Clifford algebra approach of 3D Ising model

doi:10.1007/s00006-018-0923-2

IMR OpenIR

	Clifford algebra approach of 3D Ising model
	Zhang, Zhidong 1; Suzuki, Osamu 2; March, Norman H.3
通讯作者	Zhang, Zhidong(zdzhang@imr.ac.cn)
	2019-02-01
发表期刊	ADVANCES IN APPLIED CLIFFORD ALGEBRAS
ISSN	0188-7009
卷号	29 期号:1 页码:28
摘要	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())=[I circle times I circle times circle times I circle times V()circle times I circle times circle times I], where each V() stands for the contribution of a plane of the 3D Ising lattice and interactions with its neighboring plane. The sub-transfer-matrices V() are isolated by a large number of the unit matrices, which allows us to perform a linearization process on V() (Theorem 2: Linearization Theorem). It is found that locally for each site j, the internal factor Wj 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 2n sub-spaces (and the whole system into 2nl 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 degrees B= 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.
关键词	Three-dimensional Ising model Exact solution Clifford algebra
资助者	National Natural Science Foundation of China ; State key Project of Research and Development of China ; key project of Chinese Academy of Science
DOI	10.1007/s00006-018-0923-2
收录类别	SCI
语种	英语
资助项目	National Natural Science Foundation of China[51331006] ; National Natural Science Foundation of China[51590883] ; State key Project of Research and Development of China[2017YFA0206302] ; key project of Chinese Academy of Science[KJZD-EW-M05-3]
WOS研究方向	Mathematics ; Physics
WOS类目	Mathematics, Applied ; Physics, Mathematical
WOS记录号	WOS:000452949600001
出版者	SPRINGER BASEL AG
引用统计	被引频次：64[WOS] [WOS记录] [WOS相关记录]
文献类型	期刊论文
条目标识符	http://ir.imr.ac.cn/handle/321006/130687
专题	中国科学院金属研究所
通讯作者	Zhang, Zhidong
作者单位	1.Chinese Acad Sci, Shenyang Natl Lab Mat Sci, Inst Met Res, 72 Wenhua Rd, Shenyang 110016, Liaoning, Peoples R China 2.Nihon Univ, Coll Humanities & Sci, Dept Comp & Syst Anal, Setagaya Ku, Sakurajosui 3-25-40, Tokyo 1568550, Japan 3.Univ Oxford, Oxford, England
推荐引用方式 GB/T 7714	Zhang, Zhidong,Suzuki, Osamu,March, Norman H.. Clifford algebra approach of 3D Ising model[J]. ADVANCES IN APPLIED CLIFFORD ALGEBRAS,2019,29(1):28.
APA	Zhang, Zhidong,Suzuki, Osamu,&March, Norman H..(2019).Clifford algebra approach of 3D Ising model.ADVANCES IN APPLIED CLIFFORD ALGEBRAS,29(1),28.
MLA	Zhang, Zhidong,et al."Clifford algebra approach of 3D Ising model".ADVANCES IN APPLIED CLIFFORD ALGEBRAS 29.1(2019):28.