Abstract
The 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 Z2 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.
Highlights
Introduction published maps and institutional affilIt is known that four different fundamental forces, i.e., electromagnetic, weak, and strong forces, and gravity, naturally exist in nature
In models in the topological quantum statistical mechanics, a topological phase transition occurs near the infinite temperature, which visualizes a symmetrical breaking of time inverse symmetry, accompanied with emerging of massless gauge bosons
We build the framework of topological quantum statistical mechanics (TQSM) and topological quantum field theory (TQFT) with respect to the mathematical aspects and physical features by a generalization of our findings and observations in the 3D Ising models
Summary
The topological structures of a ferromagnetic 3D Ising system at the zero magnetic field are constructed by the partition function Z (Equation (4)), which consists of the transfer matrices V1 , V2 , and V3 represented in Equations (5)–(7). The three topological phases are generated on the eigenvectors of the ferromagnetic 3D Ising model at the zero magnetic field, which represent the projection of the energy spectrum from 4D to 3D. Zhang and Suzuki [13] applied the method of the Riemann–Hilbert problem, introduced vertex operators of knot types and a flat vector bundle, and used the Gauss–Bonnet–Chern formula to generate the topological phases on the eigenvectors of the ferromagnetic 3D Ising model at the zero magnetic field. The action of these gauge lattice theories can be mapped to the kinetic (i.e., gauge) terms of quantum field theories with different symmetries for these interactions, with ultraviolet cutoff
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