Electrodynamics of the planar model: Its phase diagram, continuum limit, and mass spectrum

We consider the lattice version of the Abelian Higgs model in 1 + 1 dimensions with the Higgs field constrained to have a constant length. In statistical-mechanics terminology this is the planar Heisenberg model coupled to the electromagnetic field. Both the Euclidean Lagrangian and Minkowski Hamiltonian formulations of this lattice gauge theory are used to study its phase diagram, continuum limit, and mass spectrum. The theory contains two dimensionless coupling constants: $\kappa$, the length of the local spin variable, and $x={\left(2\mathrm{}{g}^2 a^2\right)}^{-1\mathrm{}}$, where $g$ is the electrodynamic coupling and $a$ is the lattice spacing. We find that the theory confines charges for all $\kappa$ and fixed, finite $x$. However, in the continuum limit ($a\to 0$) the theory loses its confinement property and exhibits the Higgs mechanism for ${\kappa }^2>{{\kappa }_c}^2$. The critical point ${{\kappa }_c}^2$ is estimated to lie between 0.6 and 0.8. If ${\kappa }^2<{{\kappa }_c}^2$, the continuum limit of the theory retains its confining property and appears to be simply the free electromagnetic field (a confining linear potential).