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Journal of the National Science Foundation of Sri Lanka

Ising model across planar lattices

Ranasinghe P.K.C. Malmini

Published Dec 30, 2008• 0 citations
Publisher

Abstract

Normal 0 false false false MicrosoftInternetExplorer4 st1\:*{behavior:url(#ieooui) } /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} The Ising model on a two dimensional lattice with pentagon tiling and greater number of next to neighbours interaction is equivalent to the Ising model on the centered square lattice with nearest and next to nearest neighbour non-crossing interactions. We obtain exact expressions on calculation of critical temperature with ferromagnetic and anti-ferromagnetic interactions. A direct anti-ferromagnetic bond of strength α L can give rise to the transition temperatures for appropriate α. We apply the star-triangle mapping transformation technique and decorating transform to Ising model. Keywords: Exact results, Ising model, square lattice, transformations. doi:10.4038/jnsfsr.v36i4.268 Journal of the National Science Foundation of Sri Lanka 36 (4) 303-305

DOI
10.4038/jnsfsr.v36i4.268

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Normal 0 false false false MicrosoftInternetExplorer4 st1:{behavior:url(#ieooui) } / Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} The Ising model on a two dimensional lattice with pentagon tiling and greater number of next to neighbours interaction is equivalent to the Ising model on the centered square lattice with nearest and next to nearest neighbour non-crossing interactions. We obtain exact expressions on calculation of critical temperature with ferromagnetic and anti-ferromagnetic interactions. A direct anti-ferromagnetic bond of strength α L can give rise to the transition temperatures for appropriate α. We apply the star-triangle mapping transformation technique and decorating transform to Ising model. Keywords: Exact results, Ising model, square lattice, transformations. doi:10.4038/jnsfsr.v36i4.268 Journal of the National Science Foundation of Sri Lanka 36 (4) 303-305
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 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
Normal 0 21 false false false MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabela normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} In this work we intend to develop the Mathematical Formalism of Non-Relativistic Quantum Mechanics in terms of two Basics Operators: Evolution Operator, and Density Operator and some of its properties. This article has didactic purposes.
Normal 0 21 false false false MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabela normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} Discussão do problema do átomo de hidrogênio sem levar em conta o spin das partículas interagentes, apresentando-se, posteriormente, uma solução no espaço de momentum. A nálise de uma conexão entre a equação radial para o potencial Coulombiano e a equação radial para o potencial tipo oscilador harmônico multidimensional.