An illustration of chiral fermions on a 1+1 dimensional lattice
  The vectorlike doubling of low-energy excitations is in fact a natural consequence of the pair-production around the zero-energy (E=0) due to the quantum field fluctuations of the lattice regularized vacuum. On the 1+1 dimensional lattice, we study an anomaly-free chiral model (11112) of four left-movers and one right-mover with strong interactions. Exact computations of relevant S-matrices illustrate that for high-momentum states, a negative energy-gap (E<0) develops; the bound state and its constituents, which have the same quantum numbers but opposite chiralities, fill the same energy-state so that chiral symmetries are preserved; for low-momentum states, the negative energy-gap vanishes and the bound state dissolves into its constituents near zero energy. As a consequence of the gauge-anomaly cancellation and the index theorem for flavor-singlet anomalies, the net number of zero-modes pushed down into and pumped out from the zero-energy level by the gauge field is zero. 
She-Sheng Xue