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)
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.