  I solve a quantum chain whose Hamiltonian is comprised solely of local
four-fermi operators by constructing free-fermion raising and lowering
operators. The free-fermion operators are both non-local and highly non-linear
in the local fermions. This construction yields the complete spectrum of the
Hamiltonian and an associated classical transfer matrix. The spatially uniform
system is gapless with dynamical critical exponent z=3/2, while staggering the
couplings gives a more conventional free-fermion model with an Ising
transition. The Hamiltonian is equivalent to that of a spin-1/2 chain with
next-nearest-neighbour interactions, and has a supersymmetry generated by a sum
of fermion trilinears. The supercharges are part of a large non-abelian
symmetry algebra that results in exponentially large degeneracies. The model is
integrable for either open or periodic boundary conditions but the free-fermion
construction only works for the former, while for the latter the extended
symmetry is broken and the degeneracies split.
