  Chiral field theories describe large classes of matter, from the edges of
Quantum Hall systems to the electroweak sector of the Standard Model, but
defining them on the lattice has been an ongoing challenge due to a no-go
theorem precluding free local models, the potential of symmetry anomalies, and
sign problems. Some approaches define a $1+1$d chiral field theory as the edge
of a $2+1$d system and argue that the edge decouples from the bulk, but this
can be difficult to verify due to finite size effects and strong interactions.
On the other hand, recent work has shown how to define the $2+1$d bulk theory
as an exactly solvable model with zero correlation length, in which case the
edge theory may be extracted exactly. We use these techniques to derive a
lattice field theory on a $1+1$d spacetime lattice which carries an anomalous
chiral $U(1)$ symmetry with zero chiral central charge. The lattice theory with
anomalous chiral $U(1)$ symmetry is always gapless, regardless of lattice
interactions. We demonstrate the chiral anomaly by coupling to a background
gauge field, develop a field theory which demonstrates the chiral behavior, and
show how to assemble a chiral, anomaly-free theory where the gauge field may be
taken to be dynamical.
