Topological Quantum Statistical Mechanics and Topological Quantum Field Theories
Zhang, Zhidong
Corresponding AuthorZhang, Zhidong(
Source PublicationSYMMETRY-BASEL
AbstractThe Ising model describes a many-body interacting spin (or particle) system, which can be utilized to imitate the fundamental forces of nature. Although it is the simplest many-body interacting system of spins (or particles) with Z(2) symmetry, the phenomena revealed in Ising systems may afford us lessons for other types of interactions in nature. In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan-von Neumann-Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan-von Neumann-Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry.
Keyword3D Ising model topology quantum statistical mechanics quantum field theories
Funding OrganizationNational Natural Science Foundation of China
Indexed BySCI
Funding ProjectNational Natural Science Foundation of China
WOS Research AreaScience & Technology - Other Topics
WOS SubjectMultidisciplinary Sciences
WOS IDWOS:000778136900001
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Document Type期刊论文
Corresponding AuthorZhang, Zhidong
AffiliationChinese Acad Sci, Inst Met Res, Shenyang Natl Lab Mat Sci, 72 Wenhua Rd, Shenyang 110016, Peoples R China
Recommended Citation
GB/T 7714
Zhang, Zhidong. Topological Quantum Statistical Mechanics and Topological Quantum Field Theories[J]. SYMMETRY-BASEL,2022,14(2):22.
APA Zhang, Zhidong.(2022).Topological Quantum Statistical Mechanics and Topological Quantum Field Theories.SYMMETRY-BASEL,14(2),22.
MLA Zhang, Zhidong."Topological Quantum Statistical Mechanics and Topological Quantum Field Theories".SYMMETRY-BASEL 14.2(2022):22.
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