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Finite-size scaling in high dimensions

Paper on the role of Fourier modes in FSS published in PRL

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A recent paper of graduate student Emilio Flores together with Bertrand Berche from Universite de Lorraine, and Ralph Kenna pioneered the understanding of the role of Fourier modes in the finite-size scaling at critical points of systems in high dimensions.

E.J. Flores-Sola, B. Berche, R. Kenna, and M. Weigel, Role of Fourier modes in finite-size scaling above the upper critical dimension, Phys. Rev. Lett. 115, 115701 (2016)

»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 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 either with  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. Instead, from the picture which emerges there, the Gaussian fixed point exponents are shown to play an unequivocal role above the upper critical dimension.«

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