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

Surface critical behavior of tricritical systems

Erich Eisenriegler, Hans Werner Diehl

Published Apr 1, 1988• 37 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.37.5257

Related papers

Walks, Polymers, and Other Tricritical Systems in the Presence of Walls or Surfaces

H. W Diehi, Erich Eisenriegler

Semi-infinite O ( n ) systems are considered at a bulk tricritical point. A continuum model with a |ϕ| 6 bulk nonlinearity and a |ϕ| 4 surface nonlinearity is used to analyse the critical behaviour at the special transition in d = 3 and d = 3−ε dimensions. For n = 0 the model describes the statistics of polymers at the θ-point, in the presence of a wall. It is shown that the surface critical exponents have a systematic expansion in powers of ε 1/2 . For d = 3, logarithmic corrections of an unusual form are found for surface quantities. In the special case of vanishing bulk nonlinearity the surface exponents can be expressed in terms of known bulk exponents for systems with long-range interactions.

Europhysics Letters (EPL)•Sep 15, 1987•25 citations

Surface critical behavior in the presence of linear or cubic weak surface fields

Hans Werner Diehl, A. Ciach

The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-\ensuremath{\epsilon} dimensions are generalized by incorporating an additional surface term ${\mathit{w}}_{0}$${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field ${\mathit{w}}_{0}$ generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where ${\mathit{w}}_{0}$\ensuremath{\sim}H as H\ensuremath{\rightarrow}0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field ${\mathit{w}}^{\mathrm{*}}$=O(\ensuremath{\surd}\ensuremath{\epsilon} ) exists but that at order \ensuremath{\epsilon} the usual ${\mathit{w}}^{\mathrm{*}}$=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is ${\mathrm{\ensuremath{\omega}}}_{\mathit{w}}$=\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). It is shown that the ${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.The implications of this mixing for the critical behavior near the special surface transition are discussed both for the case of Ising ferromagnets as well as that of fluids. In the case of fluids, the mixing entails that quenched surface disorder of the random bond type is sufficient to prevent the occurrence of a (multicritical) special transition for bulk dimensions d\ensuremath{\le}3 A number of other redundant surface operators are identified, and a general redundancy criterion for surface operators (analogous to Wegner's for bulk operators) is given. Finally, a fixed point with ${\mathit{w}}^{\mathrm{*}}$=iO(\ensuremath{\surd}\ensuremath{\epsilon} ) is obtained, which describes the Yang-Lee edge singularities of an equivalent (d-1)-dimensional bulk system with long-range interactions. The associated critical exponents can be determined to all orders in \ensuremath{\epsilon}.

Physical review. B, Condensed matter•Oct 1, 1991•30 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 critical behaviour atm-axial Lifshitz points: continuum models, boundary conditions and two-loop renormalization group results

Hans Werner Diehl, S. B. Rutkevich, A Gerwinski

The critical behaviour of semi-infinite d -dimensional systems with short-range interactions and an O( n ) invariant Hamiltonian is investigated at an m -axial Lifshitz point with an isotropic wave-vector instability in an m -dimensional subspace of d parallel to the surface. Continuum |ϕ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ϕ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m /2 − dimensions (with > 0) are located at λ = λ* = O( ). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l 4 /ν l 2 is the bulk anisotropy exponent.

Journal of Physics A Mathematical and General•Apr 8, 2003•8 citations

Universal relations among critical amplitudes of surface quantities

Hans Werner Diehl, Gerhard Gompper, W. Speth

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.

Physical review. B, Condensed matter•May 1, 1985•20 citations
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∥4{\ensuremath{\Vert}}^{4}\ensuremath∥4 surface interaction of strength O(\ensuremathϵ1/2{\ensuremath{\epsilon}}^{1/2}\ensuremathϵ1/2) in addition to the familiar \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath∥6{\ensuremath{\Vert}}^{6}\ensuremath∥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.
Semi-infinite O ( n ) systems are considered at a bulk tricritical point. A continuum model with a |ϕ| 6 bulk nonlinearity and a |ϕ| 4 surface nonlinearity is used to analyse the critical behaviour at the special transition in d = 3 and d = 3−ε dimensions. For n = 0 the model describes the statistics of polymers at the θ-point, in the presence of a wall. It is shown that the surface critical exponents have a systematic expansion in powers of ε 1/2 . For d = 3, logarithmic corrections of an unusual form are found for surface quantities. In the special case of vanishing bulk nonlinearity the surface exponents can be expressed in terms of known bulk exponents for systems with long-range interactions.
The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-\ensuremath{\epsilon} dimensions are generalized by incorporating an additional surface term w0{\mathit{w}}_{0}w0​${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field w0{\mathit{w}}_{0}w0​ generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where w0{\mathit{w}}_{0}w0​\ensuremath{\sim}H as H\ensuremath{\rightarrow}0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field w∗{\mathit{w}}^{\mathrm{*}}w∗=O(\ensuremath{\surd}\ensuremath{\epsilon} ) exists but that at order \ensuremath{\epsilon} the usual w∗{\mathit{w}}^{\mathrm{*}}w∗=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is ${\mathrm{\ensuremath{\omega}}}_{\mathit{w}}$=\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). It is shown that the ${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.The implications of this mixing for the critical behavior near the special surface transition are discussed both for the case of Ising ferromagnets as well as that of fluids. In the case of fluids, the mixing entails that quenched surface disorder of the random bond type is sufficient to prevent the occurrence of a (multicritical) special transition for bulk dimensions d\ensuremath{\le}3 A number of other redundant surface operators are identified, and a general redundancy criterion for surface operators (analogous to Wegner's for bulk operators) is given. Finally, a fixed point with w∗{\mathit{w}}^{\mathrm{*}}w∗=iO(\ensuremath{\surd}\ensuremath{\epsilon} ) is obtained, which describes the Yang-Lee edge singularities of an equivalent (d-1)-dimensional bulk system with long-range interactions. The associated critical exponents can be determined to all orders in \ensuremath{\epsilon}.
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 critical behaviour of semi-infinite d -dimensional systems with short-range interactions and an O( n ) invariant Hamiltonian is investigated at an m -axial Lifshitz point with an isotropic wave-vector instability in an m -dimensional subspace of d parallel to the surface. Continuum |ϕ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ϕ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m /2 − dimensions (with > 0) are located at λ = λ* = O( ). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l 4 /ν l 2 is the bulk anisotropy exponent.
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.