By means of the 1/n expansion method, the critical exponent η is discussed for the three-dimensional isotropic n-vector model with short-range interaction. Terms up to 1/n2 are obtained and the result is compared with the numerical estimate based on high temperature expansion and with Wilson's ε expansion.

Progress of Theoretical PhysicsJun 1, 197343 citations

Self-consistent screening approximation, long range interactions and the critical exponent η

Jayanta K. Bhattacharjee

Physics Letters ASep 1, 19800 citations

Physics Letters A

Wilson-type expansions of critical exponents for long-range interactions

Masuo Suzuki, Y. Yamazaki, George Igarashi

Published Dec 1, 1972• 44 citations

Publisher

DOI: 10.1016/0375-9601(72)90437-9

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Critical scaling function for long-range interactions to order 1/n

W. K. Theumann

Physics Letters AJul 1, 19752 citations

Critical Exponents for Long-Range Interactions. I

Masuo Suzuki

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 interactionThe 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

Progress of Theoretical PhysicsFeb 1, 197340 citations

The 4- $ε$ Expansion for Long-range Interacting Systems

Zhiyi Li, Kun Chen, Youjin Deng

The establishment of the Wilson-Fisher fixed point (WFP) for $O(n)$ spin models in $d=4-ε$ dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as $\propto 1/r^{d+σ}$ , are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the $ε$ -expansion calculations up to the two-loop level. We show that, as long as $σ<2$ , the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with $d/2 < σ< 2$ , the critical exponents, including the anomalous dimension, are functions of $ε$ , $δ=2-σ$ and $n$ , which reduce to the exact results in the limiting cases $ε\to 0$ , $δ\to 0$ or $n \to \infty$ . Our $(4-ε)$ -expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at $σ_*=2$ , well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.The establishment of the Wilson-Fisher fixed point (WFP) for $O(n)$ spin models in $d=4-ε$ dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as $\propto 1/r^{d+σ}$, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the $ε$-expansion calculations up to the two-loop level. We show that, as long as $σ<2$, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with $d/2 < σ< 2$, the critical exponents, including the anomalous dimension, are functions of $ε$, $δ=2-σ$ and $n$, which reduce to the exact results in the limiting cases $ε\to 0$, $δ\to 0$ or $n \to \infty$. Our $(4-ε)$-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at $σ_*=2$, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.

arXiv:2602.07818Feb 8, 20260 citations

Short-Range Spin Interaction and Critical Exponents

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Critical Exponent η up to 1/n2for the Three-Dimensional System with Short-Range Interaction

Ryuzo Abe

Progress of Theoretical PhysicsJun 1, 197343 citations

Self-consistent screening approximation, long range interactions and the critical exponent η

Jayanta K. Bhattacharjee

Physics Letters ASep 1, 19800 citations

The establishment of the Wilson-Fisher fixed point (WFP) for

O(n)

spin models in

d=4-ε

dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as

\propto 1/r^{d+σ}

, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the

ε

-expansion calculations up to the two-loop level. We show that, as long as

σ<2

, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with

d/2 < σ< 2

, the critical exponents, including the anomalous dimension, are functions of

ε

δ=2-σ

and

n

, which reduce to the exact results in the limiting cases

ε\to 0

δ\to 0

n \to \infty

. Our

(4-ε)

-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at

σ_*=2

, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.