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Modern Physics Letters B

The real space renormalization group treatment of the FCSC Ising lattice

Tuncer Kaya

Published Feb 2, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984922502232

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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.
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.
In this work, the values of critical coupling strengths of the Ising lattices which are changing their lattice structure (or non-self-dual) under decimation transformations, such as the honeycomb, the triangular and the body centered cubic Ising lattices, are obtained by a modified real space renormalization group approach (RSRG). This modification is necessary to obtain a proper relation between the coupling strengths of the original and the decimated lattices. Indeed, we have achieved to obtain a proper renormalized coupling strength relation for honeycomb and triangular lattices readily, since the decimation transformation of the honeycomb lattice produces the triangular lattice or vice versa. Here, the problem of having physically untractable interactions between degrees of freedom in the renormalized Hamiltonian, which leads eventually to inevitable approximations in the treatment, except for the 1D Ising chain, has been solved with a proper approximation. Especially for the 3D Ising lattices, the physically untractable interactions appearing in the renormalized Hamiltonian make the mathematical treatment too cumbersome. As a result, there is not enough research dealing with the 3D Ising lattices using RG theory. Our approximation is based on using the simple relation [Formula: see text], which is, of course, a very relevant first-order approximation, if [Formula: see text]. With the help of this approximation, decimation transformation process produces only pairwise interactions in the renormalized Hamiltonian instead of having four spins, six spins, or even eight spin interactions which appear naturally if all the terms are kept in the renormalized Hamiltonians of the Ising lattices in 2D and higher dimensions. Without this approximation, one cannot apply analytic RG treatment feasibly to even simple cubic lattice, let alone applying it to the body centered cubic lattice. Using this modified RG approach, the values of critical coupling strengths of the honeycomb, the triangular and the body centered cubic Ising lattices are obtained analytically as 0.6470, 0.2754 and 0.1589, respectively. Apparently, these estimations are really close to the results obtained from cumbersome exact treatments which are 0.6585, 0.2747 and 0.1575 for the honeycomb, the triangular and the body centered cubic lattices.
A two-dimensional lattice-gas model with honeycomb symmetry is investigated by using the real-space renormalization group (RSRG) approach with a number of RSRG transformations based on clusters of different symmetries and sizes (up to 42 sites). The accuracy of calculations is found to be a nonmonotonic function of the size of a cluster, and to depend strongly on its symmetry. The highest obtained accuracy with respect to the determination of the critical value of the pair interaction parameter is 0.38%. The phase diagram of the Ising antiferromagnetic spin model and of the corresponding lattice-gas model is constructed with this accuracy. The ordered phase in the lattice system is shown to appear in the very narrow density interval, 0.447<n<0.553.0.447<n<0.553.0.447<n<0.553. In addition, the coverage dependence of the chemical diffusion coefficient and mean-square density fluctuations are studied at temperatures below the critical one. Both quantities are demonstrated to exhibit singularities at the critical coverages. The type of singularities (in particular, the critical slowdown of diffusion) is in agreement with the predictions of the scaling theory and also with recent results obtained for a model treating the effect of surface reconstruction on diffusion.
In this work, an exact renormalization group treatment of honeycomb lattice leading to an exact relation between the coupling strengths of the honeycomb and the triangular lattices is presented. Using the honeycomb and the triangular duality relation, the critical coupling values of honeycomb and triangular lattices are calculated exactly by the simultaneous solution of the renormalized relation and the duality relation, without using the so-called star-triangular transformation. Apparently, the obtained coupling relation is unique. It not only takes place the role of the star triangular relation, but it is also the only exact relation obtained from renormalization group theory other than the 1D Ising chain. An exact pair correlation function expression relating the nearest neighbors and the next nearest neighbor correlation functions are also obtained for the honeycomb lattice. Utilizing this correlation relation, an exact expression of the correlation length of the honeycomb lattice is calculated analytically for the coupling constant values less than the critical value in the realm of the scaling theory. The critical exponents [Formula: see text] and [Formula: see text] are also calculated as [Formula: see text] and [Formula: see text].
In this work, an exact renormalization group treatment of honeycomb lattice leading to an exact relation between the coupling strengths of the honeycomb and the triangular lattices is presented. Using the honeycomb and the triangular duality relation, the critical coupling values of honeycomb and triangular lattices are calculated exactly by the simultaneous solution of the renormalized relation and the duality relation, without using the so-called star-triangular transformation. Apparently, the obtained coupling relation is unique. It, not only takes place the role of the star triangular relation, but also it is the only exact relation obtained from renormalization group theory other than the 1D Ising chain. An exact pair correlation functions expression relating the nearest neighbors and the next nearest neighbor correlation functions are also obtained for the honeycomb lattice. Utilizing this correlation relation, an exact expression of the correlation length of the honeycomb lattice is calculated analytically for the coupling constant values less than the critical value in the realm of the scaling theory. The critical exponents ννν and ααα are also calculated as ν=1ν=1ν=1 and α=0α=0α=0.
Using transfer-matrix extended phenomenological renormalization-group methods [M.A.Yurishchev, Nucl. Phys. B (Proc. Suppl.) 83-84, 727 (2000); hep-lat/9908019; J. Exp. Theor. Phys. 91, 332 (2000); cond-mat/0108002] the improved estimates for the critical temperature of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J⃗=(J′,J′,J){\vec J}=(J',J',J)J=(J′,J′,J) are obtained. Universality of both fundamental critical exponents yty_tyt​ and yhy_hyh​ is confirmed. We show also that the critical finite-size scaling amplitude ratios Aχ(4)Aκ/Aχ2A_{χ^{(4)}}A_κ/A_χ^2Aχ(4)​Aκ​/Aχ2​, Aκ′′/AχA_{κ^{''}}/A_χAκ′′​/Aχ​, and Aκ(4)/Aχ(4)A_{κ^{(4)}}/A_{χ^{(4)}}Aκ(4)​/Aχ(4)​ are independent of the lattice anisotropy parameter Δ=J′/JΔ=J'/JΔ=J′/J.