Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension.

Emilio José Flores-Sola (1, 2) , Bertrand Berche (1) , Ralph Kenna (2) , Martin Weigel (2)

1 Equipe 106, IJL - Groupe de Physique statistique = Statistical Physics Group [Institut Jean Lamour]
2 AMRC - Applied Mathematics Research Center, Coventry University

Renormalization-group theory has stood, for over 40 years, as one of the pillars of modern physics. As such, there should be no remaining doubt regarding its validity. However, finite-size scaling, which derives from it, has long been poorly understood above the upper critical dimension d_c in models with free boundary conditions. In addition to its fundamental significance for scaling theories, the issue is important at a practical level because finite-size, statistical-physics systems with free boundaries and above d_c are experimentally relevant for long-range interactions. Here, we address the roles played by Fourier modes for such systems and show that the current phenomenological picture is not supported for all thermodynamic observables with either free or periodic boundaries. In particular, the expectation that dangerous irrelevant variables cause Gaussian-fixed-point scaling indices to be replaced by Landau mean-field exponents for all Fourier modes is incorrect. Instead, the Gaussian-fixed-point exponents have a direct physical manifestation for some modes above the upper critical dimension.

Domaines: Mécanique statistique [cond-mat.stat-mech]

Emilio José Flores-Sola, Bertrand Berche, Ralph Kenna, Martin Weigel. Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension.. Physical Review Letters, 2016, 116, ⟨10.1103/PhysRevLett.116.115701⟩. ⟨hal-01298202⟩