95-162 Marchetti D.H.U., Faria da Veiga P.A., Hurd T.R.

THE 1/N-EXPANSION AS A PERTURBATION ABOUT THE MEAN FIELD THEORY: A One-Dimensional Fermi Model (68K, LaTeX) Mar 24, 95

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Abstract. We examine a family of probability measure on ${\bf R}^L$ with real parameter $\zeta >0$ and integer parameters $N,L >0$. Each such measure is equivalent to the lattice version of a one-dimensional chiral-invariant Fermion quantum field theory with quartic interaction, with $N$ 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 all $L$ and $\zeta$, for the fundamental expectation values. In particular the analysis proves that the model behaves like a one-dimensional Ising model as $L \to \infty$ for $N$ large, and thus remains in a pure phase for all values of $\zeta$. It is also shown that the Fermion model has a mass gap 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 transfer matrix method with Laplace method (steepest descent) for asymptotics of integrals.

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