Fisher's scaling relation above the upper critical dimension

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.

arXiv commentPublished version has 6 pages, 2 figures

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• Quantum Gravity as an Information Network: Self-Organization of a 4D Universe
• Random-length Random Walks and Finite-size scaling in high dimensions
• Geometric explanation of anomalous finite-size scaling in high dimensions
• Scaling at quantum phase transitions above the upper critical dimension
• Finite-temperature critical behaviors in 2D long-range quantum Heisenberg model
• Lee-Yang theory of the Curie-Weiss model and its rare fluctuations
• Universal Finite-Size Scaling for Percolation Theory in High Dimensions
• Finite-size Scaling of O($n$) Systems at the Upper Critical Dimensionality
• Anomalous mean-field behavior of the fully connected Ising model
• Monte Carlo based techniques for quantum magnets with long-range interactions
• Complete graph and Gaussian fixed point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries
• The length of self-avoiding walks on the complete graph
• Lee-Yang theory of Bose-Einstein condensation
• Logarithmic finite-size scaling of the self-avoiding walk at four dimensions
• Non-equilibrium relaxations: ageing and finite-size effects
• Dyson hierarchical quantum ferromagnetic Ising chain with pure or random transverse fields
• Effective-Dimension Theory of Critical Phenomena above Upper Critical Dimensions
• When correlations exceed system size: finite-size scaling in free boundary conditions above the upper critical dimension
• On a previously unpublished work with Ralph Kenna