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

Year: 2021

Authors: Galvani A., Gori G., Trombettoni A.

Autors Affiliation: SISSA, Via Bonomea 265, I-34136 Trieste, Italy; INFN, Sez Trieste, Via Bonomea 265, I-34136 Trieste, Italy; Heidelberg Univ, Inst Theoret Phys, D-69120 Heidelberg, Germany; CNR, IOM DEMOCRITOS Simulat Ctr, Via Bonomea 265, I-34136 Trieste, Italy; Univ Trieste, Dept Phys, Str Costiera 11, I-34151 Trieste, Italy.

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 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 four-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 Gori and Trombettoni [J. Stat. Mech: Theory Exp. (2020) 063210].

Journal/Review: PHYSICAL REVIEW E

Volume: 104 (2)      Pages from: 24138-1  to: 24138-12

More Information: We thank J. Cardy, N. Defenu, S. Dietrich, T. Enss, A. Gambassi, and A. Squarcini for useful discussions at various stages of this work. G.G. is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1-390900948 (the Heidelberg STRUCTURES Excellence Cluster).
KeyWords: Percolation Problem; Casimir Amplitudes; Critical Exponents; Matrix; Model
DOI: 10.1103/PhysRevE.104.024138