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physica status solidi (b)

The Local Correlation Lengths in Ising Models with Free Surfaces

W. Korneta, Zdzisław Pytel

Published Apr 1, 1988• 0 citations
Publisher

Abstract

Abstract The nearest‐neighbor Ising model of a film is considered in which couplings between spins depend on the distances to the surfaces. The local correlation lengths in directions parallel and perpendicular to the surfaces of the model near different second‐order phase transitions are studied. The random‐phase approximation is used. Analytical expressions for the local correlation lengths are obtained. Assuming that one surface of the model of a film tends to infinity, local correlation lengths in the three‐dimensional semi‐infinite Ising model are obtained.

DOI
10.1002/pssb.2221460230

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Abstract The nearest‐neighbor Ising model of a film is considered in which couplings between spins depend on the distances to the surfaces. The local correlation lengths in directions parallel and perpendicular to the surfaces of the model near different second‐order phase transitions are studied. The random‐phase approximation is used. Analytical expressions for the local correlation lengths are obtained. Assuming that one surface of the model of a film tends to infinity, local correlation lengths in the three‐dimensional semi‐infinite Ising model are obtained.
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 use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.
Recent work on local functional theories of critical inhomogeneous fluids and Ising-like magnets has shown them to be a potentially exact, or near exact, description of universal finite-size effects associated with the excess free energy and scaling of one-point functions in critical thin films. This approach is extended to predict the two-point correlation function G in critical thin films with symmetric surface fields in arbitrary dimension d. In d = 2 we show there is exact agreement with the predictions of conformal invariance for the complete spectrum of correlation lengths xi((n)) as well as the detailed position dependence of the asymptotic decay of G. In d = 3 and d>/=4 we present new numerical predictions for the universal finite-size correlation length and scaling functions determining the structure of G across the thin film. Highly accurate analytical closed form expressions for these universal properties are derived in arbitrary dimension.