String tension scaling in models of the confined phase

We introduce a D-dimensional Hamiltonian formalism for the study of Polyakov loop models of finite temperature gauge theories in D+1 dimensions. Polyakov loop string tensions are obtained from energy eigenstates of the Hamiltonian. For D=1, the gauge theory reduces to quantum mechanics on the gauge group; for D>1, the Hamiltonian includes hopping terms that link sites on the transverse lattice. The deconfined phase is associated with a ground state which breaks Z(N) symmetry, and Svetitsky-Yaffe critical universality emerges naturally for D>1. A minimal model is proposed which naturally reproduces approximate Casimir scaling for a range of couplings. Different classes of potentials lead to different pictures of how confinement is realized. Such potential energy terms also modify string tension scaling laws, as we demonstrate using two potentials: one representing the perturbative thermal contributions from gluons, and the other arising from magnetic monopoles in certain confining supersymmetric theories.