Disorder Solutions for Generalized Ising Model with Multispin Interaction
Abstract
This study demonstrates a development of convenient formulae for obtaining the value of the free energy in the thermodynamic limit on a set of exact disorder solutions depending on four parameters for a 2D generalized Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the four interactions for the planar model, and for the 3D generalized Ising model in an external magnetic field with all kinds of interactions in the tetrahedron formed by four spins: at the origin of the coordinates and the closest to it along three coordinate axes in the first coordinate octant. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and a cyclic closure of the set of all points (in natural ordering). For both the planar model and the 3D model, elementary transfer matrices with non-negative matrix elements are constructed, while the free energy in the thermodynamic limit is equal to the Napierian logarithm of the maximum eigenvalue of the transfer matrix. This maximum eigenvalue can be found for a special kind of eigenvector with positive components. The region of existence of these solutions is described. The examples show the existence of nontrivial solutions of the resulting systems of equations for plane and three-dimensional generalized Ising models. The system of equations and the value of free energy in the thermodynamic limit will remain the same for plane and three-dimensional models with Hamiltonians, in which the value of the maximum in the natural ordering of the spin is replaced by the value of the spin at almost any other point in the lattice, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and repeatability of the components of the found eigenvectors, disappearing when we go beyond the framework of the obtained set of exact solutions, are the reason for the search for phase transitions in the vicinity of this set of disordered solutions.
References
[2] Wu F.Y. The Potts model. Reviews of Modern Physics. 1982; 54(1):235-268. (In Eng.) DOI: 10.1103/RevModPhys.54.235
[3] Baxter R.J. Exactly solved models in statistical mechanics. New York: Academic Press, 1982. (In Eng.)
[4] Verhagen A.M.W. An exactly soluble case of the triangular Ising model in a magnetic field. Journal of Statistical Physics. 1976; 15(3):219-231. (In Eng.) DOI: 10.1007/BF01012878
[5] Ruján P. Order and disorder lines in systems with competing interactions. III. Exact results from stochastic crystal growth. Journal of Statistical Physics. 1984; 34(3-4):615-646. (In Eng.) DOI: 10.1007/BF01018562
[6] Stephenson J. Ising-Model Spin Correlations on the Triangular Lattice. IV. Anisotropic Ferromagnetic and Antiferromagnetic Lattices. Journal of Mathematical Physics. 970; 11(2):420-431. (In Eng.) DOI: 10.1063/1.1665155
[7] Welberry T.R., Galbraith R. A two-dimensional model of crystal-growth disorder. Journal of Applied Crystallography. 1973; 6:87-96. (In Eng.) DOI: 10.1107/S0021889873008216
[8] Enting I.G. Triplet order parameters in triangular and honeycomb Ising models. Journal of Physics A: Mathematical and General. 1977; 10(10):1737-1743. (In Eng.) DOI: 10.1088/0305-4470/10/10/008
[9] Welberry T.R., Miller G.H. A Phase Transition in a 3D Growth-Disorder Model. Acta Crystallographica Section A: Foundations and Advances. 1978; A34:120-123. (In Eng.) DOI: 10.1107/S0567739478000212
[10] Forrester P.J., Baxter R.J. Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan identities. Journal of Statistical Physics. 1985; 38(3-4):435-472. (In Eng.) DOI: 10.1007/BF01010471
[11] Jaekel M.T., Maillard J.M. A criterion for disorder solutions of spin models. Journal of Physics A: Mathematical and General. 1985; 18(8):1229-1238. (In Eng.) DOI: 10.1088/0305-4470/18/8/023
[12] Baxter R.J. Disorder points of the IRF and checkerboard Potts models. Journal of Physics A: Mathematical and General. 1984; 17(17):L911-L917. (In Eng.) DOI: 10.1088/0305-4470/17/17/001
[13] Jaekel M.T., Maillard J.M. A disorder solution for a cubic Ising model. Journal of Physics A: Mathematical and General. 1985; 18(4):641-651. (In Eng.) DOI: 10.1088/0305-4470/18/4/013
[14] Jaekel M.T., Maillard J.M. Disorder solutions for Ising and Potts models with a field. Journal of Physics A: Mathematical and General. 1985; 18(12):2271-2277. (In Eng.) DOI: 10.1088/0305-4470/18/12/025
[15] Wu F.Y. Two-Dimensional Ising Model with Crossing and Four-Spin Interactions and a Magnetic Field i(π/2)kT. Journal of Statistical Physics. 1986; 44(3-4):455-463. (In Eng.) DOI: 10.1007/BF01011305
[16] Wu F.Y. Exact Solution of a Triangular Ising Model in a Nonzero Magnetic Field. Journal of Statistical Physics. 1985; 40(5-6):613-620. (In Eng.) DOI: 10.1007/BF01009892
[17] Bessis J.D., Drouffe J.M., Moussa P. Positivity constraints for the Ising ferromagnetic model. Journal of Physics A: Mathematical and General. 1976; 9(12):2105-2124. (In Eng.) DOI: 10.1088/0305-4470/9/12/015
[18] Phase Transitions and Critical Phenomena. Vol. 1: Exact Results. Domb C., Green H.S. (eds). New York: Academic Press, 1972. 506 pp. (In Eng.)
[19] Dhar D., Maillard J.M. Susceptibility of the checkerboard Ising model. Journal of Physics A: Mathematical and General. 1985; 18(7):L383-L388. (In Eng.) DOI: 10.1088/0305-4470/18/7/010
[20] Minlos R.A., Khrapov P.V. Cluster properties and bound states of the Yang-Mills model with compact gauge group. I. Theoretical and Mathematical Physics. 1984; 61(3):1261-1265. (In Eng.) DOI: 10.1007/BF01035013
[21] Khrapov P.V. Cluster expansion and spectrum of the transfer matrix of the two-dimensional ising model with strong external field. Theoretical and Mathematical Physics. 1984; 60(1):734-735. (In Eng.) DOI: 10.1007/BF01018259
[22] Katrakhov V.V., Kharchenko Yu.N. Two-dimensional fourline models of the Ising model type. Theoretical and Mathematical Physics. 2006; 149(2):1545-1558. (In Eng.) DOI: 10.1007/s11232-006-0137-y
[23] Khrapov P.V. Disorder Solutions for Generalized Ising and Potts Models with Multispin Interaction. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2019; 15(1):33-44. (In Eng.) DOI: 10.25559/SITITO.15.201901.33-44
[24] Levenberg K. A Method for the Solution of Certain Non-Linear Problems in Least Squares. Quarterly of Applied Mathematics. 1944; 2:164-168. (In Eng.) DOI: 10.1090/qam/10666
[25] Perron O. Zur Theorie der Matrices. Mathematische Annalen. 1907; 64(2):248-263. (In Eng.) DOI: 10.1007/BF01449896
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication policy of the journal is based on traditional ethical principles of the Russian scientific periodicals and is built in terms of ethical norms of editors and publishers work stated in Code of Conduct and Best Practice Guidelines for Journal Editors and Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE). In the course of publishing editorial board of the journal is led by international rules for copyright protection, statutory regulations of the Russian Federation as well as international standards of publishing.
Authors publishing articles in this journal agree to the following: They retain copyright and grant the journal right of first publication of the work, which is automatically licensed under the Creative Commons Attribution License (CC BY license). Users can use, reuse and build upon the material published in this journal provided that such uses are fully attributed.
