Physical Review Letters

Nonuniversality, Exponent Asymmetry, and Surface Magnetization in an Inhomogeneous Square Ising Lattice

Henk W. J. Blöte, H. J. Hilhorst

Published Nov 21, 1983• 30 citations

Abstract

A semi-infinite nearest-neighbor square Ising system whose couplings at a distance $l$ from the boundary differ from homogeneity by an amount $\ensuremath{\delta}K\ensuremath{\sim}\ensuremath{-}\frac{A}{l}$ is investigated. On the basis of the Pfaffian method we obtain the critical behavior at the surface of this system. The exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$, ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$, ${\ensuremath{\beta}}_{1}$, ${\ensuremath{\gamma}}_{11}$, and ${\ensuremath{\delta}}_{11}$ display rich nonuniversal behavior as a function of the amplitude $A$. For $A$ below a critical value, there is exponent asymmetry and a spontaneous surface magnetization when the bulk ($l=\ensuremath{\infty}$) is critical.

DOI: 10.1103/physrevlett.51.2015

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Physical Review Letters

Nonuniversality, Exponent Asymmetry, and Surface Magnetization in an Inhomogeneous Square Ising Lattice

Henk W. J. Blöte, H. J. Hilhorst

Published Nov 21, 1983• 30 citations

Publisher

Abstract

A semi-infinite nearest-neighbor square Ising system whose couplings at a distance

l

from the boundary differ from homogeneity by an amount

\ensuremath{\delta}K\ensuremath{\sim}\ensuremath{-}\frac{A}{l}

is investigated. On the basis of the Pfaffian method we obtain the critical behavior at the surface of this system. The exponents

{\ensuremath{\eta}}_{\ensuremath{\parallel}}

{\ensuremath{\nu}}_{\ensuremath{\parallel}}

{\ensuremath{\beta}}_{1}

{\ensuremath{\gamma}}_{11}

, and

{\ensuremath{\delta}}_{11}

display rich nonuniversal behavior as a function of the amplitude

A

. For

A

below a critical value, there is exponent asymmetry and a spontaneous surface magnetization when the bulk (

l=\ensuremath{\infty}

) is critical.

DOI: 10.1103/physrevlett.51.2015

Related papers

Surface critical behaviour of the smoothly inhomogeneous planar Ising model: the Pfaffian method

Henk W. J. Blöte, H. J. Hilhorst

A semi-infinite nearest-neighbour square Ising lattice is investigated whose couplings J(l) at a distance l from the boundary differ from homogeneity by an amount j(l)-J(infinity) approximately -A/l. On the basis of the Pfaffian method the critical behaviour at the surface of this system is investigated. The exponents eta 11 , nu 11 , beta 1 , gamma 11 and delta 11 all display rich non-universal behaviour as a function of the amplitude A. For A below a critical value there is a spontaneous surface magnetisation when the bulk (l = infinity) is critical and an asymmetry between the exponents on either side of the critical point.A semi-infinite nearest-neighbour square Ising lattice is investigated whose couplings J(l) at a distance l from the boundary differ from homogeneity by an amount j(l)-J(infinity) approximately -A/l. On the basis of the Pfaffian method the critical behaviour at the surface of this system is investigated. The exponents eta 11 , nu 11 , beta 1 , gamma 11 and delta 11 all display rich non-universal behaviour as a function of the amplitude A. For A below a critical value there is a spontaneous surface magnetisation when the bulk (l = infinity) is critical and an asymmetry between the exponents on either side of the critical point.

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Physical review. B, Condensed matterJan 1, 197841 citations

Edge critical behavior at the surface transition of Ising magnets

Michel Pleimling, Walter Selke

arXiv:cond-mat/9806175Physical review. B, Condensed matterJan 1, 19999 citations

A semi-infinite nearest-neighbor square Ising system whose couplings at a distance

l

from the boundary differ from homogeneity by an amount

\ensuremath{\delta}K\ensuremath{\sim}\ensuremath{-}\frac{A}{l}

is investigated. On the basis of the Pfaffian method we obtain the critical behavior at the surface of this system. The exponents

{\ensuremath{\eta}}_{\ensuremath{\parallel}}

{\ensuremath{\nu}}_{\ensuremath{\parallel}}

{\ensuremath{\beta}}_{1}

{\ensuremath{\gamma}}_{11}

, and

{\ensuremath{\delta}}_{11}

display rich nonuniversal behavior as a function of the amplitude

A

. For

A

below a critical value, there is exponent asymmetry and a spontaneous surface magnetization when the bulk (

l=\ensuremath{\infty}

) is critical.

We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large

m

m

being the distance from the edge. The case

A<0

of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when

A>0

. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.

We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance

m

from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{\ensuremath{-}y}$ for large

m

is examined in detail. For

y<1

the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For

y=1

A>{A}_{c}

(

{A}_{c}

being a positive critical value), and for

y<1

A>0

, there is a spontaneous surface magnetization at the bulk critical temperature.

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.