# Clifford algebra approach of 3D Ising model

- 22 Downloads

## Abstract

We develop a Clifford algebra approach for 3D Ising model. We first note the main difficulties of the problem for solving exactly the model and then emphasize two important principles (i.e., Symmetry Principle and Largest Eigenvalue Principle) that will be used for guiding the path to the desired solution. By utilizing some mathematical facts of the direct product of matrices and their trace, we expand the dimension of the transfer matrices **V** of the 3D Ising system by adding unit matrices **I** (with compensation of a factor) and adjusting their sequence, which do not change the trace of the transfer matrices **V** (Theorem 1: Trace Invariance Theorem). The transfer matrices **V** are re-written in terms of the direct product of sub-transfer-matrices \(Sub(V^{(\delta )})=[I\otimes I\otimes \cdots \otimes I\otimes V^{(\delta )}\otimes I\otimes \cdots \otimes I]\), where each \(V^{(\delta )}\) stands for the contribution of a plane of the 3D Ising lattice and interactions with its neighboring plane. The sub-transfer-matrices \(V^{(\delta )}\) are isolated by a large number of the unit matrices, which allows us to perform a linearization process on \(V^{(\delta )}\) (Theorem 2: Linearization Theorem). It is found that locally for each site j, the internal factor \(\hbox {W}_{\mathrm{j}}\) in the transfer matrices can be treated as a boundary factor, which can be dealt with by a procedure similar to the Onsager–Kaufman approach for the boundary factor U in the 2D Ising model. This linearization process splits each sub-transfer matrix into \(2^{\mathrm{n}}\) sub-spaces (and the whole system into \(2^{\mathrm{nl}}\) sub-spaces). Furthermore, a local transformation is employed on each of the sub-transfer matrices (Theorem 3: Local Transformation Theorem). The local transformation trivializes the non-trivial topological structure, while it generalizes the topological phases on the eigenvectors. This is induced by a gauge transformation in the Ising gauge lattice that is dual to the original 3D Ising model. The non-commutation of operators during the processes of linearization and local transformation can be dealt with to be commutative in the framework of the Jordan-von Neumann–Wigner procedure, in which the multiplication \(A\circ B=\frac{1}{2}\left( {AB+BA} \right) \) in Jordan algebras is applied instead of the usual matrix multiplication AB (Theorem 4: Commutation Theorem). This can be realized by time-averaging t systems of the 3D Ising models with time evaluation. In order to determine the rotation angle for the local transformation, the star–triangle relationship of the 3D Ising model is employed for Curie temperature, which is the solution of generalized Yang-Baxter equations in the continuous limit. Finally, the topological phases generated on the eigenvectors are determined, based on the relation with the Ising gauge lattice theory.

## Keywords

Three-dimensional Ising model Exact solution Clifford algebra## Notes

### Acknowledgements

This work has been supported by the National Natural Science Foundation of China under Grant numbers 51331006 and 51590883, by the State key Project of Research and Development of China (No. 2017YFA0206302), and by the key project of Chinese Academy of Science under grant number KJZD-EW-M05-3. ZDZ acknowledges Prof. J.H.H. Perk for helpful discussion on properties of the transfer matrices, Prof.Julian Ławrynowicz for discussion on linearization process. ZDZ also is grateful to Fei Yang for understanding, encouragement, support and discussion.

## References

- 1.Adler, S.L.: Quaternion Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)zbMATHGoogle Scholar
- 2.Bohr, N.: Das Quantenpostulat und die neuere Entwicklung der Atomistik. Naturwissenschaften
**16**, 245–257 (1928)ADSCrossRefGoogle Scholar - 3.de Leo, S.: Quaternions and special relativity. J. Math. Phys.
**37**, 2955–2968 (1996)ADSMathSciNetCrossRefGoogle Scholar - 4.de Leo, S., Rodrigues Jr., W.A.: Quantum mechanics: from complex to complexified quaternions. Int. J. Theor. Phys.
**36**, 2725–2757 (1997)MathSciNetCrossRefGoogle Scholar - 5.Department of Physics of Beijing University (eds.): Quantum Statistical Mechanics. Beijing University Press, Beijing (1987)Google Scholar
- 6.Finkelstein, D., Jauch, J.M., Schiminovich, S., Speiser, D.: Foundations of quaternion quantum mechanics. J. Math. Phys.
**3**, 207–220 (1962)ADSMathSciNetCrossRefGoogle Scholar - 7.Francesco, P.D., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Springer, New York (1996)zbMATHGoogle Scholar
- 8.Heisenberg, W.: The development of interpretation of the quantum theory. In: Niels Bohr and the Development of Physics. Essays dedicated to Niels Bohr on the occasion of his seventieth birthday. Ed. by W. Pauli with the assistance of L. Rosenfeld and V.F. Weisskopf, Pergamon Press, London 1955, pp. 12-29 (1955)Google Scholar
- 9.Ising, E.: Beitrag zur Theorie des Ferromagnetismus. Z. Phys.
**31**, 253–258 (1925)ADSCrossRefGoogle Scholar - 10.Istrail, S.: Universality of intractability for the partition function of the Ising model across non-planar lattices. In: Proceedings of the \(32^{{\rm nd}}\) ACM Symposium on the Theory of Computing (STOC00), ACM Press, p. 87–96, Portland, Oregon, May 21–23 (2000)Google Scholar
- 11.Jaekel, M.T., Maillard, J.M.: Symmetry-relations in exactly soluble models. J. Phys. A
**15**, 1309–1325 (1982)ADSMathSciNetCrossRefGoogle Scholar - 12.Jordan, P.: Über eine Klasse nichtassoziativer hyperkomplexer Algebren. Nachr. d. Ges. d. Wiss. Göttingen, 569–575 (1932)Google Scholar
- 13.Jordan, P.: Über Verallgemeinerungsmöglichkeiten des Formalismus der Quantenmechanik. Nachr. d. Ges. d. Wiss. Göttingen, 209–217 (1933)Google Scholar
- 14.Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math.
**35**, 29–64 (1934)MathSciNetCrossRefGoogle Scholar - 15.Kauffman, L.H.: Knots and Physics, 3rd edn. World Scientific Publishing Co. Pte. Ltd, Singapore, (2001) (2001)Google Scholar
- 16.Kauffman, L.H.: The mathematics and physics of knots. Rep. Prog. Phys.
**68**, 2829–2857 (2005)ADSMathSciNetCrossRefGoogle Scholar - 17.Kauffman, L.H.: Knot Theory and Physics, in the Encyclopedia of Mathematical Physics. In: Francoise, J.P., Naber, G.L, Tsun, T.S. (eds), Elsevier, Amsterdam (2007)Google Scholar
- 18.Kaufman, B.: Crystal statistics II: Partition function evaluated by spinor analysis. Phys. Rev.
**76**, 1232–1243 (1949)ADSCrossRefGoogle Scholar - 19.Kogut, J.B.: An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys.
**51**, 659–713 (1979)ADSMathSciNetCrossRefGoogle Scholar - 20.Ławrynowicz, J., Marchiafava, S., Niemczynowicz, A.: An approach to models of order–disorder and Ising lattices. Adv. Appl. Clifford Algebra.
**20**, 733–743 (2010)MathSciNetCrossRefGoogle Scholar - 21.Ławrynowicz, J., Suzuki, O., Niemczynowicz, A.: On the ternary approach to Clifford structures and Ising lattices. Adv. Appl. Clifford Algebra.
**22**, 757–769 (2012)MathSciNetCrossRefGoogle Scholar - 22.Ławrynowicz, J., Nowak-Kepczyk, M., Suzuki, O.: Fractals and chaos related to Ising–Onsager–Zhang lattices versus the Jordan–von Neumann–Wigner procedures. Quat. Approach Int. J. Bifurcat. Chaos.
**22**, 1230003 (2012)MathSciNetCrossRefGoogle Scholar - 23.Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase transitions. 2. Lattice gas and Ising model. Phys. Rev.
**87**, 410–419 (1952)ADSMathSciNetCrossRefGoogle Scholar - 24.Lou, S.L., Wu, S.H.: Three-dimensional Ising model and transfer matrices. Chin. J. Phys.
**38**, 841–854 (2000)MathSciNetGoogle Scholar - 25.March, N.H., Angilella, G.G.N.: Exactly solvable models in many-body theory. World Sci. Singap. Chap.
**9**, 147–191 (2016)Google Scholar - 26.Marchiafava, S., Rembieliński, J.: Quantum of quaternions. J. Math. Phys.
**33**, 171–173 (1992)ADSMathSciNetCrossRefGoogle Scholar - 27.Nayak, C., Simon, S.H., Stern, A., Freedman, M., Sarma, S.D.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys.
**80**, 1083–1159 (2008)ADSMathSciNetCrossRefGoogle Scholar - 28.Newell, G.F., Montroll, E.W.: On the theory of the Ising model with ferromagnetism. Rev. Mod. Phys.
**25**, 353–389 (1953)ADSMathSciNetCrossRefGoogle Scholar - 29.Onsager, L.: Crystal statistics I: a two-dimensional model with an order-disorder transition. Phys. Rev.
**65**, 117–149 (1944)ADSMathSciNetCrossRefGoogle Scholar - 30.Perk, J.H.H.: Comment on ‘Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices’. Philos. Magn.
**89**, 761–764 (2009)ADSCrossRefGoogle Scholar - 31.Perk, J.H.H.: Rejoinder to the response to the comment on ’Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices’. Philos. Magn.
**89**, 769–770 (2009)ADSCrossRefGoogle Scholar - 32.Perk, J.H.H.: Comment on “Mathematical structure of the three-dimensional (3D) Ising model”. Chin. Phys. B
**22**, 131507-1–131507-5 (2013)CrossRefGoogle Scholar - 33.Savit, R.: Duality in field theory and statistical systems. Rev. Mod. Phys.
**52**, 453–487 (1980)ADSMathSciNetCrossRefGoogle Scholar - 34.Schultz, T.D., Mattis, D.C., Lieb, E.H.: Two-dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys.
**36**, 856–871 (1964)ADSMathSciNetCrossRefGoogle Scholar - 35.Stroganov, YuG: Tetrahedron equation and spin integrable models on a cubic lattice. Theor. Math. Phys.
**110**, 141–167 (1997)MathSciNetCrossRefGoogle Scholar - 36.Suzuki, O., Zhang, Z.D.: A method of Riemann–Hilbert problem for Zhang’s conjecture 1 in 3D Ising model, to be publishedGoogle Scholar
- 37.Witten, E.: Gauge-theories and integrable lattice models. Nucl. Phys. B
**322**, 629–697 (1989)ADSMathSciNetCrossRefGoogle Scholar - 38.Witten, E.: Gauge-theories, vertex models and quantum groups. Nucl. Phys. B
**330**, 285–346 (1989)ADSMathSciNetCrossRefGoogle Scholar - 39.Wu, F.Y., McCoy, B.M., Fisher, M.E., Chayes, L.: Comment on a recent conjectured solution of the three-dimensional Ising model. Philos. Magn.
**88**, 3093–3095 (2008)ADSCrossRefGoogle Scholar - 40.Wu, F.Y., McCoy, B.M., Fisher, M.E., Chayes, L.: Rejoinder to the response to ’Comment on a recent conjectured solution of the three-dimensional Ising model’. Philos. Magn.
**88**, 3103 (2008)ADSCrossRefGoogle Scholar - 41.Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transitions.1. Theory of condensation. Phys. Rev.
**87**, 404–409 (1952)ADSMathSciNetCrossRefGoogle Scholar - 42.Zamolodchikov, A.B.: Tetrahedra equations and integrable systems in three-dimensional space. Sov. Phys. JETP
**52**, 325–336 (1980)ADSGoogle Scholar - 43.Zamolodchikov, A.B.: Tetrahedron equations and the relativistic S-matrix of straight-strings in 2+1 dimensions. Commun. Math. Phys.
**79**, 489–505 (1981)ADSMathSciNetCrossRefGoogle Scholar - 44.Zhang, Z.D.: Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices. Philos. Magn.
**87**, 5309–5419 (2007)ADSCrossRefGoogle Scholar - 45.Zhang, Z.D.: Mathematical structure of the three-dimensional (3D) Ising model. Chin. Phys. B
**22**, 030513-1–030513-15 (2013)ADSGoogle Scholar - 46.Zhang, Z.D.: Response to “Comment on a recent conjectured solution of the three-dimensional Ising model”. Philos. Magn.
**88**, 3097–3101 (2008)ADSCrossRefGoogle Scholar - 47.Zhang, Z.D.: Response to the Comment on ’Conjectures on exact solution of threedimensional (3D) simple orthorhombic Ising lattices’. Philos. Magn.
**89**, 765–768 (2009)ADSCrossRefGoogle Scholar - 48.Zhang, Z.D.: Mathematical structure and the conjectured exact solution of three-dimensional (3D) Ising model. Acta. Metall. Sin.
**52**, 1311–1325 (2016)Google Scholar - 49.Zhang, Z.D.: The nature of three dimensions: non-local behavior in the three-dimensional (3D) Ising model. J. Phys. Conf. Ser.
**827**, 012001-1–012001-10 (2017)Google Scholar - 50.Zhang, Z.D.: Topological effects and critical phenomena in the three-dimensional (3D) Ising model, Chapter 27 in “Many-body approaches at different scales: a tribute to Norman H. March on the occasion of his 90th birthday’. In: Angilella, G. G. N., Amovilli, C. (eds), Springer, New York (2018)Google Scholar