Relevant operators in exactly solvable lattice Hamiltonians

A recent suggestion that the Hamiltonian of lattice QCD may be rendered exactly solvable by quadrature after addition of an irrelevant operator is critically analyzed. It is shown that a precisely analogous procedure may be applied to the nonlinear $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ $\sigma$ model. An exact solution may be performed in the large-$N$ limit in this latter case, and the modifying operator found to be a relevant perturbation of the original theory, but only after summing to all orders of perturbation theory.