Physical review. B, Condensed matter

Magnetization profile near a wall in the planar ising ferromagnet

Douglas B. Abraham, David A. Huse

Published Oct 1, 1988• 23 citations

Abstract

The local magnetization in a semi-infinite planar Ising ferromagnet with an edge field ${H}_{1}$ is examined as a function of distance from the edge. The asymptotic form of this magnetization profile for distances large compared to the bulk correlation length has a nontrivial dependence on the edge field. The exact scaling function for this dependence in the limit of small edge field can be derived from a simple droplet argument.

DOI: 10.1103/physrevb.38.7169

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Surface critical behavior in Ising magnets: Exact results

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Physical review. B, Condensed matter

Magnetization profile near a wall in the planar ising ferromagnet

Douglas B. Abraham, David A. Huse

Published Oct 1, 1988• 23 citations

Publisher

Abstract

The local magnetization in a semi-infinite planar Ising ferromagnet with an edge field

{H}_{1}

is examined as a function of distance from the edge. The asymptotic form of this magnetization profile for distances large compared to the bulk correlation length has a nontrivial dependence on the edge field. The exact scaling function for this dependence in the limit of small edge field can be derived from a simple droplet argument.

DOI: 10.1103/physrevb.38.7169

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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

Magnetization Profile in the D= 2 Semi-Infinite Ising Model and Crossover Between Ordinary and Normal Transition

Peter Czerner, Uwe Ritschel

We study the two-dimensional semi-infinite Ising model with a free surface at or near bulk criticality. Special attention is paid to the influence of a boundary magnetic field h1 on the surface-near regime and the crossover between the fixed points at h1=0 and h1=∞. Near the surface, a smallh1 causes a steeply increasing magnetization m(z)~z3/8 log z as the distance z increases away from the surface. By means of a phenomenological scaling analysis, this phenomenon can be related to the well-known logarithmic dependence of the surface magnetization m1 on h1. Our analysis provides a deeper understanding of the existing exact results on m(z) and relates the short-distance phenomena in d=2 to those in higher dimensions. Both the results of the scaling analysis and the exact analytic profiles are corroborated by Monte Carlo simulations.We study the two-dimensional semi-infinite Ising model with a free surface at or near bulk criticality. Special attention is paid to the influence of a boundary magnetic field h1 on the surface-near regime and the crossover between the fixed points at h1=0 and h1=∞. Near the surface, a smallh1 causes a steeply increasing magnetization m(z)~z3/8 log z as the distance z increases away from the surface. By means of a phenomenological scaling analysis, this phenomenon can be related to the well-known logarithmic dependence of the surface magnetization m1 on h1. Our analysis provides a deeper understanding of the existing exact results on m(z) and relates the short-distance phenomena in d=2 to those in higher dimensions. Both the results of the scaling analysis and the exact analytic profiles are corroborated by Monte Carlo simulations.

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Peter Czerner, Uwe Ritschel, Typeset Using Revtex

We theoretically investigate the spatial dependence of the order parameter of the two-dimensional semi-infinite Ising model with a free surface at or above the bulk critical temperature. Special attention is paid to the influence of a surface magnetic field h1 and the crossover between the fixed points at h1 = 0 and h1 = ∞. The sharp increase of the magnetization m(z) close to the boundary generated by a small h1, which was found by the present authors in the three-dimensional model, is also seen in two dimensions. There, however, the universal short-distance power law is modified by a logarithm, m(z) ∼ z κ ln z, where κ, the difference between the scaling dimensions of h1 and the bulk magnetization, has the exact value 3/8. By means of a phenomenological scaling analysis, the short-distance behavior can be related to the logarithmic dependence of the surface magnetization m1 on h1. Our results, which are corroborated by Monte Carlo simulations, provide a deeper understanding of the existing exact results concerning the local magnetization and relate the short-distance phenomena in two dimensions to those in higher dimensionality.We theoretically investigate the spatial dependence of the order parameter of the two-dimensional semi-infinite Ising model with a free surface at or above the bulk critical temperature. Special attention is paid to the influence of a surface magnetic field h1 and the crossover between the fixed points at h1 = 0 and h1 = ∞. The sharp increase of the magnetization m(z) close to the boundary generated by a small h1, which was found by the present authors in the three-dimensional model, is also seen in two dimensions. There, however, the universal short-distance power law is modified by a logarithm, m(z) ∼ z κ ln z, where κ, the difference between the scaling dimensions of h1 and the bulk magnetization, has the exact value 3/8. By means of a phenomenological scaling analysis, the short-distance behavior can be related to the logarithmic dependence of the surface magnetization m1 on h1. Our results, which are corroborated by Monte Carlo simulations, provide a deeper understanding of the existing exact results concerning the local magnetization and relate the short-distance phenomena in two dimensions to those in higher dimensionality.

Jan 1, 199611 citations

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Anna Maciołek

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Journal of Physics A Mathematical and GeneralJul 21, 199625 citations

Effect of bulk magnetic field on critical Ising films

A. Drzewiński, Anna Maciołek, A. Ciach

Two-dimensional Ising films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature ${T}_{c}$ for two choices of surface fields (a) ${H}_{1}{=H}_{L}=0$ (ordinary transition), and (b) ${H}_{1}{=H}_{L}=\ensuremath{\infty}$ (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization $\ensuremath{\Gamma},$ the longitudinal correlation length ${\ensuremath{\xi}}_{\ensuremath{\Vert}},$ and for the analog of the solvation force ${f}_{\mathrm{solv}}$ are found and discussed. When ${H}_{1}=0$ the scaling function for ${f}_{\mathrm{solv}}$ has two symmetric minima at $y=\mathrm{sgn}(H)L|H{|}^{\ensuremath{\nu}/\ensuremath{\Delta}}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$ with an amplitude at the minimum about 3.8 times the value at $H=0,$ the Casimir amplitude. For the normal transition the scaling function for ${f}_{\mathrm{solv}}$ has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.Two-dimensional Ising films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature ${T}_{c}$ for two choices of surface fields (a) ${H}_{1}{=H}_{L}=0$ (ordinary transition), and (b) ${H}_{1}{=H}_{L}=\ensuremath{\infty}$ (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization $\ensuremath{\Gamma},$ the longitudinal correlation length ${\ensuremath{\xi}}_{\ensuremath{\Vert}},$ and for the analog of the solvation force ${f}_{\mathrm{solv}}$ are found and discussed. When ${H}_{1}=0$ the scaling function for ${f}_{\mathrm{solv}}$ has two symmetric minima at $y=\mathrm{sgn}(H)L|H{|}^{\ensuremath{\nu}/\ensuremath{\Delta}}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$ with an amplitude at the minimum about 3.8 times the value at $H=0,$ the Casimir amplitude. For the normal transition the scaling function for ${f}_{\mathrm{solv}}$ has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMay 1, 200019 citations

Surface critical behavior in Ising magnets: Exact results

Douglas B. Abraham, Anna Maciołek

Physical Review EJun 29, 20062 citations

The local magnetization in a semi-infinite planar Ising ferromagnet with an edge field

{H}_{1}

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.

Two-dimensional Ising films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature

{T}_{c}

for two choices of surface fields (a)

{H}_{1}{=H}_{L}=0

(ordinary transition), and (b) ${H}_{1}{=H}_{L}=\ensuremath{\infty}$ (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization $\ensuremath{\Gamma},$ the longitudinal correlation length ${\ensuremath{\xi}}_{\ensuremath{\Vert}},$ and for the analog of the solvation force

{f}_{\mathrm{solv}}

are found and discussed. When

{H}_{1}=0

the scaling function for

{f}_{\mathrm{solv}}

has two symmetric minima at $y=\mathrm{sgn}(H)L|H{|}^{\ensuremath{\nu}/\ensuremath{\Delta}}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$ with an amplitude at the minimum about 3.8 times the value at

H=0,

the Casimir amplitude. For the normal transition the scaling function for

{f}_{\mathrm{solv}}

has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.