Abstract
In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models, which are in this class, have been intensively studied. Here, numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5, and 7. Hyperscaling is clearly violated in dimensions 3 and 4, as well as above the upper critical dimension . Estimates are obtained for the “violation of hyperscaling exponent” values in the various models.
Highlights
It has been tacitly or explicitly assumed that Edwards–Anderson Ising Spin Glasses (ISGs), where the quenched interactions are random, follow the same basic scaling and Universality rules as in the canonical Ising models, whose properties are understood in great detail through Renormalization
A textbook definition of hyperscaling is: “Identities obtained from the generalised homogeneity assumption involving the space dimension D are known as hyperscaling relations.” [1]
Though not conventionally written this way, in the standard Ising models above D = 4 equivalent modified hyperscaling relations 2 − α = ( D − θ )ν = 2 and 2∆ = ( D − θ )ν + γ = 3 can be seen by inspection to be consistent with the mean field exponents plus a violation exponent θ = D − 4
Summary
It has been tacitly or explicitly assumed that Edwards–Anderson Ising Spin Glasses (ISGs), where the quenched interactions are random, follow the same basic scaling and Universality rules as in the canonical Ising models, whose properties are understood in great detail through Renormalization. The hyperscaling relations valid in canonical Ising models below the upper critical dimension are:. The breakdown of hyperscaling in the 3D Random Field Ising model (RFIM) has been extensively studied [5,6,7,8]. Though not conventionally written this way, in the standard Ising models above D = 4 equivalent modified hyperscaling relations 2 − α = ( D − θ )ν = 2 and 2∆ = ( D − θ )ν + γ = 3 can be seen by inspection to be consistent with the mean field exponents plus a violation exponent θ = D − 4. Ising spin glasses (ISGs) are systems with quenched randomness in which hyperscaling might be expected to break down, from a generalization of Schwartz’s argument. We are not aware of any tests of hyperscaling in ISGs
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