Double-well threshold of an auxiliary Ising potential ($J>\frac{1}{2}\tanh (\kappa /2)$)

The zero-field Hubbard–Stratonovich transformation of an exponentially coupled Ising model produces $U(m)=A{m}^2-\log [2\cosh (2m)]$, where $A=\tanh (\kappa /2)/J>0$. Since $U^{\prime \prime }(0)=2A-4$ 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.