In the long-wavelength

O (N)

description of an antiferromagnet,

v

g

controls the stiffness or strength of quantum fluctuations. The field

λ

is an auxiliary Lagrange multiplier enforcing the fixed-length constraint on the order parameter. At the translation-invariant saddle,

iλ = m^{2}

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.

ii

Solution

T \to 0

, the Matsubara sum becomes

\int d ω / (2 π)

. Put

p_{0} = ω / v

and

M = m / v

. The factors of

v

cancel from both sides, leaving

\frac{1}{g} = \int^{Λ} \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{1}{p ^{2} + M ^{2}} .

Using four-dimensional spherical coordinates,

\frac{1}{g} = \frac{2 π ^{2}}{( 2 π ) ^{4}} \int_{0}^{Λ} \frac{p ^{3} d p}{p ^{2} + M ^{2}} = \frac{1}{16 π ^{2}} [Λ^{2} - M^{2} lo g (1 + \frac{Λ ^{2}}{M ^{2}})] .

At the onset of spontaneous symmetry breaking, the symmetric-phase gap closes. Thus

\frac{1}{g _{c}} = \frac{Λ ^{2}}{16 π ^{2}}, g_{c} = \frac{16 π ^{2}}{Λ ^{2}} .

For

g < g_{c}

the constraint is instead satisfied by an ordered condensate; a positive symmetric gap exists for

g > g_{c}

Solved by gpt-5.6-sol high.

Subtracting the critical equation from the massive equation gives

\frac{1}{g _{c}} - \frac{1}{g} = \frac{M ^{2}}{16 π ^{2}} lo g (1 + \frac{Λ ^{2}}{M ^{2}}), M = \frac{m}{v} .

g ↓ g_{c}

from above,

\frac{g - g _{c}}{g _{c}^{2}} \sim \frac{M ^{2}}{16 π ^{2}} lo g \frac{Λ ^{2}}{M ^{2}} .

Therefore, up to constants inside the slowly varying logarithm,

m \sim 4 π v [\frac{g - g _{c}}{g _{c}^{2} lo g ( 1/ ( g - g _{c} ))}]^{1/2} .

Solved by gpt-5.6-sol high.

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