Critical behaviour of three-dimensional Ising ferromagnets at imperfect surfaces: Bounds on the surface critical exponent

Abstract

The critical behaviour of three-dimensional semi-infinite Ising ferromagnets at planar surfaces with (i) random surfacebond disorder or (ii) a terrace of monatomic height and macroscopic size is considered. The Griffiths-Kelly-Sherman correlation inequalities are shown to impose constraints on the order-parameter density at the surface, which yield upper and lower bounds for the surface critical exponent β1. If the surface bonds do not exceed the threshold for supercritical enhancement of the pure system, these bounds force β1 to take the value β In a recent paper Pleimling and Selke (PS) [1] reported the results of a detailed Monte Carlo analysis of the effects of two types of surface imperfections on the surface critical behaviour of d = 3 dimensional semiinfinite Ising models with planar surfaces and ferromagnetic nearest-neighbour (NN) interactions: (i) random surface-bond disorder and (ii) a terrace of monatomic height and macroscopic size on the surface. For type (i), both the ordinary and special transitions were studied. They found that the asymptotic temperature dependence of the disorder-averaged surface magnetization on approaching the bulk critical temperature T c from below could be represented by a power law ∼ |τ | β1 with τ ≡ (T − T c )/T c , where β 1 agreed, within the available numerical accuracy, with the respective values β ord 1 ≃ 0.8 and β sp 1 ≃ 0.2 of the pure system's ordinary and special transitions. For type (ii), where the interaction constants were chosen such that only an ordinary transition could occur, the same value β ord 1 of the perfect system was found for β 1 .Their findings for the case of (i) are in conformity with the relevance/irrelevance criteria of Diehl and Nüsser [2,3] according to which the pure system's surface critical behaviour should be expected to be stable or unstable with respect to short-range correlated random surfacebond disorder depending on whether the surface specific heat C 11 [4] of the pure system remains finite or diverges at the transition. It is fairly well established [5,6] that C 11 approaches a finite constant at the ordinary transition, but has a leading thermal singularity ∼ |τ |at the special transition, where Φ is the surface crossover exponent. In the latter case, the condition for irrelevance, [2,3] seem to work quite well in practice. Yet, from a mathematical point of view, they are rather weak because they are nothing but a necessary (though not sufficient) condition for stability of the pure system's critical behaviour.In this note, I shall employ the Griffiths-Kelly-Sherman (GKS) inequalities [12] to obtain upper and lower bounds on the surface magnetization densities of both types of imperfect systems, bounds that are given by surface magnetizations of analogous systems without such imperfections. Their known asymptotic temperature dependence near T c will then be exploited to obtain restrictions on the surface critical behaviour of the imperfect systems considered. For some cases of interest studied by PS [1], the equality β 1 = β o...

Critical phenomena at perfect and non-perfect surfaces

“…These numerical findings provide support for the conjecture of [227] that surface enhancement disorder is irrelevant at the ordinary transition. The robustness of the critical exponent β 1 against dilution observed in the numerical study was later proven by Diehl [249] who derived upper and lower bounds on the surface magnetisation of the diluted surface and showed in a rigorous way that β 1 takes the same value in the diluted system as in the perfect system.…”

“…However, in contrast to the case of random couplings, corrections to scaling are here distinctively different from those of the perfect surface. Diehl also considered this type of imperfections in [249] and showed in a rigorous way that the critical exponent of the step-edge magnetisation is identical to that of the magnetisation of a perfect surface. Note that at the surface transition local critical magnetic exponents are expected to be non-universal close to the step-edge.…”

Logarithmic corrections in the two-dimensional Ising model in a random surface field

“…In an inhomogeneous system the local critical behaviour near localized or extended defects may differ considerably from the bulk critical behaviour in the regular lattice (for a review, see [1]). One possible source of inhomogeneity is quenched (i.e., time-independent) randomness, which can be localized at the surface of the system (fluctuating surface coupling constants [2][3][4][5][6], microscopic terraces at the surface [7]) or at a grain boundary in the bulk of the system. It is known experimentally [8][9][10] that impurities may diffuse from inside the sample and segregate on the surface or at grain boundaries.…”

Crossover between special and ordinary transitions in random semi-infinite Ising-like systems

“…At the present time is very well developed theory of critical behavior of individual surface universality classes for pure isotropic systems [7][8][9][10]5,11] and systems with quenched surfaceenhancement disorder [12][13][14]. General irrelevance-relevance criteria of the Harris type for the systems with quenched short-range correlated surface-bond disorder were predicted in [12] and confirmed by Monte-Carlo calculations [13,15].…”