Hyperscaling breakdown and Ising Spin Glasses: the Binder cumulant

Abstract

Among the Renormalization Group Theory scaling rules relating critical exponents, there are hyperscaling rules involving the dimension of the system. It is well known that in Ising models hyperscaling breaks down above the upper critical dimension. It was shown by M. Schwartz [Europhys. Lett. {\bf 15}, 777 (1991)] that the standard Josephson hyperscaling rule can also break down in Ising systems with quenched random interactions. A related Renormalization Group Theory hyperscaling rule links the critical exponents for the normalized Binder cumulant and the correlation length in the thermodynamic limit. An appropriate scaling approach for analyzing measurements from criticality to infinite temperature is first outlined. Numerical data on the scaling of the normalized correlation length and the normalized Binder cumulant are shown for the canonical Ising ferromagnet model in dimension three where hyperscaling holds, for the Ising ferromagnet in dimension five (so above the upper critical dimension) where hyperscaling breaks down, and then for Ising spin glass models in dimension three where the quenched interactions are random. For the Ising spin glasses there is a breakdown of the normalized Binder cumulant hyperscaling relation in the thermodynamic limit regime, with a return to size independent Binder cumulant values in the finite-size scaling regime around the critical region.

11 pages, 22 figures

References in corpus (11)

• Universality in three-dimensional Ising spin glasses: A Monte Carlo study
• The critical behavior of three-dimensional Ising spin glass models
• An Extended Scaling Scheme for Critically Divergent Quantities in Ferromagnets and Spin Glasses
• Critical behavior of the three-dimensional bond-diluted Ising spin glass: Finite-size scaling functions and Universality
• Finite size scaling in Ising-like systems with quenched random fields: Evidence of hyperscaling violation
• The high-temperature expansions of the higher susceptibilities for the Ising model in general dimension d
• Extended Scaling in High Dimensions
• Extended Scaling for the high dimension and square lattice Ising Ferromagnets
• The Ising Spin Glass in dimension four
• Evidence for non-universal scaling in dimension four Ising spin glasses
• Critical scaling to infinite temperature