Physical Review Letters

Critical behavior of the nonlinearσmodel with a free surface: The ‘‘ordinary’’ transition in 2+ε dimensions

Hans Werner Diehl, Adi Nusser

Published Jun 30, 1986• 36 citations

Abstract

We analyze the critical behavior of isotropic ferromagnets with a free surface by means of field-theoretic renormalization methods in d=2+\ensuremath{\epsilon} dimensions. Renormalization-group equations for surface quantities are derived which imply scaling to all orders in \ensuremath{\epsilon}. Explicit results to order ${\mathrm{\ensuremath{\epsilon}}}^{2}$ are presented for the critical surface exponents of the ordinary transition. Interpolation with the known results near four dimensions yields estimates for d=3 which seem to be remarkably accurate.

DOI: 10.1103/physrevlett.56.2834

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Physical Review Letters

Critical behavior of the nonlinearσmodel with a free surface: The ‘‘ordinary’’ transition in 2+ε dimensions

Hans Werner Diehl, Adi Nusser

Published Jun 30, 1986• 36 citations

Publisher

Abstract

{\mathrm{\ensuremath{\epsilon}}}^{2}

are presented for the critical surface exponents of the ordinary transition. Interpolation with the known results near four dimensions yields estimates for d=3 which seem to be remarkably accurate.

DOI: 10.1103/physrevlett.56.2834

Related papers

Effects of surface exchange anisotropies on magnetic critical and multicritical behavior at surfaces

Hans Werner Diehl, Erich Eisenriegler

The critical behavior of a semi-infinite $n$ -vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with ${m}_{e}$ (easy-magnetization) components take a critical value above which ${m}_{e}$ -component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis ( ${m}_{e}=1$ ) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.The critical behavior of a semi-infinite $n$-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with ${m}_{e}$ (easy-magnetization) components take a critical value above which ${m}_{e}$-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (${m}_{e}=1$) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.

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Hans Werner Diehl, S. Dietrich

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Hikaru Kawamura

Critical behavior associated with the helical or noncollinear ordering for isotropic n >3-component spins is studied near d =2 dimensions based on an O ( n )× O (2) nonlinear σ model and the renormalization-group ε= d -2 expansion. A chiral fixed point is found for general n characterized by the critical exponents that are consistent with the previous 1/ n expansion calculation. For n =3, it is argued that because of the nontrivial global topology of the order-parameter space, ∝ 1 ( S O (3))= Z 2 , and the possible existence of a finite-temperature topological transition at d =2, the standard ε= d -2 expansion technique is not applicable.Critical behavior associated with the helical or noncollinear ordering for isotropic n >3-component spins is studied near d =2 dimensions based on an O ( n )× O (2) nonlinear σ model and the renormalization-group ε= d -2 expansion. A chiral fixed point is found for general n characterized by the critical exponents that are consistent with the previous 1/ n expansion calculation. For n =3, it is argued that because of the nontrivial global topology of the order-parameter space, ∝ 1 ( S O (3))= Z 2 , and the possible existence of a finite-temperature topological transition at d =2, the standard ε= d -2 expansion technique is not applicable.

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{\mathrm{\ensuremath{\epsilon}}}^{2}

are presented for the critical surface exponents of the ordinary transition. Interpolation with the known results near four dimensions yields estimates for d=3 which seem to be remarkably accurate.

The critical behavior of a semi-infinite

n

-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with

{m}_{e}

(easy-magnetization) components take a critical value above which

{m}_{e}

-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (

{m}_{e}=1

) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.

d<6

bulk dimensions is clarified. Two distinct renormalization schemes are utilized. The first is a massless one based on minimal subtraction of dimensional poles and the dimensionality expansion about

d=6.

To overcome its problems in going below

d=4

dimensions, a massive one for fixed dimensions

d<~4

is constructed. The resulting renormalization group (or Callan-Symanzik) equations are exploited to obtain the scaling forms of surface quantities like the dynamic structure factor. In conjunction with boundary operator expansions scaling relations follow that relate the critical indices of the dynamic and static infrared singularities of surface quantities to familiar static bulk and surface exponents. To test the predicted scaling forms and scaling-law expressions for the critical exponents involved, accurate computer-simulation data are presented for the dynamic surface structure factor. These are in conformity with our predictions.

The critical behavior of a semi-infinite

n

-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions near the special transition is studied. The renormalization of the relevant surface operators is discussed and explicitly carried out in two-loop order. It is found that the crossover exponent ${\ensuremath{\varphi}}_{s}$ related to the square of the order parameter in the surface does not satisfy the relation ${\ensuremath{\varphi}}_{s}=1\ensuremath{-}\ensuremath{\nu}$ due to Bray and Moore.