Non Trivial Saddle Points and Band Structure of Bound States of the Two Dimensional O(N) Vector Model

TL;DR

This paper finds exact non-trivial space-dependent saddle points in 0+1 and 1+1 dimensional O(N) scalar field theories, revealing a band structure of bound states and collective excitations.

Contribution

It introduces a novel method combining saddle point techniques and Schrödinger operator properties to identify exact solutions and interpret them as collective excitations with band structures.

Findings

Exact non-trivial saddle points found in O(N) models.

Bound states exhibit a band structure similar to molecular spectra.

Results connect to classical and previous quantum analyses of the O(N) model.

Abstract

We discuss O(N) invariant scalar field theories in 0+1 and 1+1 space-time dimensions. Combining ordinary ``Large N" saddle point techniques and simple properties of the diagonal resolvent of one dimensional Schr\"odinger operators we find {\it exact} non-trivial (space dependent) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the 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. We comment on the relation of our results to the classical work of Dashen, Hasslacher and Neveu and to a previous analysis of bound states in the O(N) model by Abbott.}