Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem

We study the connection between magnetization profiles of models described by a scalar field with marginal interaction term in bounded domain and solutions so-called Yamabe problem same domain, which amounts to finding metric having constant curvature. Taking slab as reference we first at upper critical dimensions $d=3$, $4$, $6$ for different (scale invariant) boundary conditions. By studying saddle-point equations magnetization, find general formulas terms Weierstrass elliptic functions, extending exact results known literature new ones case percolation. The zeros poles can be put direct then show that, any dimension $d$, are solution corresponding integer equation $d$ specific $4$-dimensional Ising model fixed conditions compared Monte Carlo simulations, good agreement. These explicitly confirm recent presented [1].