On the role of Fourier modes in finite-size scaling above the upper critical dimension

TL;DR

This paper investigates the role of Fourier modes in finite-size scaling above the upper critical dimension, revealing that the traditional phenomenological picture is incomplete and highlighting the importance of a previously overlooked sector of the renormalization group.

Contribution

It demonstrates that the standard understanding of finite-size scaling above the upper critical dimension is incorrect for certain observables and introduces the significance of an unphysical RG sector.

Findings

The current phenomenological picture does not hold for all thermodynamic observables.

A sector of the renormalization group, previously considered unphysical, is crucial for correct finite-size scaling.

Finite-size systems with free boundaries above $d_c$ are experimentally relevant and require revised theoretical understanding.

Abstract

Renormalization-group theory stands, since 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. Besides its fundamental significance for scaling theories, the issue is important at a practical level because finite-size, statistical-physics systems, with free boundaries above $d_c$, are experimentally accessible with 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 either with free or periodic boundaries. Instead, the correct picture emerges from a sector of the renormalization group hitherto considered unphysical.