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 $ν$ and $α$ are also calculated as $ν=1$ and $α=0$ .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 $ν$ and $α$ are also calculated as $ν=1$ and $α=0$.

arXiv:1910.05521Oct 12, 20190 citations

The critical values of coupling strength of the non-self-dual Ising lattices under decimation transformation

Tuncer Kaya

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.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.

Modern Physics Letters BDec 30, 20212 citations

Star-triangle duality estimates for triangular and honeycomb permutation models

Masayuki Ohzeki

We study a duality analysis in conjunction with the star-triangle transformation for symmetric-group permutation models on the triangular and honeycomb lattices. The calculation is motivated by the permutation-model description of random tensor networks and by earlier duality analyses of replicated spin glasses. The essential point is that the finite-basis unit is not a bare bond but a star-triangle block. Our analysis estimates the critical bond dimension for the honeycomb lattice to be 2.634929344884, and the associated single-bond duality relation yields the triangular-lattice estimate 1.475661534848.We study a duality analysis in conjunction with the star-triangle transformation for symmetric-group permutation models on the triangular and honeycomb lattices. The calculation is motivated by the permutation-model description of random tensor networks and by earlier duality analyses of replicated spin glasses. The essential point is that the finite-basis unit is not a bare bond but a star-triangle block. Our analysis estimates the critical bond dimension for the honeycomb lattice to be 2.634929344884, and the associated single-bond duality relation yields the triangular-lattice estimate 1.475661534848.

cond-mat.dis-nnCross-listedarXiv:2607.14917Progress of Theoretical and Experimental PhysicsJul 17, 20260 citations

The real space renormalization group treatment of the FCSC Ising lattice

Tuncer Kaya

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.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.

Modern Physics Letters BFeb 2, 20230 citations

A new approach to real space renormalization group treatment of Ising model for square and simple cubic lattice

Tuncer Kaya

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.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.

arXiv:1801.05385International Journal of Modern Physics BAug 14, 20187 citations

On Ising models

Hoede, C., Zandvliet, Henricus J.W.

For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.

Jun 1, 20060 citations

ν

and

α

are also calculated as

ν=1

and

α=0

2-\sqrt{2} \approx 0.5858,

comparable to the exponent

4/7 \approx 0.5714

found from renormalization arguments.