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

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].

Version published in Physical Review E. 12 pages, 9 figures, 2 tables

References in corpus (7)

• Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations
• Monte Carlo simulation results for critical Casimir forces
• Critical behaviour of the Ising model on the 4-dimensional lattice
• Thinning of superfluid films below the critical point
• Thermodynamic Casimir effects involving interacting field theories with zero modes
• Relation between the thermodynamic Casimir effect in Bose-gas slabs and critical Casimir forces
• Topological excitations in statistical field theory at the upper critical dimension

Cited by in corpus (2)

• Critical Casimir Effect: Exact Results
• Critical $1$- and $2$-point spin correlations for the $O(2)$ model in $3d$ bounded domains