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

TL;DR

This paper links magnetization profiles at the upper critical dimension to solutions of the Yamabe problem, deriving explicit formulas and confirming findings through simulations, thus bridging statistical physics and differential geometry.

Contribution

It establishes a novel connection between magnetization profiles in critical models and the integer Yamabe problem, providing explicit solutions and extending previous results.

Findings

Explicit formulas for magnetization profiles using Weierstrass elliptic functions

Connection between boundary conditions and elliptic function singularities

Good agreement between theoretical profiles and Monte Carlo simulations

Abstract

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