The 1/N-expansion as a perturbation about the mean field theory: A one-dimensional fermi model

Abstract

We examine a family of probability measures onR ^L with real parameter ζ>0 and integer parametersN,L>0. Each such measure is equivalent to the lattice version of a one-dimensional discrete chiral-invariant fermionic quantum field theory with quartic interaction, withN the number of flavours. After applying the Matthews-Salam formula, the model becomes a statistical mechanical model of a chain of continuous Gaussian spins, coupled with a certain non-standard weight function. Finally, the model can also be considered as a probability measure on the set of tridiagonal matrices with fixed off-diagonal and random diagonal entries.

Our analysis shows how to develop an asymptotic expansion in 1/N, valid for allL and ζ, for the fundamental expectation values. From this it follows that the two point fermion correlation function decays with a mass which agrees to the leading order in 1/N with the mean field value calculated by the argument of Gross-Neveu. The analytical technique we develop in essence combines a transfer matrix method with the Laplace method (steepest descent) for asymptotics of integrals.