= Double-well threshold of an auxiliary Ising potential
{title2=$J>\tfrac12\tanh(\kappa/2)$}
The zero-field <Hubbard–Stratonovich transformation> of an exponentially coupled <Ising model> produces $U(m)=Am^2-\log[2\cosh(2m)]$, where $A=\tanh(\kappa/2)/J>0$. Since $U''(0)=2A-4$ and $U$ tends to positive infinity, a symmetric double well exists exactly when $A<2$. Its nonzero stationary points solve $Am=\tanh(2m)$; the ratio $\tanh(2m)/m$ decreases from 2 to 0 for $m>0$, proving that there is exactly one positive minimum. At $A=2$ the minimum is a single quartic minimum. A small conjugate field adds $-h\tanh(2m)+O(h^2)$ and tilts the wells, while a sufficiently large field can remove the metastable well. A finite barrier permits <quantum tunnelling>; two classical minima alone do not establish <spontaneous symmetry breaking>.
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