# Strong anisotropy in two-dimensional surfaces with generic scale invariance: Gaussian and related models.

TL;DR: A SA ansatz is formulated that, albeit equivalent to existing ones borrowed from equilibrium critical phenomena, is more naturally adapted to the type of observables that are measured in experiments on the dynamics of thin films, such as one- and two-dimensional height structure factors.

Abstract: Among systems that display generic scale invariance, those whose asymptotic properties are anisotropic in space (strong anisotropy, SA) have received relatively less attention, especially in the context of kinetic roughening for two-dimensional surfaces. This is in contrast with their experimental ubiquity, e.g., in the context of thin-film production by diverse techniques. Based on exact results for integrable (linear) cases, here we formulate a SA ansatz that, albeit equivalent to existing ones borrowed from equilibrium critical phenomena, is more naturally adapted to the type of observables that are measured in experiments on the dynamics of thin films, such as one- and two-dimensional height structure factors. We test our ansatz on a paradigmatic nonlinear stochastic equation displaying strong anisotropy like the Hwa-Kardar equation [Phys. Rev. Lett. 62, 1813 (1989)], which was initially proposed to describe the interface dynamics of running sand piles. A very important role to elucidate its SA properties is played by an accurate (Gaussian) approximation through a nonlocal linear equation that shares the same asymptotic properties.

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TL;DR: In this paper, an experimental assessment of surface kinetic roughening properties that are anisotropic in space was performed for two specific instances of silicon surfaces irradiated by ion-beam sputtering under diverse conditions.

Abstract: We report an experimental assessment of surface kinetic roughening properties that are anisotropic in space. Working for two specific instances of silicon surfaces irradiated by ion-beam sputtering under diverse conditions (with and without concurrent metallic impurity codeposition), we verify the predictions and consistency of a recently proposed scaling Ansatz for surface observables like the two-dimensional (2D) height power spectral density (PSD). In contrast with other formulations, this ansatz is naturally tailored to the study of two-dimensional surfaces, and allows us to readily explore the implications of anisotropic scaling for other observables, such as real-space correlation functions and PSD functions for 1D profiles of the surface. Our results confirm that there are indeed actual experimental systems whose kinetic roughening is strongly anisotropic, as consistently described by this scaling analysis. In the light of our work, some types of experimental measurements are seen to be more affected by issues like finite space resolution effects, etc. that may hinder a clear-cut assessment of strongly anisotropic scaling in the present and other practical contexts.

21 citations

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Abstract: Applying the standard field theory renormalization group to the model of landscape erosion introduced by Pastor-Satorras and Rothman yields unexpected results: the model is multiplicatively renormalizable only if it involves infinitely many coupling constants (i.e., the corresponding renormalization group equations involve infinitely many β-functions). We show that the one-loop counterterm can nevertheless be expressed in terms of a known function V (h) in the original stochastic equation and its derivatives with respect to the height field h. Its Taylor expansion yields the full infinite set of the one-loop renormalization constants, β-functions, and anomalous dimensions. Instead of a set of fixed points, there arises a two-dimensional surface of fixed points that quite probably contains infrared attractive regions. If that is the case, then the model exhibits scaling behavior in the infrared range. The corresponding critical exponents turn out to be nonuniversal because they depend on the coordinates of the fixed point on the surface, but they satisfy certain universal exact relations.

12 citations

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TL;DR: It is concluded that the occurrence of strong anisotropy in two-dimensional surfaces requires dynamics to be conserved and is not generic in parameter space but requires, rather, specific forms of the terms appearing in the equation of motion.

Abstract: We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditions for occurrence of strong anisotropy in the scaling properties of two-dimensional surfaces displaying generic scale invariance. In that study, a natural scaling ansatz was proposed for strongly anisotropic systems, which arises naturally when analyzing data from, e.g., thin-film production experiments. The ansatz was tested in Gaussian (linear) models of surface dynamics and in nonlinear models, like the Hwa-Kardar (HK) equation [ Phys. Rev. Lett.62, 1813 (1989)], which are susceptible of accurate approximations through the former. In contrast, here we analyze nonlinear equations for which such approximations fail. Working within generically scale-invariant situations, and as representative case studies, we formulate and study a generalization of the HK equation for conserved dynamics and reconsider well-known systems, such as the conserved and the nonconserved anisotropic Kardar-Parisi-Zhang equations. Through the combined use of dynamic renormalization group analysis and direct numerical simulations, we concludethattheoccurrenceofstronganisotropyintwo-dimensionalsurfacesrequiresdynamicstobeconserved. We find that, moreover, strong anisotropy is not generic in parameter space but requires, rather, specific forms of the terms appearing in the equation of motion, whose justification needs detailed information on the dynamical process that is being modeled in each particular case.

11 citations

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TL;DR: In this article, the standard field theoretic renormalization group is applied to the model of landscape erosion introduced by R. Pastor-Satorras and D. H. Rothman.

Abstract: Standard field theoretic renormalization group is applied to the model of landscape erosion introduced by R. Pastor-Satorras and D. H. Rothman [Phys. Rev. Lett. 80: 4349 (1998); J. Stat. Phys. 93: 477 (1998)] yielding unexpected results: the model is multiplicatively renormalizable only if it involves infinitely many coupling constants ( i.e., the corresponding renormalization group equations involve infinitely many beta-functions). Despite this fact, the one-loop counterterm can be derived albeit in a closed form in terms of the certain function $V(h)$, entering the original stochastic equation, and its derivatives with respect to the height field $h$. Its Taylor expansion gives rise to the full infinite set of the one-loop renormalization constants, beta-functions and anomalous dimensions. Instead of a set of fixed points, there is a two-dimensional surface of fixed points that is likely to contain infrared attractive region(s). If that is the case, the model exhibits scaling behaviour in the infrared range. The corresponding critical exponents are nonuniversal through the dependence on the coordinates of the fixed point on the surface, but satisfy certain universal exact relations.

10 citations

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TL;DR: In this paper, the authors show that while the KPZ equation follows the Tracy-Widom distribution, its derivative displays Gaussian behavior, hence being in a different universality class.

Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a paradigmatic model of nonequilibrium low-dimensional systems with spatiotemporal scale invariance, recently highlighting universal behavior in fluctuation statistics. Its space derivative, namely the noisy Burgers equation, has played a very important role in its study, predating the formulation of the KPZ equation proper, and being frequently held as an equivalent system. We show that, while differences in the scaling exponents for the two equations are indeed due to a mere space derivative, the field statistics behave in a remarkably different way: while the KPZ equation follows the Tracy-Widom distribution, its derivative displays Gaussian behavior, hence being in a different universality class. We reach this conclusion via direct numerical simulations of the equations, supported by a dynamic renormalization group study of field statistics.

10 citations

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01 Jan 1972

TL;DR: The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results as discussed by the authors, and the major aim of this serial is to provide review articles that can serve as standard references for research workers in the field.

Abstract: The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It has moved into a central place in condensed matter studies. Statistical physics, and more specifically, the theory of transitions between states of matter, more or less defines what we know about 'everyday' matter and its transformations. The major aim of this serial is to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments.

12,039 citations

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01 Jan 1995

TL;DR: The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html as discussed by the authors.

Abstract: This book brings together two of the most exciting and widely studied subjects in modern physics: namely fractals and surfaces. To the community interested in the study of surfaces and interfaces, it brings the concept of fractals. To the community interested in the exciting field of fractals and their application, it demonstrates how these concepts may be used in the study of surfaces. The authors cover, in simple terms, the various methods and theories developed over the past ten years to study surface growth. They describe how one can use fractal concepts successfully to describe and predict the morphology resulting from various growth processes. Consequently, this book will appeal to physicists working in condensed matter physics and statistical mechanics, with an interest in fractals and their application. The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html

3,891 citations

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2,551 citations

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TL;DR: The examination of phase transitions and critical phenomena has dominated statistical physics for the latter half of this century as discussed by the authors, and beautiful experimental results have elucidated the singularities (critical behavior) that occur in phase transitions.

Abstract: The examination of phase transitions and critical phenomena has dominated statistical physics for the latter half of this century--there is a great theoretical challenge in solving special statistical mechanical models. Additionally, beautiful experimental results have elucidated the singularities (critical behavior) that occur in phase transitions. Although a few spectacular successes in the exact solution of simple models have occurred, interesting systems have mostly proven to be intractable from the analytic perspective. Because many physically disparate systems show identical critical behavior, identifying and understanding general principles of universality and their relationship to microscopic symmetries are of fundamental importance

2,269 citations