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Physical Review Letters

Critical Exponents for Long-Range Interactions

Michael Ellis Fisher, Shang-keng Ma, Bernie G. Nickel

Published Oct 2, 1972• 1,241 citations
Publisher

Abstract

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 $\frac{1}{n}$ 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.

DOI
10.1103/physrevlett.29.917

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Ferromagnets with the number of spin components n>2n>2n>2 and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on ddd and nnn but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to ddd and nnn.
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
Critical exponents are functions of ddd, the dimension, and nnn, the number of components of the order parameter involved in the second-order phase transition. They can be expanded in power series of 1n\frac{1}{n}n1​ when nnn is large. We present details of calculating by perturbation theory the critical exponents above Tc{T}_{c}Tc​ for arbitrary ddd and to O(1n)O(\frac{1}{n})O(n1​) for short-range interacting systems and also for long-range interacting systems.
Recent calculations on ferromagnets with nnn-component spins and with long-range forces (the interaction between spin decays as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$, where ddd is the dimensionality of the system and $\ensuremath{\sigma}>0$) have given exponents which are apparently discontinuous at $\ensuremath{\sigma}=2$, i.e., when the transition to short-range interactions is made. By solving Wilson's exact recursion relations to order ${\ensuremath{\epsilon}}^{2}(\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d)$ we show that the exponents $\ensuremath{\eta}$, $\ensuremath{\gamma}$, and $\ensuremath{\phi}$ are continuous functions of $\ensuremath{\sigma}$ and that there exists a region of long-range potentials defined by $2>\ensuremath{\sigma}>2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}$ where the exponents assume their short-range values.
The influence of long-range interactions decaying in d dimensions as ${1/R}^{d+\ensuremath{\sigma}}$ on the critical behavior of systems with negativeFisher's correlation-function exponent for short-range interactions, ${\ensuremath{\eta}}_{\mathrm{SR}}<0,$ is reexamined. Such systems, typically described by ${\ensuremath{\varphi}}^{3}$-field theories, are, e.g., the Potts model in the percolation limit, the Edwards-Anderson spin-glass model, and the Yang-Lee edge singularity. In contrast to preceding studies, it is shown by means of Wilson's momentum-shell renormalization-group recursion relations that the long-range interactions dominate as long as $\ensuremath{\sigma}<2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}.$ Exponents change continuously to their short-range values at the boundary of this region.
The critical behavior of one-dimensional ferromagnetic chains with isotropic long-range interactions between nnn-component spins is investigated near the critical range of ${r}^{\ensuremath{-}2}$. Different formulations of the renormalization group are used for n>1n>1n>1 and n=1n=1n=1. It is shown that the behavior is quite different in the two cases, with very close analogies to the short-range problem near two dimensions. Critical exponents are calculated to first order.