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arXiv:hep-th/0310239

Critical Exponents from Dimensional Variation

Hendrik Ballhausen

Published Oct 26, 2003• 1 citation
arXivHTMLPDF

Abstract

Recent work on exact renormalization group flow equations has pointed out the possibility to study critical phenomena in continuous dimension D of space. In an investigation of the O(N) model the dimension N of the fields may be seen as a continuous parameter as well. One may conjecture that a variation of D or N in the vicinity of a second order phase transition yields the same critical exponents as a variation of the temperature. A numerical computation confirms this.

DOI
10.48550/arxiv.hep-th/0310239

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Recent work on exact renormalization group flow equations has pointed out the possibility to study critical phenomena in continuous dimension D of space. In an investigation of the O(N) model the dimension N of the fields may be seen as a continuous parameter as well. One may conjecture that a variation of D or N in the vicinity of a second order phase transition yields the same critical exponents as a variation of the temperature. A numerical computation confirms this.
A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent ννν, which should hold for the large class of continuous transitions associated with ddd-dimensional Landau-Ginzburg-Wilson (LGW) Φ4Φ^4Φ4 theories with a multicomponent scalar field φφφ and a unique φ⋅φφ\cdot φφ⋅φ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If Δφ=(d−2+η)/2Δ_φ=(d-2+η)/2Δφ​=(d−2+η)/2 is the dimension of the order-parameter field φφφ, and Δε=d−1/νΔ_\varepsilon=d-1/νΔε​=d−1/ν is the RG dimension of the energy operator ε\varepsilonε, which can be identified with [φ⋅φ][φ\cdot φ][φ⋅φ] (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality Δε≥2ΔφΔ_\varepsilon \ge 2 Δ_φΔε​≥2Δφ​, which implies ν≥(2−η)−1ν\ge (2-η)^{-1}ν≥(2−η)−1 and γ=(2−η)ν≥1γ= (2-η)ν\ge 1γ=(2−η)ν≥1. These inequalities are supported by general arguments for ferromagnetic lattice models, by εεε-expansion results for generic LGW Φ4Φ^4Φ4 theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW Φ4Φ^4Φ4 theories. In particular, since unitarity requires η≥0η\ge 0η≥0, the above inequality implies ν≥1/2ν\ge 1/2ν≥1/2 for unitary theories. This lower bound is more restrictive than ν>1/dν> 1/dν>1/d, derived by noting that ν=1/dν=1/dν=1/d characterizes the singular finite-size behavior at first-order transitions.
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.
Abstract Until the early 1970s all calculations of critical exponents were made either by exactly solving a model for its thermodynamic properties and then examining its behaviour in the critical regime (as with the Ising model in one and two dimensions) or by direct numerical simulations (as with the three-dimensional Ising model) or by extrapolation from approximate solutions which become invalid in the vicinity of a phase transition (high-temperature expansions, for example). However, in a now-famous paper published in 1966, Kadanoff presented arguments which, he maintained, would allow one to simplify calculations in the critical regime to the point at which critical exponents could be extracted, without ever working out the partition function for the problem. However, though Kadanoff’s ideas showed great physical intuition about the processes giving rise to critical phenomena, they lacked the mathematical precision which might have given one faith in his results.
Effective critical exponents for the λϕ^4 scalar field theory are calculated as a function of the renormalization group block size k_o^{-1} and inverse critical temperature β_c. Exact renormalization group equations are presented up to first order in the derivative expansion and numerical solutions are obtained with and without polynomial expansion of the blocked potential. For a finite temperature system in d dimensions, it is shown that \barβ_c = β_c k_o determines whether the d-dimensional (\barβ_c << 1) or (d+1)-dimensional (\barβ_c >> 1) fixed point governs the phase transition. The validity of a polynomial expansion of the blocked potential near criticality is also addressed.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.