Fisher's scaling relation above the upper critical dimension

TL;DR

This paper clarifies the apparent discrepancy in Fisher's scaling relation above the upper critical dimension by distinguishing between different length scales, showing the relation holds on the correlation scale but requires modification on the system scale.

Contribution

It introduces a scale-dependent interpretation of Fisher's relation above the upper critical dimension, resolving longstanding numerical and theoretical inconsistencies.

Findings

Fisher's relation holds on the correlation length scale above d_c.

On the system length scale, the anomalous dimension is negative, requiring a modified relation.

Similar scale-dependent relations are derived at the upper critical dimension involving logarithmic corrections.

Abstract

Fisher's fluctuation-response relation is one of four famous scaling formulae and is consistent with a vanishing correlation-function anomalous dimension above the upper critical dimension d_c. However, it has long been known that numerical simulations deliver a negative value for the anomalous dimension there. Here, the apparent discrepancy is attributed to a distinction between the system-length and correlation- or characteristic-length scales. On the latter scale, the anomalous dimension indeed vanishes above d_c and Fisher's relation holds in its standard form. However, on the scale of the system length, the anomalous dimension is negative and Fisher's relation requires modification. Similar investigations at the upper critical dimension, where dangerous irrelevant variables become marginal, lead to an analogous pair of Fisher relations for logarithmic-correction exponents. Implications of a similar distinction between length scales in percolation theory above d_c and for the Ginzburg criterion are briefly discussed.