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Physical Review E

Surface critical behavior in Ising magnets: Exact results

Douglas B. Abraham, Anna Maciołek

Published Jun 29, 2006• 2 citations
Publisher

Abstract

An exact derivation of the surface magnetizations of the two phases m(1)I and m(1)II coexisting below the bulk critical temperature for the semi-infinite two-dimensional Ising ferromagnet subject to a surface field is given. The surface critical behavior of the difference m(1)I - m(1)II is that of the ordinary transition only in a limit of a weak surface field below the wetting temperature.

DOI
10.1103/physreve.73.066129

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An exact derivation of the surface magnetizations of the two phases m(1)I and m(1)II coexisting below the bulk critical temperature for the semi-infinite two-dimensional Ising ferromagnet subject to a surface field is given. The surface critical behavior of the difference m(1)I - m(1)II is that of the ordinary transition only in a limit of a weak surface field below the wetting temperature.
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 mmm, mmm being the distance from the edge. The case A<0A<0A<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>0A>0A>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 obtain an exact solution of the d = 2 infinite Ising strip with surface fields of opposite signs , . Earlier predictions by Parry et al are confirmed of the crucial importance of wetting for a system confined between two parallel walls that exert competitive surface fields. The transition of the magnetization profile m(z) from the partial wetting regime below the critical wetting temperature at a single wall with a surface field , to the soft-mode single phase in the temperature range , where is the critical temperature of the bulk system, is described in detail. A scaling ansatz is verified for the singular part of the surface excess free energy, and for the profile, . The modification of the magnetization profile near one wall, due to wetting, is studied. Near the (-) wall at the profile is found to be a linear function of the scaled distance but its slope is different than for the case of perfect asymmetry system hence these two cases belong to the different universality regime.
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.
With simple, exact arguments we show that the surface magnetization m1{m}_{1}m1​ at the extraordinary and normal transitions and the surface energy density ${\ensuremath{\epsilon}}_{1}$ at the ordinary, extraordinary, and normal transitions of semi-infinite ddd-dimensional Ising systems have leading thermal singularities $B\ifmmode\pm\else\textpm\fi{}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, with the same critical exponent and amplitude ratio as the bulk free energy fb(t,0){f}_{b}(t,0)fb​(t,0). The derivation is carried out in three steps: (i) By tracing out the surface spins, the semi-infinite Ising model with supercritical surface enhancement ggg and vanishing surface magnetic field h1{h}_{1}h1​ is mapped exactly onto a semi-infinite Ising model with subcritical surface enhancement, a nonzero surface field, and irrelevant additional surface interactions. This establishes the equivalence of the extraordinary (h1=0,g>0{h}_{1}=0, g>0h1​=0,g>0) and normal (${h}_{1}\ensuremath{\ne}0, g<0$) transitions. (ii) The magnetization m1{m}_{1}m1​ at the interface of an infinite system with uniform temperature ttt and a nonzero magnetic field hhh in the half-space z>0z>0z>0 only is shown to be proportional to ${f}_{b}(t,0)\ensuremath{-}{f}_{b}(t,h)$. (iii) The energy density ${\ensuremath{\epsilon}}_{1}$ at the interface of an infinite system with temperatures t+{t}_{+}t+​ and ttt in the half-spaces z>0z>0z>0 and z<0z<0z<0 and no magnetic field is shown to be proportional to ${f}_{b}(t,0)\ensuremath{-}{f}_{b}({t}_{+},0)$.
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.