Physical review. B, Solid state

Renormalization-group calculation of the critical properties of a free magnetic surface

Nenad M. Svrakic, Michael Wortis

Published Jan 1, 1977• 67 citations

Abstract

A semi-infinite Ising model is studied by renormalization-group techniques in the position-space formulation due to Niemeijer and van Leeuwen. We show how to set up the calculation of thermodynamic properties (both critical and noncritical) of a sample with a free surface by iteration of a set of recursion relations for spatially inhomogeneous couplings and magnetic fields. Surface and bulk properties are clearly distinguished. The method is illustrated by a calculation (using a two-cell cluster approximation) of the magnetic-surface critical exponent in two dimensions, where exact results are available for comparison.

DOI: 10.1103/physrevb.15.396

Related papers

Renormalization-group approach to the Ising model with a free surface

Theodore W. Burkhardt, Erich Eisenriegler

Physical review. B, Solid stateOct 1, 1977124 citations

Critical Exponents for the Three-Dimensional Ising Model from the Real-Space Renormalization Group in Two Dimensions

Zvi Friedman

Physical Review LettersMay 31, 197651 citations

A real-space renormalisation group study of a semi-infinite Ising model

H. C. Tseng, H H Chen

A modified Migdal-Kadanoff renormalisation group scheme, called the cluster-decimation approximation, is generalised and applied to study surface critical properties of the Ising model with a free surface. The authors determine the fixed points and calculate surface critical exponents of this model. Comparisons of their results with other real-space renormalisation group estimates are given.

Journal of Physics A Mathematical and GeneralAug 21, 19890 citations

Surface thermodynamic functions of the Ising model from a renormalization group

Nenad M. Svrakic, Rahul Anil Pandit, Michael Wortis

Physical review. B, Condensed matterAug 1, 198033 citations

Renormalization-group approach to surface critical behavior in the Ising model

Theodore W. Burkhardt, Erich Eisenriegler

Physical review. B, Condensed matterJan 1, 197841 citations

Surface critical behavior of the smoothly inhomogeneous Ising model

Theodore W. Burkhardt, Ihnsouk Guim

Physical review. B, Condensed matterJan 1, 198430 citations

Physical review. B, Solid state

Renormalization-group calculation of the critical properties of a free magnetic surface

Nenad M. Svrakic, Michael Wortis

Published Jan 1, 1977• 67 citations

Publisher

Abstract

DOI: 10.1103/physrevb.15.396

Related papers

Renormalization-group approach to the Ising model with a free surface

Theodore W. Burkhardt, Erich Eisenriegler

We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.

Physical review. B, Solid stateOct 1, 1977124 citations

Critical Exponents for the Three-Dimensional Ising Model from the Real-Space Renormalization Group in Two Dimensions

Zvi Friedman

The equivalence of the critical behavior of the $d$ -dimensional Ising model at a finite temperature to that of a $d$ - 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.The equivalence of the critical behavior of the $d$-dimensional Ising model at a finite temperature to that of a $d$- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.

Physical Review LettersMay 31, 197651 citations

A real-space renormalisation group study of a semi-infinite Ising model

H. C. Tseng, H H Chen

Journal of Physics A Mathematical and GeneralAug 21, 19890 citations

Surface thermodynamic functions of the Ising model from a renormalization group

Nenad M. Svrakic, Rahul Anil Pandit, Michael Wortis

The surface thermodynamic functions of the semi-infinite square and simple-cubic $s=\frac{1}{2}$ nearest-neighbor Ising models are computed numerically from renormalization groups based on simple cell clusters. It is shown how the exact high- and low-temperature limits may be incorporated into the calculations by (i) building in proper site coordination via an Ursell expansion for the free energy and (ii) treating the ground state exactly. In $d=2$ a 4 \ifmmode\times\else\texttimes\fi{} 4 cell-cluster approximation is in excellent agreement with the exact results of Fisher and Ferdinand. In $d=3$ a 2 \ifmmode\times\else\texttimes\fi{} 8 cell-cluster approximation gives plausible results with enough structure to be interesting.The surface thermodynamic functions of the semi-infinite square and simple-cubic $s=\frac{1}{2}$ nearest-neighbor Ising models are computed numerically from renormalization groups based on simple cell clusters. It is shown how the exact high- and low-temperature limits may be incorporated into the calculations by (i) building in proper site coordination via an Ursell expansion for the free energy and (ii) treating the ground state exactly. In $d=2$ a 4 \ifmmode\times\else\texttimes\fi{} 4 cell-cluster approximation is in excellent agreement with the exact results of Fisher and Ferdinand. In $d=3$ a 2 \ifmmode\times\else\texttimes\fi{} 8 cell-cluster approximation gives plausible results with enough structure to be interesting.

Physical review. B, Condensed matterAug 1, 198033 citations

Renormalization-group approach to surface critical behavior in the Ising model

Theodore W. Burkhardt, Erich Eisenriegler

A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The $d=2$ 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.A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The $d=2$ 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.

Physical review. B, Condensed matterJan 1, 197841 citations

Surface critical behavior of the smoothly inhomogeneous Ising model

Theodore W. Burkhardt, Ihnsouk Guim

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 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.

Physical review. B, Condensed matterJan 1, 198430 citations

The equivalence of the critical behavior of the

d

-dimensional Ising model at a finite temperature to that of a

d

- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.

The surface thermodynamic functions of the semi-infinite square and simple-cubic

s=\frac{1}{2}

nearest-neighbor Ising models are computed numerically from renormalization groups based on simple cell clusters. It is shown how the exact high- and low-temperature limits may be incorporated into the calculations by (i) building in proper site coordination via an Ursell expansion for the free energy and (ii) treating the ground state exactly. In

d=2

a 4 \ifmmode\times\else\texttimes\fi{} 4 cell-cluster approximation is in excellent agreement with the exact results of Fisher and Ferdinand. In

d=3

a 2 \ifmmode\times\else\texttimes\fi{} 8 cell-cluster approximation gives plausible results with enough structure to be interesting.

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.

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.