Physical Review Letters

Evaluation of η in Wilson's Incomplete-Integration Method: Independence of Cutoff Parameters to Order ε2

Joseph Rudnick

Published Feb 17, 1975• 15 citations

Abstract

The expression for the coefficient of ${\ensuremath{\epsilon}}^{2}$ in the $\ensuremath{\epsilon}$ expansion for the critical exponent $\ensuremath{\eta}$ of the Ising model has been obtained by Shukla and Green within the incomplete-integration method of Wilson. It was left by them in the form of an unevaluated integral, which appears to depend on parameters characterizing details of the renormalization-group procedure. The integral is evaluated here and found to equal $\frac{1}{54}$, regardless of the values of those parameters.

DOI: 10.1103/physrevlett.34.438

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Physical Review Letters

Evaluation of η in Wilson's Incomplete-Integration Method: Independence of Cutoff Parameters to Order ε2

Joseph Rudnick

Published Feb 17, 1975• 15 citations

Publisher

Abstract

The expression for the coefficient of

{\ensuremath{\epsilon}}^{2}

in the

\ensuremath{\epsilon}

expansion for the critical exponent

\ensuremath{\eta}

of the Ising model has been obtained by Shukla and Green within the incomplete-integration method of Wilson. It was left by them in the form of an unevaluated integral, which appears to depend on parameters characterizing details of the renormalization-group procedure. The integral is evaluated here and found to equal

\frac{1}{54}

, regardless of the values of those parameters.

DOI: 10.1103/physrevlett.34.438

Related papers

ε -Expansion Solution of Wilson's Exact Renormalization-Group Equation

Geoffrey R. Golner, Eberhard K. Riedel

The $\ensuremath{\epsilon}$-expansion formula for the critical exponent $\ensuremath{\eta}$ derived by Shukla and Green from Wilson's exact renormalization-group equation is found to yield the standard result $\ensuremath{\eta}=\frac{1}{54}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, independent of the choice of the incomplete-integration function. The result is obtained analytically in the limit $a\ensuremath{\rightarrow}\ensuremath{\infty}$ and numerically for arbitrary $a$ , where $a$ denotes the normalization constant of the Gaussian fixed-point functional.The $\ensuremath{\epsilon}$-expansion formula for the critical exponent $\ensuremath{\eta}$ derived by Shukla and Green from Wilson's exact renormalization-group equation is found to yield the standard result $\ensuremath{\eta}=\frac{1}{54}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, independent of the choice of the incomplete-integration function. The result is obtained analytically in the limit $a\ensuremath{\rightarrow}\ensuremath{\infty}$ and numerically for arbitrary $a$, where $a$ denotes the normalization constant of the Gaussian fixed-point functional.

Physical Review LettersJan 20, 197515 citations

Calculation of the Critical Exponent η via Renormalization-Group Recursion Formulas

Geoffrey R. Golner

This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.

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ε Expansion in Semi-infinite Ising Systems

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The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents in a semi-infinite Ising model to first order in $\ensuremath{\epsilon}$. It is found that potentials renormalize just as in the bulk, and all surface information is contained in the wave-function renormalization. $\ensuremath{\nu}$ is $\frac{1}{2}+\frac{1}{12}\ensuremath{\epsilon}$ and $\ensuremath{\eta}$ is 0, just as in infinite systems. ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ are, respectively, $1\ensuremath{-}\frac{1}{6}\ensuremath{\epsilon}$ and $2\ensuremath{-}\frac{1}{3}\ensuremath{\epsilon}$ and the surface gap exponent ${\ensuremath{\Delta}}_{1}$ is $\frac{1}{2}\ensuremath{-}\frac{1}{12}\ensuremath{\epsilon}$ if scaling is assumed.The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents in a semi-infinite Ising model to first order in $\ensuremath{\epsilon}$. It is found that potentials renormalize just as in the bulk, and all surface information is contained in the wave-function renormalization. $\ensuremath{\nu}$ is $\frac{1}{2}+\frac{1}{12}\ensuremath{\epsilon}$ and $\ensuremath{\eta}$ is 0, just as in infinite systems. ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ are, respectively, $1\ensuremath{-}\frac{1}{6}\ensuremath{\epsilon}$ and $2\ensuremath{-}\frac{1}{3}\ensuremath{\epsilon}$ and the surface gap exponent ${\ensuremath{\Delta}}_{1}$ is $\frac{1}{2}\ensuremath{-}\frac{1}{12}\ensuremath{\epsilon}$ if scaling is assumed.

Physical Review LettersDec 10, 197347 citations

Unified theory of critical exponents generated by the Ising Hamiltonian for discrete dimensionalities 2, 3 and 4 in terms of the critical exponentη

Norman Henry March

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Physics and Chemistry of LiquidsSep 4, 20153 citations

Diagrammatic renormalization-group method

Joseph Rudnick

A method is presented for the convergent evaluation of the Feynman diagrams for the free energy of an Ising-type system. The method involves three first-order Wilson-type differential equations whose solution near and at the fixed point yield the critical parameters $y$ and $\ensuremath{\eta}$. The method is discussed only for a system with no long-range order. That is, we consider only the case $H=0$ , $T>{T}_{c}$ . The $\ensuremath{\epsilon}$ expansions for $y$ and $\ensuremath{\eta}$ are evaluated to second order, and $y$ and $\ensuremath{\eta}$ are found exactly for the spherical model.A method is presented for the convergent evaluation of the Feynman diagrams for the free energy of an Ising-type system. The method involves three first-order Wilson-type differential equations whose solution near and at the fixed point yield the critical parameters $y$ and $\ensuremath{\eta}$. The method is discussed only for a system with no long-range order. That is, we consider only the case $H=0$, $T>{T}_{c}$. The $\ensuremath{\epsilon}$ expansions for $y$ and $\ensuremath{\eta}$ are evaluated to second order, and $y$ and $\ensuremath{\eta}$ are found exactly for the spherical model.

Physical review. B, Solid stateJan 1, 197525 citations

Optimizing Wilson’s momentum shell renormalization group

A. Tröster

Previous attempts to accurately compute critical exponents from Wilson's momentum shell renormalization prescription suffered from the difficulties posed by the presence of an infinite number of irrelevant couplings. Taking the example of the one-dimensional long-ranged Ising model, we calculate the momentum shell renormalization flow in the plane spanned by the coupling constants $({u}_{0},{r}_{0})$ for different values of the momentum shell thickness parameter $b$ by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the $b$ dependence of the fixed-point couplings and the resulting exponents ${y}_{\ensuremath{\tau}}$ and $\ensuremath{\omega}$ in the vicinity of a shell parameter ${b}^{\ensuremath{\ast}}<1$ characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of $b$ yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for $b={b}^{\ensuremath{\ast}}$.Previous attempts to accurately compute critical exponents from Wilson's momentum shell renormalization prescription suffered from the difficulties posed by the presence of an infinite number of irrelevant couplings. Taking the example of the one-dimensional long-ranged Ising model, we calculate the momentum shell renormalization flow in the plane spanned by the coupling constants $({u}_{0},{r}_{0})$ for different values of the momentum shell thickness parameter $b$ by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the $b$ dependence of the fixed-point couplings and the resulting exponents ${y}_{\ensuremath{\tau}}$ and $\ensuremath{\omega}$ in the vicinity of a shell parameter ${b}^{\ensuremath{\ast}}<1$ characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of $b$ yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for $b={b}^{\ensuremath{\ast}}$.

Physical Review BMar 31, 20108 citations

The expression for the coefficient of

{\ensuremath{\epsilon}}^{2}

in the

\ensuremath{\epsilon}

expansion for the critical exponent

\ensuremath{\eta}

\frac{1}{54}

, regardless of the values of those parameters.

a

, where

a

denotes the normalization constant of the Gaussian fixed-point functional.

y

and $\ensuremath{\eta}$. The method is discussed only for a system with no long-range order. That is, we consider only the case

H=0

T>{T}_{c}

. The $\ensuremath{\epsilon}$ expansions for

y

and $\ensuremath{\eta}$ are evaluated to second order, and

y

and $\ensuremath{\eta}$ are found exactly for the spherical model.

({u}_{0},{r}_{0})

for different values of the momentum shell thickness parameter

b

by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the

b

dependence of the fixed-point couplings and the resulting exponents ${y}_{\ensuremath{\tau}}$ and $\ensuremath{\omega}$ in the vicinity of a shell parameter ${b}^{\ensuremath{\ast}}<1$ characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of

b

yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for $b={b}^{\ensuremath{\ast}}$.