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arXiv:2402.00427 · Entropy

Ralph Kenna’s Scaling Relations in Critical Phenomena

Leïla Moueddene, Arnaldo Donoso, Bertrand Berche

Published Feb 29, 2024• 8 citations
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Abstract

In this note, we revisit the scaling relations among "hatted critical exponents", which were first derived by Ralph Kenna, Des Johnston, and Wolfhard Janke, and we propose an alternative derivation for some of them. For the scaling relation involving the behavior of the correlation function, we will propose an alternative form since we believe that the expression is erroneous in the work of Ralph and his collaborators.

DOI
10.3390/e26030221

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In this note, we revisit the scaling relations among "hatted critical exponents", which were first derived by Ralph Kenna, Des Johnston, and Wolfhard Janke, and we propose an alternative derivation for some of them. For the scaling relation involving the behavior of the correlation function, we will propose an alternative form since we believe that the expression is erroneous in the work of Ralph and his collaborators.
We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models at the upper critical behavior (for short and long range φ n -theories) and below it.This allows us to check the scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and Fisher zeroes) of these models.Moreover, we have obtained an explicit method to compute the ϙ exponent [defined by ξ ∼ L(log L) ϙ] and, finally, we have found a new derivation of the scaling law associated with it.
Here proposed is a revised version of the scaling theory of equilibrium critical phenomena. This revised theory is based on hypotheses which differ only slightly from those set forth in original formulations. The conclusions reached are essentially equivalent to those of Stell's weak scaling theory. One obtains relations which require prior knowledge of three exponents for the derivation of the rest. An additional exponent, denoted w, enters certain relations containing exponents which pertain to the correlation function. This exponent must be greater than or equal to one in order to satisfy certain generally accepted inequalities. An argument for why w should be greater than or equal to one is formulable within the framework of the original hypotheses of scaling theory. While the revised theory has the drawback of containing an extra unknown, it is nevertheless capable of reconciling scaling theory with other theories of critical exponents.
There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.
A single assumption is made about the dynamic correlation function, that close to the critical point and for large r and t, it is a generalized homogeneous function. After proving and using certain simple theorems on this class of functions, contact is made with previous formulations of dynamic and static scaling. It is found that 1. this single assumption is sufficient to obtain all previous results of both the dynamic and static scaling hypotheses, 2. simple expressions, which can be written down ‘by inspection’ for all critical point exponents, enable all the exponent relations to be derived immediately, 3. we have additional ways of plotting data in the form of ‘scaling functions’, 4. it becomes possible to easily and accurately calculate the scaling function for models (such as the Heisenberg model) for which no low‐temperature series expansions exist, and 5. the two‐exponent relations (e. g., dν=2−α) are not consequences of our approach unless we make an additional assumption, an assumption made implicitlyby many other workers. If we don't make this additional assumption, the two‐exponent relations are found to have correction terms which transform them into ordinary thermodynamic scaling laws.