title: Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension.
abstract: 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.
authors.affiliation: Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom.|Institut Jean Lamour, CNRS/UMR 7198, Groupe de Physique Statistique, Université de Lorraine, BP 70239, F-54506 Vandœuvre-les-Nancy Cedex, France.|Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (𝕃4), D-04009 Leipzig, Germany.
authors.forename: Emilio
authors.initials: E
authors.lastname: Flores-Sola
authors.affiliation: Institut Jean Lamour, CNRS/UMR 7198, Groupe de Physique Statistique, Université de Lorraine, BP 70239, F-54506 Vandœuvre-les-Nancy Cedex, France.|Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (𝕃4), D-04009 Leipzig, Germany.
authors.forename: Bertrand
authors.initials: B
authors.lastname: Berche
authors.affiliation: Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom.|Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (𝕃4), D-04009 Leipzig, Germany.
authors.forename: Ralph
authors.initials: R
authors.lastname: Kenna
authors.affiliation: Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom.|Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (𝕃4), D-04009 Leipzig, Germany.
authors.forename: Martin
authors.initials: M
authors.lastname: Weigel