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For a constant effective density of states and attractive coupling $\lambda$, let the linearized BCS theory gap equation integrate $\tanh (\xi /(2k_B{T}_c))/(2\xi )$ over $-\sigma <\xi <\theta$, with positive cutoffs. Its even integrand makes the result the mean of two positive-cutoff logarithms. When $k_B{T}_c\ll \sigma ,\theta$, $1/\lambda \simeq \frac{1}{2}\ln (1.13\theta /(k_B{T}_c))+\frac{1}{2}\ln (1.13\sigma /(k_B{T}_c))$, giving the displayed geometric mean of cutoffs.

If a band bottom truncates the lower side, $\sigma =\min (\mu ,\theta )$ for positive chemical potential $\mu$. The weak-coupling limit at fixed $\mu /\theta >0$ must not be interchanged with an uncontrolled $\mu \to 0$ extrapolation: the lower logarithm requires $\mu \gg k_B{T}_c$ when $\mu <\theta$.

A critical coupling is a coupling value at which a gap equation first admits a massless solution. Tuning through it can separate symmetric gapped and symmetry-broken phases.

A gap equation determines a physical mass or excitation gap self-consistently from an interaction and a regulated fluctuation integral.

The Large-N expansion organizes an $O(N)$-symmetric theory in powers of $1/N$. At leading order, an auxiliary constraint field is determined by a self-consistent saddle-point or gap equation.

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.