Clifford algebra approach of 3D Ising model

  • Zhidong ZhangEmail author
  • Osamu Suzuki
  • Norman H. March


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.


Three-dimensional Ising model Exact solution Clifford algebra 



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.


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Authors and Affiliations

  1. 1.Shenyang National Laboratory for Materials Science, Institute of Metal ResearchChinese Academy of SciencesShenyangPeople’s Republic of China
  2. 2.Department of Computer and System Analysis, College of Humanities and SciencesNihon UniversityTokyoJapan
  3. 3.Oxford UniversityOxfordEngland

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