Non-trivial saddle points and band structure of bound states of the two-dimensional O(N) vector model

We discuss O(N) invariant scalar field theories in 0 + 1 space-time dimensions (quantum mechanics) and in 1 + 1 space-time dimensions (field theory). Combining ordinary "Large N" saddle point techniques and simple properties of the diagonal resolvent of one-dimensional Schrödinger operators we find non-trivial (non-constant) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the theory. In the "Large N" limit these saddle points are exact for the quantum mechanical case, but only approximate in the two-dimensional theory. In the latter case they are the leading contributions to the time evolution kernel at short times, or equivalently, the leading contribution to the high temperature expansion of partition function stemming from space dependent static configurations in case of the Euclidean theory. We interpret these novel saddle points as collective O(N) singlet excitations of the field theory, each embracing a host of finer quantum states arranged in O(N) multiplets, in an analogous manner to the band structure of molecular spectra.