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

Physical review. B, Condensed matter

Universal relations among critical amplitudes of surface quantities

Hans Werner Diehl, Gerhard Gompper, W. Speth

Published May 1, 1985• 20 citations
Publisher

Abstract

Universal relations among critical amplitudes of surface quantities are discussed for d-dimensional semi-infinite systems near their bulk critical temperature ${T}_{c}^{b}$. Based on the field-theoretic renormalization-group approach, a general derivation of the expected multi-scale-factor universality is given and various universal ratios involving critical amplitudes of the ordinary and special transitions both above and below ${T}_{c}^{b}$ are calculated to first order in \ensuremath{\epsilon}=4-d. In addition, an error in the published one-loop result for the universal order-parameter profile at the ordinary transition of the semi-infinite Ising model is corrected.

DOI
10.1103/physrevb.31.5841

Related papers

Universal critical amplitude ratios and scaling functions for the semi-infinite system with a surface

Yutaka Okabe, Kaoru Ohno

We have shown that several critical amplitude ratios are universal in the critical phenomena near the surface. The general statement on the number of the scaling relations among surface critical exponents and the surface universal amplitude ratios is given. The universal scaling function for the equation of state in the presence of surface is discussed. We have calculated the amplitude ratios and the scaling function for the ordinary, special, and extraordinary transitions in the mean-field approximation.

Physical review. B, Condensed matter•Dec 1, 1984•9 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

Universal amplitude ratios of the renormalization group: Two-dimensional tricritical Ising model

Davide Fioravanti A, Giuseppe Mussardo, Pascal Simon

The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.

arXiv:cond-mat/0008216•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Dec 18, 2000•31 citations

Surface critical behavior of tricritical systems

Erich Eisenriegler, Hans Werner Diehl

We study the surface critical behavior of semi-infinite systems with an n-component order parameter and short-range interactions at a bulk tricritical point. The special surface transition is investigated in detail for space dimensionalities d=3-\ensuremath{\epsilon}\ensuremath{\le}3 using field-theoretic renormalization-group methods. For \ensuremath{\epsilon}>0 its infrared-stable fixed point is described by a Hamiltonian which contains a \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{4}$ surface interaction of strength O(${\ensuremath{\epsilon}}^{1/2}$) in addition to the familiar \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{6}$ bulk interaction of strength O(\ensuremath{\epsilon}). For d\ensuremath{\equiv}3, logarithmic corrections of a non-power-law form appear. The renormalization of a proper continuum model is expounded and the implications of the results are discussed, especially with regard to the adsorption of polymers from a \ensuremath{\theta} solution.

Physical review. B, Condensed matter•Apr 1, 1988•37 citations

Universal relations among critical amplitude. Calculations up to order ε2 for systems with continuous symmetry

Claude Bervillier

A general derivation of the two-scale-factor universality is given using the renormalized ${\ensuremath{\varphi}}^{4}$ theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order ${\ensuremath{\epsilon}}^{2}$ for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and ${R}_{\ensuremath{\xi}}^{+}$, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMn${\mathrm{F}}_{3}$ cases, are repeated with our improved numerical results.

Physical review. B, Solid state•Dec 1, 1976•151 citations
Universal relations among critical amplitudes of surface quantities are discussed for d-dimensional semi-infinite systems near their bulk critical temperature Tcb{T}_{c}^{b}Tcb​. Based on the field-theoretic renormalization-group approach, a general derivation of the expected multi-scale-factor universality is given and various universal ratios involving critical amplitudes of the ordinary and special transitions both above and below Tcb{T}_{c}^{b}Tcb​ are calculated to first order in \ensuremath{\epsilon}=4-d. In addition, an error in the published one-loop result for the universal order-parameter profile at the ordinary transition of the semi-infinite Ising model is corrected.
We have shown that several critical amplitude ratios are universal in the critical phenomena near the surface. The general statement on the number of the scaling relations among surface critical exponents and the surface universal amplitude ratios is given. The universal scaling function for the equation of state in the presence of surface is discussed. We have calculated the amplitude ratios and the scaling function for the ordinary, special, and extraordinary transitions in the mean-field approximation.
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 scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.
We study the surface critical behavior of semi-infinite systems with an n-component order parameter and short-range interactions at a bulk tricritical point. The special surface transition is investigated in detail for space dimensionalities d=3-\ensuremath{\epsilon}\ensuremath{\le}3 using field-theoretic renormalization-group methods. For \ensuremath{\epsilon}>0 its infrared-stable fixed point is described by a Hamiltonian which contains a \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{4}$ surface interaction of strength O(${\ensuremath{\epsilon}}^{1/2}$) in addition to the familiar \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{6}$ bulk interaction of strength O(\ensuremath{\epsilon}). For d\ensuremath{\equiv}3, logarithmic corrections of a non-power-law form appear. The renormalization of a proper continuum model is expounded and the implications of the results are discussed, especially with regard to the adsorption of polymers from a \ensuremath{\theta} solution.
A general derivation of the two-scale-factor universality is given using the renormalized ${\ensuremath{\varphi}}^{4}$ theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order ${\ensuremath{\epsilon}}^{2}$ for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and ${R}_{\ensuremath{\xi}}^{+}$, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMnF3{\mathrm{F}}_{3}F3​ cases, are repeated with our improved numerical results.