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Journal of Physics A Mathematical and General

Depinning transition and non-universal behaviour of defects in the two-dimensional Ising model: a unified treatment

Nenad M. Svrakic

Published Apr 1, 1983• 13 citations
Publisher

Abstract

The author develops a unified renormalisation group (RG) framework for studying depinning and/or non-universal behaviour associated with linear defects in two-dimensional Ising models. Non-universality is characterised by a fixed line in the RG recursion relations. A generalised version of the depinning transition is proposed and a nearly exact phase diagram is calculated.

DOI
10.1088/0305-4470/16/5/008

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The author develops a unified renormalisation group (RG) framework for studying depinning and/or non-universal behaviour associated with linear defects in two-dimensional Ising models. Non-universality is characterised by a fixed line in the RG recursion relations. A generalised version of the depinning transition is proposed and a nearly exact phase diagram is calculated.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
The existence of nonuniversal, as well as surface (universal) critical behavior near a defect plane is demonstrated on a one-dimensional quantum lattice gas model, belonging to the two-dimensional Ising universality class. According to numerical results the weak defect region is attracted by a line of fixed points starting at the bulk critical point, while the strong defect region is controlled by a separate fixed point describing the surface transition of the model. This new kind of phase diagram is argued to be general for two-dimensional lattice gas models.
The Niemeijer-van Leeuwen formulation of the renormalisation group has been used to discuss the quenched dilute Ising model and the Ising spin-glass in two dimensions. The authors chose a five-spin cell which, in the pure case, treats the Ising antiferromagnet equally as well as the ferromagnet. This choice of cell leads to the existence of a spin-glass ground state which was found to be stable for the whole composition range. The spin-glass transition was found to be sharp and the corresponding critical exponents were calculated. For the dilute Ising model the behaviour of the reduced critical temperature T c (p)/T c (1), the percolation concentration and the correlation length exponent were found to be in good agreement with the exact results and series calculations.