[Paper Review] Universality and geometry dependence in the class of the nonlinear molecular beam epitaxy equation

Ismael S. S. Carrasco, Tiago J. Oliveira|arXiv (Cornell University)|Jun 20, 2016

Climate variability and models3 citations

TL;DR

This study investigates universality and geometry dependence in the nonlinear molecular beam epitaxy (nMBE) class via numerical simulations in 1+1 and 2+1 dimensions. It finds that height distributions and correlations are universal yet geometry-dependent, with identical critical exponents for flat and curved interfaces, indicating the nMBE class splits into subclasses—similar to the KPZ class—while spatial covariance exhibits a minimum only in the flat subclass.

ABSTRACT

We report extensive numerical simulations of growth models belonging to the nonlinear molecular beam epitaxy (nMBE) class, with flat and curved geometries. In both $d=1+1$ and $2+1$, we find that growth regime height distributions (HDs), spatial and temporal covariances are universal, but geometry-dependent, while the critical exponents are the same for flat and curved interfaces. Therefore the nMBE class does split into subclasses, as also does the Kardar-Parisi-Zhang (KPZ) class. Applying the KPZ ansatz to nMBE models, we estimate the cumulants of the $1+1$ HDs. Spatial covariance for flat subclass is hallmarked by a minima, which is not present in the curved one. Temporal correlations are shown to decay following well-known conjectures.

Motivation & Objective

• To investigate the universality of growth statistics in the nonlinear molecular beam epitaxy (nMBE) class across different geometries.
• To determine whether flat and curved interfaces in nMBE models exhibit distinct scaling behaviors despite shared critical exponents.
• To assess the validity of the KPZ ansatz in estimating height distribution cumulants for 1+1 dimensional nMBE systems.
• To analyze spatial and temporal correlation structures and their dependence on interface geometry.
• To test conjectures on temporal correlation decay in nMBE growth dynamics.

Proposed method

• Conducting extensive numerical simulations of nMBE models in both flat and curved geometries across d=1+1 and d=2+1 dimensions.
• Computing height distributions (HDs), spatial covariances, and temporal correlations from simulated growth data.
• Applying the KPZ ansatz to estimate cumulants of the 1+1 dimensional height distribution for comparison with simulation results.
• Analyzing spatial covariance functions to detect structural differences, such as the presence of a minimum in flat geometry.
• Evaluating temporal correlation decay against established conjectures in the literature.
• Comparing critical exponents between flat and curved interface subclasses to test universality.

Experimental results

Research questions

• RQ1 Are height distributions and correlation functions in the nMBE class universal across different geometries?
• RQ2 Does the presence of curvature in the interface geometry alter the scaling behavior of spatial and temporal correlations?
• RQ3 Do flat and curved nMBE interfaces share the same critical exponents, indicating a single universality class?
• RQ4 Is the spatial covariance in the flat nMBE subclass characterized by a minimum, as observed in the KPZ class?
• RQ5 How well do theoretical conjectures on temporal correlation decay describe the dynamics in nMBE models?

Key findings

• Height distributions (HDs), spatial and temporal covariances in the nMBE class are universal but geometry-dependent.
• The critical exponents for flat and curved interfaces are identical, indicating shared dynamic scaling behavior.
• Spatial covariance in the flat nMBE subclass exhibits a distinct minimum, which is absent in the curved subclass.
• Temporal correlations in nMBE models decay in accordance with established theoretical conjectures.
• The KPZ ansatz provides a consistent estimate for the cumulants of the 1+1 dimensional height distribution.
• The nMBE class splits into subclasses based on geometry, analogous to the KPZ class, despite shared critical exponents.