chirality and circulates in only one direction around the boundary. We comment on how our results are consistent with Nielsen-Ninomiya theorem. This work removes one potential obstacle facing the program outlined in D. B. Kaplan, preceding Letter, for regulating chiral gauge theories.

Figure 1

The 1800 ordered eigenvalues ${\omega }_n$ on the $y$ axis versus $n$ on the $x$ axis for the free Wilson fermion Hamiltonian with $a=M=r=1$ on a $30\times 30$ lattice, with a magnified view of the crossing point in the lower panel. Cyan points are for purely periodic boundary conditions, corresponding to a spatial manifold with no boundary, and exhibit a gap. Magenta indicates mixed periodic and open boundary conditions, as specified by Shamir [10] for domain wall fermions, supporting a Weyl fermion on each of its two $S^1$ surfaces with opposite chiralities, exact zero modes, and evident degeneracy. The nondegenerate black points are for purely open boundary conditions, representing the situation described in Ref. [3]: a manifold with a single boundary (a square, in this case) supporting a single Weyl fermion without an exact zero mode.

Figure 2

The energy for every mode plotted versus the expectation value of the angular momentum $J$ on a radius $R=34$ disk of square lattice with open boundary conditions. The straight line is a plot of the function $\omega =-⟨J⟩/R$, where $⟨J⟩/R$ can be identified as the edge state momentum along the boundary.

Figure 3

The fermion number charge densities $\rho$ (left) and angular fermion number current density $j_{\theta }$ (right) on a radius $R=34$ disk of square lattice with open boundary conditions for energy eigenstates with energies $\omega =0.0831$, 0.5199, 1.0001 in lattice units (top to bottom). Both the $\rho$ and $j_{\theta }$ plots display sharp edge localization for the two modes in the Weyl gap $|\omega |<1$, and delocalization for the mode slightly above $\omega =1$. The Weyl modes exhibit strictly negative $j_{\theta }$ corresponding to clockwise rotation around the lattice boundary, while the bulk mode does not display any net chirality. The data has been rescaled by the maximum magnitude of the density value.