A new critical exponent koppa and its logarithmic counterpart koppa-hat

Abstract

It is well known that standard hyperscaling breaks down above the upper critical dimension d_c, where the critical exponents take on their Landau values. Here we show that this is because, in standard formulations in the thermodynamic limit, distance is measured on the correlation-length scale. However, the correlation-length scale and the underlying length scale of the system are not the same at or above the upper critical dimension. Above d_c they are related algebraically through a new critical exponent koppa, while at d_c they differ through logarithmic corrections governed by an exponent koppa-hat. Taking proper account of these different length scales allows one to extend hyperscaling to all dimensions.

Invited contribution to a special edition of CMP: Phase transitions and critical phenomena: universality and non-universal features. 13 pages, 6 figures

References in corpus (3)

• Scaling Relations for Logarithmic Corrections
• Self-Consistent Scaling Theory for Logarithmic Correction Exponents
• Non-vanishing boundary effects and quasi-first order phase transitions in high dimensional Ising models

Cited by in corpus (5)

• Finite-size scaling above the upper critical dimension
• Finite-size scaling above the upper critical dimension in Ising models with long-range interactions
• Fisher's scaling relation above the upper critical dimension
• Critical and tricritical behavior of the $d=3$ Blume-Capel model: Results from small-scale Monte Carlo simulations
• Dyson hierarchical quantum ferromagnetic Ising chain with pure or random transverse fields