Effective critical exponents for dimensional crossover and quantum systems from an environmentally friendly renormalization group

Abstract

Series for the Wilson functions of an "environmentally friendly" renormalization group are computed to two loops, for an O(N ) vector model, in terms of the "floating coupling", and resummed by the Padé method to yield crossover exponents for finite size and quantum systems. The resulting effective exponents obey all scaling laws, including hyperscaling in terms of an effective dimensionality, d ef f = 4 − γ λ , which represents the crossover in the leading irrelevant operator, and are in excellent agreement with known results.

Probability distribution functions for small scale turbulence

“…Perhaps what is required is a radically new way of applying field theory ideas to turbulence, which is an approach that is being explored in some recent work [9,20,22,29]. It is hoped that the present work will add to this debate by showing that statistical field theory methods, based on T'(v), can, in fact, account for many features of small scale phenomena.…”

Renormalization-Group Calculation of the Dependence on Gravity of the Surface Tension and Bending Rigidity of a Fluid Interface

“…At the same time, those equations are simple statements of the relationship between measurable quantities and the direct correlation function C ͑ R͒. Without knowing anything about the latter, those relationships are empty as far as predicting actual dependences on thermodynamic quantities such m and T and external variables such as g. Our purpose below is to show that with the use of the standard theory of critical phenomena and the renormalization group [25][26][27], adapted to the present case, one can fill the gap and obtain the dependence on gravity.…”

“…Using standard renormalization-group calculations [25,27] one can find, for arbitrary dimension, the renormalized N-vertex functions. Our interest now is the twopoint function, which can be shown to bẽ…”

Thickness Dependence of Effective Critical Exponents in Three-Dimensional Ising Plates

“…Once the exponents β* (D) and δ* (D) are known, it is straightforward to calculate the exponent γ* (D), using the scaling equality This scaling relation has been checked for thin 'sing films all over the crossover region by renormalization group analysis [4] and more recently by Monte Carlo simulations [151. The behavior of η* (D) so obtained is shown in Fig.…”

“…Second, the evolution of the effective critical exponents with temperature in the close vicinity of Τ for systems with the thin film geometry and with free surfaces has been studied by means of series expansion [3], the renormalization group [4] and high accuracy Monte Carlo calculations in 'sing systems [5] and in the Χ-Υ model [6]. Attention was given to the study of the crossover phenomena in the effective critical exponents from three-dimensional to two-dimensional as the system approaches the critical temperature.…”