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arXiv:2403.14955 · Journal of Statistical Physics

Faithfulness of Real-Space Renormalization Group Maps

Katsuya O. Akamatsu, Naoki Kawashima

Published Aug 27, 2024• 0 citations
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Abstract

Abstract The behavior of $$b=2$$ b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical $$q=2,3,4$$ q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice $$L_g$$ L g and allowing the source configuration size $$L_0$$ L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that $$b=2$$ b = 2 weighted majority-like RSRG maps acting on the $$q=2$$ q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.

DOI
10.1007/s10955-024-03323-7

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Abstract The behavior of b=2b=2b=2 b = 2 real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical q=2,3,4q=2,3,4q=2,3,4 q = 2 , 3 , 4 Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice LgL_gLg​ L g and allowing the source configuration size L0L_0L0​ L 0 to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that b=2b=2b=2 b = 2 weighted majority-like RSRG maps acting on the q=2q=2q=2 q = 2 Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.
We introduce and discuss a real-space renormalization group (RSRG) procedure on very small lattices, which in principle does not require any of the usual approximations, e.g. a cut-off in the expansion of the Hamiltonian in powers of the field. The procedure is carried out numerically on very small lattices (4x4 to 2x2) and implemented for the Ising Model and the q=3,4,5 Potts Models. Nevertheless, the resulting estimates of the correlation length exponent and the magnetization exponent are typically within 3% to 7% of the exact values. The 4-state Potts Model generates a third magnetic exponent which seems to be unknown in the literature. A number of questions about the meaning of certain exponents and the procedure itself arise from its use of symmetry principles and its application to the q=5 Potts Model.
Real Space Renormalization Group (RSRG) treatment of Ising model for square and simple cubic lattice is investigated and critical coupling strengths of these lattices are obtained. The mathematical complications, which appear inevitable in the decimated partition function due to Block-spin transformation, is treated with a relevant approximation. The approximation is based on the approximate equivalence of [Formula: see text] for small [Formula: see text], where K is the nearest neighbor coupling strength and [Formula: see text] is the nearest neighbor spins degrees of freedom around a central spin. The values of the critical coupling strengths are obtained as 0.4830 for square lattice and 0.2225 for simple cubic (SC) lattice. The corresponding critical exponents values [Formula: see text] and [Formula: see text] are also calculated within very acceptable agreement with those values obtained from numerical works.
We present a real-space renormalization-group (RG) study of the nnn-component classical Heisenberg model in the large-nnn limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.
A large-scale renormalization group study of the Ising model for the square, honeycomb, triangular, simple cubic, and body-centered (BC) cubic lattices has been performed recently by us. We complement those studies with the structurally more complicated face-centered cubic lattice Ising model. The results obtained from the real space renormalization group (RSRG) treatment of the face-centered simple cubic (FCSC) Ising lattice have been presented in this work. The difficulty due to its non-self-dual decimation transformation property and high numbers of nearest neighbors in the treatment is overcome with some relevant approximations. The approximation is based on keeping only the pairwise interactions in the decimated lattice which is apparently in the form of a tetragonal structure. Within this approximation, the renormalized coupling strength is related to the coupling constant of the original lattice. The critical coupling strength for the decimated tetragonal structure is calculated as 0.0905.
The authors discuss results from two types of real space renormalization group (RSRG) calculations applied to the random field Ising model in three dimensions. Starting from a lattice of size L, the RSRG is used to reduce the lattice to a size L=2, on which the trace is done exactly. In this way, thermodynamic properties, such as the magnetization and susceptibility, can be determined approximately. They find that, for a given size, the susceptibility increases as the temperature, T, is reduced down to the transition temperature, T c , and becomes essentially independent of temperature below T c . Both in the vicinity of T c and at lower temperatures, there are large sample-to-sample fluctuations in the susceptibility which grow with increasing system size. They interpret these results in terms of the droplet theory of the transition.
Extending the parameter space of the three-dimensional (d=3) Ising model, we search for a regime of eliminated corrections to finite-size scaling. For that purpose, we consider a real-space renormalization group (RSRG) with respect to a couple of clusters simulated with the transfer-matrix (TM) method. Imposing a criterion of "scale invariance," we determine a location of the nontrivial RSRG fixed point. Subsequent large-scale TM simulation around the fixed point reveals eliminated corrections to finite-size scaling. As anticipated, such an elimination of corrections admits systematic finite-size-scaling analysis. We obtained the estimates for the critical indices as nu=0.6245(28) and y(h)=2.4709(73). As demonstrated, with the aid of the preliminary RSRG survey, the transfer-matrix simulation provides rather reliable information on criticality even for d=3, where the tractable system size is restricted severely.