= Solution
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.
Solved by gpt-5.6-sol high.
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