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Physical Review Letters

Position-Space Renormalization-Group Transformations: Some Proofs and Some Problems

Robert B. Griffiths, Paul A. Pearce

Published Oct 2, 1978• 85 citations
Publisher

Abstract

It is shown that certain types of position-space, or "cell," renormalization-group transformations from an Ising-model object Hamiltonian onto another Ising-model image Hamiltonian are well defined and smooth in the thermodynamic limit, provided the object Hamiltonian has a large enough magnetic field. In certain other cases there is evidence (though not a rigorous proof) that no thermodynamic limit exists, or, if it exists, the transformation is not smooth.

DOI
10.1103/physrevlett.41.917

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It is shown that certain types of position-space, or "cell," renormalization-group transformations from an Ising-model object Hamiltonian onto another Ising-model image Hamiltonian are well defined and smooth in the thermodynamic limit, provided the object Hamiltonian has a large enough magnetic field. In certain other cases there is evidence (though not a rigorous proof) that no thermodynamic limit exists, or, if it exists, the transformation is not smooth.
We suggest that the Monte Carlo renormalization group, when combined with the type of cell-spin transformation introduced by van Leeuwen, should be a powerful tool in the study of Ising models with n> 2n>~2n> 2. Numerical results are presented for the Baxter model and the Ising model with nearest- and next-nearest-neighbor interactions on a square lattice.
The renormalization-group (RG) transformation for the one-dimensional quantum Ising model has been performed exactly for two different block transformations with arbitrary size of the block (L). The recursion equations have been generalized in the large-L limit for other models using finite-size scaling ideas. We show that the success of the real-space RG method for the quantum Ising model is accidental. For other models the RG transformation generally mixes up the surface and bulk properties of the model such that strong block-size corrections appear.
The description of first-order phase transitions using the linear renormalization group is illustrated with the two-dimensional Ising model. Different linear renormalization groups should be used to describe the system in the neighborhood of criticality and away from it. In the latter case, the renormalization-group trajectories are lines of constant spin expectation value, which map a system below critical temperature in a small magnetic field to a system of free spins at a finite magnetic field. A finite-lattice approximation is also presented. The equation of state and several thermodynamic quantities are obtained; a smooth matching of the behavior near and away from criticality occurs naturally.
A method for computing low--temperature series for renormalized operators in the two--dimensional Ising model is proposed. Series for the renormalized magnetization and nearest--neighbor correlation function are given for the majority rule transformation on 2 \\Theta 2 blocks and random tie--breaker. These series are applied to the study at very low temperature of the first--order phase transition undergone by this model. We analyze how truncation in the renormalized Hamiltonian leads to spurious discontinuities of the Renormalization Group transformation. Keywords: Renormalization group, position--space renormalization--group transformations, Ising model, low--temperature expansions. E-mail: salas@mafalda.physics.nyu.edu 1 Introduction The behavior of the Renormalization Group (RG) in the vicinity of first order phase transitions has been a very controversial matter for the last 20 years. In 1975 Nienhuis and Nauenberg [1] proposed that the RG transformations behave near first--or...
Here is the first part of the summary of my work on random Ising model using real-space renormalization group (RSRG), also known as a Migdal-Kadanoff one. This approximate renormalization scheme was applied to the analysis thermodynamic properties of the model, and of probabilistic properties of a pair correlator, which is a fluctuating object in disordered systems. PACS numbers: 02.50.-r, 05.20.-y, 05.70.Fh, 64.60.ae, 75.10.-b, 75.10.Hk, 87.10.+e Keywords: statistical physics, magnetism, spin glass, neural networks