Critical exponents are functions of $d$, the dimension, and $n$, the number of components of the order parameter involved in the second-order phase transition. They can be expanded in power series of $\frac{1}{n}$ when $n$ is large. We present details of calculating by perturbation theory the critical exponents above ${T}_{c}$ for arbitrary $d$ and to $O(\frac{1}{n})$ for short-range interacting systems and also for long-range interacting systems.
Critical exponents are functions of d, the dimension, and n, the number of components of the order parameter involved in the second-order phase transition. They can be expanded in power series of n1 when n is large. We present details of calculating by perturbation theory the critical exponents above Tc for arbitrary d and to O(n1) for short-range interacting systems and also for long-range interacting systems.
Critical exponents for a d-dimensional system with an isotropic n-component order parameter and long-range attractive interactions decaying as $\frac{1}{{r}^{d+\ensuremath{\sigma}}} (\ensuremath{\sigma}>0)$ are derived, using the renormalization group approach, as power series in $\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d>0$ ($\ensuremath{\sigma}\ensuremath{\ne}2$, fixed) or $\ensuremath{\Delta}\ensuremath{\sigma}=\ensuremath{\sigma}\ensuremath{-}\frac{1}{2}d>0$ (d fixed) and, separately, to order n1 for all d and $\ensuremath{\sigma}\ensuremath{\ne}2$. For $\ensuremath{\epsilon}<0$ the exponents have fixed ("classical") values; when $\ensuremath{\epsilon}=\ensuremath{\Delta}\ensuremath{\sigma}=0$ fractional powers of $\mathrm{ln}(\frac{\ensuremath{\Delta}T}{{T}_{c}})$ appear; when $\ensuremath{\sigma}>2$ the exponents assume their short-range values.
Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.
his book explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various second-order phase transitions of three-dimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition. Beginning with an introduction to critical phenomena, this book develops the functional-integral description of quantum field theories, their perturbation expansions, and a m
It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in
The critical exponents r and a are derived in powers of 1/n to first order for the n- vector model (or n-component system) with long-range interactions.The results show that the exponents have discontinuity with respect to the potential-range " at <T=2 for d>2 (d: dimensionality).It is also shown that they depend upon d and "• rather independently.The connection of the present results with the ones derived in E-expansion is briefly discussed.The 1/n-expansion of other critical exponents such as {J, iJ, "' and 71 are also discussed.§ 1. Introduction and summary Critical exponents characteristic of critical phenomena depend upon the dimensionality d, symmetry (or the degrees of freedom, n) and potential-range s (or effective potential-range rJ).Among these three parameters, the d-and n- dependences of critical exponents have been investigated 1 l-4 l mainly by using an expansion method combined with renormalization group techniques. 5> However, the potential-range s (or rJ) is also a very important parameter for critical phenomena, in particular for the purpose of discussing critical behavior in systems of restricted dimensionality, in which there would occur no phase transition otherwise.6>One of our main purposes in this paper is to investigate how the critical exponents depend upon the three parameters d, n and (J, particularly to clarify the mutual relationship between dimensionality d and potential-range s (or rJ).For this purpose, the critical exponents for general d and s (or rJ) are calculated, (I) by applying 1/n-expansion 7 > to n-vector model with potential-range s, and (II) by extending s-expansion 1 >• 2 > to a generalized effective Hamiltonian of Wilson-type 8 >• 6 l (essentially equivalent to the n-vector model with long-range interactions).Following Wilson 2 l and Fisher,ll for any real value d(>1), we make use of the formulae (2n)-a S d"-kf(k~) =Ka f"ka-~(k~)dk,Downloaded from https://academic.oup.com/ptp/article/49/2/424/1856924 by guest on 12 August 2023Critical Exponents for Long-Range Interactions.I with Ka = 2-<"-t>n-"12 [T(d/2) J-t.The effective potential-range (J is defined by rJ=s for O<s<2 and rJ=2 for s>2 through the interaction
The critical-point exponent $\ensuremath{\eta}$ for a critical point of order O in dimensions less than ${d}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}$, is calculated to leading nonvanishing order in the parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}{d}_{\mathcal{O}}\ensuremath{-}d$. The result is given for n-component isotropically interacting magnetic systems. For Ising systems, n=1, the result is ${\ensuremath{\eta}}_{\mathcal{O}}=\frac{{\ensuremath{\epsilon}}_{\mathcal{O}}^{2}4{(\mathcal{O}\ensuremath{-}1)}^{2}}{{({2\mathcal{O}}{\mathcal{O}})}^{3}}$. As O increases, the coefficient of ${\ensuremath{\epsilon}}_{\mathcal{O}}^{2}$ rapidly becomes very small, varying as ${2}^{\ensuremath{-}6\mathcal{O}}$ for O large. In the limit of large n, ${\ensuremath{\eta}}_{\mathcal{O}}$ for odd order points approaches a constant and, for even order points, is proportional to n1.