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Physical review. B, Solid state

Exact renormalization-group equations in one dimension: The Ising model

Richard G. Priest

Published May 1, 1975• 8 citations
Publisher

Abstract

A solution of the one-dimensional spin-one-half Ising model by the renormalization-group method is given. The form of the scaling functions is calculated and the exact dependence of the thermodynamic functions on the scaling fields is calculated.
DOI
10.1103/physrevb.11.3461

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arXiv:cond-mat/9907468•Jul 29, 1999•0 citations
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).
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.
We present an exact differential renormalization-group method for the square Ising model. This method is analogous to that introduced by Hilhorst et al. for the triangular Ising model. Our approach is based upon the construction of such a renormalization transformation which remains confined to the space of four spatially dependent nearest-neighbor interactions. We derive the renormalization-group equations in the form of a set of four linear first-order partial differential equations for the interactions. We obtain a class of solutions of these equations and determine a nontrivial fixed point. The analysis of the flow around the fixed point shows that this fixed point is stable with respect to perturbations within the critical surface and is unstable in the temperaturelike direction. A unique eigenvalue, yT=1{y}_{T}=1yT​=1, from the eigenvalue equation for the temperature direction is found, in accordance with the exact Onsager solution.
Calculation Method for the Three-Dimensional Ising Ferromagnet Thermodynamics within the Frames of ρ6ρ^6ρ6 Model
Calculation of thermodynamic functions of the three-dimensional Ising ferromagnet above and below critical temperature is performed in the approximation of sixfold basis distribution (ρ6ρ^6ρ6 model). Comparison with the results for the ρ4ρ^4ρ4 model indicates that dependence of the thermodynamic functions on the renormalization group parameter sss becomes weaker. The optimal interval of the renormalization group parameter values is determined.