Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

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

Peter Czerner, Uwe Ritschel, Typeset Using Revtex

Published Jan 1, 1996• 11 citations
OpenAlex

Abstract

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.

Related papers

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.

arXiv:cond-mat/9609120•International Journal of Modern Physics B•Jul 10, 1997•16 citations

Near-Surface Long-Range Order at the Ordinary Transition: Scaling Analysis and Monte Carlo Results

Peter Czerner, Uwe Ritschel

Motivated by recent experimental activities on surface critical phenomena, we present a detailed theoretical study of the near-surface behavior of the local order parameter m(z) in Ising-like spin systems. Special attention is paid to the crossover regime between “ordinary” and “normal” transition in the three-dimensional semi-infinite Ising model, where a finite magnetic field H1 is imposed on the surface which itself exhibits a reduced tendency to order spontaneously. As the theoretical foundation, the spatial behavior of m(z) is discussed by means of phenomenological scaling arguments, and a finite-size scaling analysis is performed. Then we present Monte Carlo results for m(z) obtained with the Swendsen-Wang algorithm. In particular the sharp power-law increase of the magnetization, m(z) ≈ H1z1−η∼ord, predicted for a smallH1 by previous work of the authors, is corroborated by the numerical results. The relevance of these findings for experiments on critical adsorption in systems where a small effective surface field occurs is pointed out.

•Jan 1, 1996•9 citations

Crossover between ordinary and normal transitions in two dimensional critical Ising films

Anna Maciołek, A. Ciach, A. Drzewiński

We investigate two dimensional critical Ising films of width L with surface fields H(1)=H(L) in the crossover between ordinary (H(1)=0) and normal (H(1)=infinity) transitions. Using exact transfer-matrix diagonalization and density matrix renormalization-group (DMRG) methods, we calculate magnetization profiles m(z), the excess magnetization Gamma, and the analog of the solvation force f(solv) as functions of H1 for several L. Scaling functions of the above quantities deviate substantially from their asymptotic forms at fixed points for a broad region of the scaling variable LH21 approximately L/l(1), where l(1) is the length induced by the surface field H1. The scaling function for /f(solv)/ has a deep minimum near LH(2)(1)=1, which is about one order of magnitude smaller than its value at both fixed points (the "Casimir" amplitude). For weak H1 (l(1)>L) the magnetization profile has a maximum at the center of the film, and f(solv) decays much faster than L-2. For stronger H1 (1<l(1)<L), the magnetization has two maxima at a distance approximately l(1) from the walls, and the solvation force decays much slower than L-2. For L>>l(1) the solvation force decays according to the universal power law f(solv) approximately L(-2). The results of the approximate DMRG method show remarkable agreement with the exact ones.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 1999•28 citations

Critical behavior at supercritical surface enhancement: Temperature singularity of surface magnetization and order-parameter profile to one-loop order

Hans Werner Diehl, Martin Smock

The surface critical behavior of semi-infinite systems belonging to the Ising universality class with short-range interactions is investigated for supercritical surface enhancement -c>0 and vanishing surface field ${\mathit{h}}_{1}$. Renormalization-group improved perturbation theory is applied to the standard semi-infinite scalar ${\mathrm{\ensuremath{\varphi}}}^{4}$ model in d=4-\ensuremath{\epsilon} dimensions to compute the order-parameter profile to one-loop order both for temperatures T with \ensuremath{\tau}\ensuremath{\equiv}(T-${\mathit{T}}_{\mathit{c}\mathit{b}}$)/${\mathit{T}}_{\mathit{c}\mathit{b}}$\ensuremath{\gtrsim}0 and \ensuremath{\tau}\ensuremath{\lesssim}0. The associated scaling functions are found to cross smoothly over from their short-distance behavior for distances z\ensuremath{\ll}${\ensuremath{\xi}}_{\mathit{b}}$ (=bulk correlation length) to their long-distance behavior for z\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{b}}$ without showing the peculiar nonmonotonic behavior asserted by Peliti and Leibler [J. Phys. C 16, 2635 (1983)]. Furthermore, the short-distance behavior of the profiles is shown to be fully consistent with a \ensuremath{\Vert}\ensuremath{\tau}${\mathrm{\ensuremath{\Vert}}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ singularity of the surface magnetization ${\mathit{m}}_{1}$ plus a regular background term; that is, in contrast to results published recently by other authors, the amplitudes ${\mathit{A}}_{+}$ and ${\mathit{A}}_{\mathrm{\ensuremath{-}}}$ of the contributions ${\mathit{A}}_{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\tau} to ${\mathit{m}}_{1}$ linear in \ensuremath{\tau}>0 or \ensuremath{\tau}0 agree to one-loop order. Finally, we confirm that the universal profiles for the critical adsorption of fluids (governed by the critical-adsorption fixed point at c=+\ensuremath{\infty} and ${\mathit{h}}_{1}$=\ensuremath{\infty}) agree with the previous ones pertaining to the negative-c-transition fixed point at c=-\ensuremath{\infty} and ${\mathit{h}}_{1}$=0.

Physical review. B, Condensed matter•Mar 1, 1993•93 citations

Critical Adsorption in Systems with Weak Surface Field: The Renormalization-Group Approach

A. Ciach, Uwe Ritschel

We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.

•Aug 17, 2016•12 citations

Surface effects on magnetic phase transitions

Kurt Binder, Pierre C. Hohenberg

The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange ($0<{J}_{s}<J$), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below ${T}_{c}$, thus providing a possible explanation for previous experiments. For sufficiently strong negative ${J}_{s}$, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.

Physical review. B, Solid state•Mar 1, 1974•566 citations
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 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.
Motivated by recent experimental activities on surface critical phenomena, we present a detailed theoretical study of the near-surface behavior of the local order parameter m(z) in Ising-like spin systems. Special attention is paid to the crossover regime between “ordinary” and “normal” transition in the three-dimensional semi-infinite Ising model, where a finite magnetic field H1 is imposed on the surface which itself exhibits a reduced tendency to order spontaneously. As the theoretical foundation, the spatial behavior of m(z) is discussed by means of phenomenological scaling arguments, and a finite-size scaling analysis is performed. Then we present Monte Carlo results for m(z) obtained with the Swendsen-Wang algorithm. In particular the sharp power-law increase of the magnetization, m(z) ≈ H1z1−η∼ord, predicted for a smallH1 by previous work of the authors, is corroborated by the numerical results. The relevance of these findings for experiments on critical adsorption in systems where a small effective surface field occurs is pointed out.
We investigate two dimensional critical Ising films of width L with surface fields H(1)=H(L) in the crossover between ordinary (H(1)=0) and normal (H(1)=infinity) transitions. Using exact transfer-matrix diagonalization and density matrix renormalization-group (DMRG) methods, we calculate magnetization profiles m(z), the excess magnetization Gamma, and the analog of the solvation force f(solv) as functions of H1 for several L. Scaling functions of the above quantities deviate substantially from their asymptotic forms at fixed points for a broad region of the scaling variable LH21 approximately L/l(1), where l(1) is the length induced by the surface field H1. The scaling function for /f(solv)/ has a deep minimum near LH(2)(1)=1, which is about one order of magnitude smaller than its value at both fixed points (the "Casimir" amplitude). For weak H1 (l(1)>L) the magnetization profile has a maximum at the center of the film, and f(solv) decays much faster than L-2. For stronger H1 (1<l(1)<L), the magnetization has two maxima at a distance approximately l(1) from the walls, and the solvation force decays much slower than L-2. For L>>l(1) the solvation force decays according to the universal power law f(solv) approximately L(-2). The results of the approximate DMRG method show remarkable agreement with the exact ones.
The surface critical behavior of semi-infinite systems belonging to the Ising universality class with short-range interactions is investigated for supercritical surface enhancement -c>0 and vanishing surface field h1{\mathit{h}}_{1}h1​. Renormalization-group improved perturbation theory is applied to the standard semi-infinite scalar ${\mathrm{\ensuremath{\varphi}}}^{4}$ model in d=4-\ensuremath{\epsilon} dimensions to compute the order-parameter profile to one-loop order both for temperatures T with \ensuremath{\tau}\ensuremath{\equiv}(T-Tcb{\mathit{T}}_{\mathit{c}\mathit{b}}Tcb​)/Tcb{\mathit{T}}_{\mathit{c}\mathit{b}}Tcb​\ensuremath{\gtrsim}0 and \ensuremath{\tau}\ensuremath{\lesssim}0. The associated scaling functions are found to cross smoothly over from their short-distance behavior for distances z\ensuremath{\ll}${\ensuremath{\xi}}_{\mathit{b}}$ (=bulk correlation length) to their long-distance behavior for z\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{b}}$ without showing the peculiar nonmonotonic behavior asserted by Peliti and Leibler [J. Phys. C 16, 2635 (1983)]. Furthermore, the short-distance behavior of the profiles is shown to be fully consistent with a \ensuremath{\Vert}\ensuremath{\tau}${\mathrm{\ensuremath{\Vert}}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ singularity of the surface magnetization m1{\mathit{m}}_{1}m1​ plus a regular background term; that is, in contrast to results published recently by other authors, the amplitudes A+{\mathit{A}}_{+}A+​ and ${\mathit{A}}_{\mathrm{\ensuremath{-}}}$ of the contributions ${\mathit{A}}_{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\tau} to m1{\mathit{m}}_{1}m1​ linear in \ensuremath{\tau}>0 or \ensuremath{\tau}0 agree to one-loop order. Finally, we confirm that the universal profiles for the critical adsorption of fluids (governed by the critical-adsorption fixed point at c=+\ensuremath{\infty} and h1{\mathit{h}}_{1}h1​=\ensuremath{\infty}) agree with the previous ones pertaining to the negative-c-transition fixed point at c=-\ensuremath{\infty} and h1{\mathit{h}}_{1}h1​=0.
We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.
The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange (0<Js<J0<{J}_{s}<J0<Js​<J), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below Tc{T}_{c}Tc​, thus providing a possible explanation for previous experiments. For sufficiently strong negative Js{J}_{s}Js​, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.