  A model with a half boson degree of freedom per lattice site in one dimension
is developed. The boson is protected from developing a gap by translation
symmetry: while the left movers are at zero quasi-momentum, the associated
right movers are at the midpoint of the quasi-momentum period. The model has
different properties depending on if a periodic lattice has an even or an odd
number of sites and similar features are found for open boundary conditions. A
special case of the non-linear half boson model where even and odd lattice
sites contribute differently to the Hamiltonian gives rise to the Toda chain
and a more symmetric generalization of the Toda chain is found. Upon periodic
identifications of the half bosons degrees of freedom under a shift, the total
Hilbert space has a finite dimension and can be encoded in finitely many qubits
per unit length. This way one finds interesting critical spin chains, examples
of which include the critical Ising model in a transverse magnetic field and
the 3-state Potts model at criticality. Extensions to higher dimensions are
considered. Models obtained this way automatically produce dynamical systems of
gapless fractons.
