Hamiltonian formulation of the mean spherical model in 1 + 1 dimensions

The exact spectrum of the mean spherical model in 1 + 1 dimensions is calculated using a lattice Hamiltonian formulation. This model is equivalent to the $O\left(N\right)\mathrm{}$ Heisenberg model in the limit $N\to \infty$, and exists only in a disordered phase. I investigate the singularity structure of the theory, and discuss implications for both weak- and strong-coupling approximations. I compare my results to approximate calculations for the $O\left(3\right)\mathrm{}$ and $O\left(4\right)\mathrm{}$ models by Hamer, Kogut, and Susskind, and find that the $N\to \infty$ limit reproduces many of their semiquantitative results concerning the transition from weak to strong coupling.