Progress of Theoretical Physics

1/n Expansion Up to Order 1/n2. II: Critical Exponent for d=3

Yutaka Okabe, M. Oku

Published Nov 1, 1978• 18 citations

Abstract

By means of 1/n expansion, the low-temperature critical behavior of the classical n-vector model is studeid. The discussion is restricted to a three-dimensional lattice with a short-range interaction. The critical exponent β is computed up to oreder 1/n2.

DOI: 10.1143/ptp.60.1277

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

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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.

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1/n Expansion Up to Order 1/n2. III: Critical Exponents and for d=3

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1/n Expansion Up to Order 1/n2. IV: Critical Amplitude Ratio R

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Critical Exponents for Long-Range Interactions. I

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Critical Exponents for Long-Range Interactions. II

Masuo Suzuki

A general formulation for the expansion of the partition function in inverse powers of spin dimensionality n is given in the classical n-vector model in the presence of an inhomogeneous magnetic field. The critical exponents δ and η have been evaluated through the first order of 1/n, as functions of dimensionality d and potential-range σ. The universality and scaling relations on the critical exponents have been partially verified.

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Validity of the long-range expansion in the n -vector model

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Progress of Theoretical Physics

1/n Expansion Up to Order 1/n2. II: Critical Exponent for d=3

Yutaka Okabe, M. Oku

Published Nov 1, 1978• 18 citations

Publisher

Abstract

DOI: 10.1143/ptp.60.1277

Related papers

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

Ryuzo Abe

Progress of Theoretical PhysicsJun 1, 197343 citations

1/n Expansion Up to Order 1/n2. III: Critical Exponents and for d=3

Yutaka Okabe, M. Oku

Progress of Theoretical PhysicsNov 1, 197828 citations

1/n Expansion Up to Order 1/n2. IV: Critical Amplitude Ratio R

M. Oku, O. Yutaka

As one of the series of papers on 1/n expansion of the classical n-vector model up to O(1/n2), the universal amplitude ratio Rχ=1"DB8-1 is investigated up to O(1/n2) in the present paper. The 1/n2 term for Rχ is calculated numerically for d=3. In case of the critical exponent β, γ or ν, the result of the second-order calculation is not satisfactory for small values of n. On the contrary, the situation is much better for Rχ. The numerical value obtained is Rχ=1.20 for n=3 at d=3, which is almost equal to that of high-temperature expansion, Rχ=1.23.As one of the series of papers on 1/n expansion of the classical n-vector model up to O(1/n2), the universal amplitude ratio Rχ=1"DB8-1 is investigated up to O(1/n2) in the present paper. The 1/n2 term for Rχ is calculated numerically for d=3. In case of the critical exponent β, γ or ν, the result of the second-order calculation is not satisfactory for small values of n. On the contrary, the situation is much better for Rχ. The numerical value obtained is Rχ=1.20 for n=3 at d=3, which is almost equal to that of high-temperature expansion, Rχ=1.23.

Progress of Theoretical PhysicsFeb 1, 19795 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

Critical Exponents for Long-Range Interactions. II

Masuo Suzuki

Progress of Theoretical PhysicsApr 1, 197328 citations

Validity of the long-range expansion in the n -vector model

M. A. Gusmão, W. K. Theumann

The critical behavior of an $n$ -component system in $d$ dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.The critical behavior of an $n$-component system in $d$ dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.

Physical review. B, Condensed matterDec 1, 198320 citations

The critical behavior of an

n

-component system in

d

dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.