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.

27 pages, 4 figures