 95162 Marchetti D.H.U., Faria da Veiga P.A., Hurd T.R.
 THE 1/NEXPANSION AS A PERTURBATION ABOUT THE MEAN FIELD THEORY:
A OneDimensional Fermi Model
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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 onedimensional chiralinvariant Fermion quantum field
theory with quartic interaction, with $N$ the number of flavours. After
applying the MatthewsSalam formula, the model becomes a statistical mechanical
model of a chain of continuous Gaussian spins, coupled with a certain
nonstandard weight function. Finally, the model can also be considered as a
probability measure on the set of tridiagonal matrices with fixed offdiagonal
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 onedimensional 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 GrossNeveu. The analytical technique we develop
in essence combines transfer matrix method with Laplace method (steepest
descent) for asymptotics of integrals.
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