Physical Review Letters

ε Expansion in Semi-infinite Ising Systems

T. C. Lubensky, Morton H. Rubin

Published Dec 10, 1973• 47 citations

Abstract

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.

DOI: 10.1103/physrevlett.31.1469

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Critical phenomena in semi-infinite systems. I. ε expansion for positive extrapolation length

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Physical Review Letters

ε Expansion in Semi-infinite Ising Systems

T. C. Lubensky, Morton H. Rubin

Published Dec 10, 1973• 47 citations

Publisher

Abstract

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}

\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}

\frac{1}{2}\ensuremath{-}\frac{1}{12}\ensuremath{\epsilon}

if scaling is assumed.

DOI: 10.1103/physrevlett.31.1469

Related papers

Critical phenomena in semi-infinite systems. I. ε expansion for positive extrapolation length

T. C. Lubensky, Morton H. Rubin

The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for $n$ -dimensional classical spins on a semi-infinite lattice with surface exchange such that the extrapolation length is positive. It is found that to first order in $\ensuremath{\epsilon}$, all surface exponents can be calculated from bulk exponents and a single surface exponent, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\frac{(\frac{1}{2})\ensuremath{\epsilon}(n+2)}{(n+8)}$, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ introduced by Binder and Hohenberg are, respectively, $1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ and $2(1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}})$. A form for the fixed-point spin correlation valid for all dimensions containing only the exponents $\ensuremath{\eta}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from $\ensuremath{\eta}$, $\ensuremath{\nu}$, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ if scaling is assumed.The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for $n$-dimensional classical spins on a semi-infinite lattice with surface exchange such that the extrapolation length is positive. It is found that to first order in $\ensuremath{\epsilon}$, all surface exponents can be calculated from bulk exponents and a single surface exponent, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\frac{(\frac{1}{2})\ensuremath{\epsilon}(n+2)}{(n+8)}$, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ introduced by Binder and Hohenberg are, respectively, $1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ and $2(1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}})$. A form for the fixed-point spin correlation valid for all dimensions containing only the exponents $\ensuremath{\eta}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from $\ensuremath{\eta}$, $\ensuremath{\nu}$, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ if scaling is assumed.

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Alan J. Bray, M A Moore

The critical behaviour of a semi-infinite system with O(n) spin symmetry is investigated using four techniques: (i) mean-field theory, (ii) exact solution for n= infinity , (iii) the epsilon expansion, and (iv) scaling arguments. The surface equation of state is computed using mean-field theory, and the four phase transitions defined by Lubensky and Rubin, the 'ordinary', 'surface', 'extraordinary', and ' lambda = infinity ' ('special') transitions, are identified. The first three may be observed in the bulk system by adding a suitable 'surface' perturbation which destroys translational invariance. Scaling arguments give their critical exponents exactly in terms of bulk exponents. Critical exponents for the ' lambda = infinity ' transition are determined for n= infinity and also to O( epsilon ). Spin-spin correlation functions at T=T c for the 'ordinary', 'extraordinary' and ' lambda = infinity ' transitions are calculated exactly for n= infinity .The critical behaviour of a semi-infinite system with O(n) spin symmetry is investigated using four techniques: (i) mean-field theory, (ii) exact solution for n= infinity , (iii) the epsilon expansion, and (iv) scaling arguments. The surface equation of state is computed using mean-field theory, and the four phase transitions defined by Lubensky and Rubin, the 'ordinary', 'surface', 'extraordinary', and ' lambda = infinity ' ('special') transitions, are identified. The first three may be observed in the bulk system by adding a suitable 'surface' perturbation which destroys translational invariance. Scaling arguments give their critical exponents exactly in terms of bulk exponents. Critical exponents for the ' lambda = infinity ' transition are determined for n= infinity and also to O( epsilon ). Spin-spin correlation functions at T=T c for the 'ordinary', 'extraordinary' and ' lambda = infinity ' transitions are calculated exactly for n= infinity .

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Accurate critical exponents for Ising like systems in non-integer dimensions

J. C. Le Guillou, Jean Zinn‐Justin

In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1

Journal de physiqueJan 1, 1987189 citations

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}

\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}

\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 to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for

n

-dimensional classical spins on a semi-infinite lattice with surface exchange such that the extrapolation length is positive. It is found that to first order in $\ensuremath{\epsilon}$, all surface exponents can be calculated from bulk exponents and a single surface exponent, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\frac{(\frac{1}{2})\ensuremath{\epsilon}(n+2)}{(n+8)}$, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ introduced by Binder and Hohenberg are, respectively, $1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ and $2(1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}})$. A form for the fixed-point spin correlation valid for all dimensions containing only the exponents $\ensuremath{\eta}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from $\ensuremath{\eta}$, $\ensuremath{\nu}$, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ if scaling is assumed.

d=2

square lattice and the

d=3

bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for

d=3

are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.