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Every term in $\tilde{S}$ is proportional to $N$. The Large-N expansion therefore suppresses fluctuations of the auxiliary field by powers of $1/N$, and the leading partition function comes from a saddle-point approximation.

Vary with respect to $i\lambda$ and use $\delta \mathrm{Tr}\log K=\mathrm{Tr}(K^{-1}\delta K)$. At a translation-invariant saddle $i\lambda =m^2$, the gap equation isThe frequencies are the bosonic Matsubara frequencies imposed by periodicity around the imaginary-time thermal circle.

In the long-wavelength $O(N)$ description of an antiferromagnet, $v$ is the antiferromagnetic spin wave velocity and $g$ controls the stiffness or strength of quantum fluctuations. The field $\lambda$ is an auxiliary Lagrange multiplier enforcing the fixed-length constraint on the order parameter. At the translation-invariant saddle, $i\lambda =m^2$ shifts every propagator denominator and $m$ is the excitation gap, which explains the name gap equation.

The Large-N expansion makes fluctuations of the auxiliary field relatively small. Its leading saddle-point condition is then self-consistent and becomes the displayed gap equation.