An overview of the mathematical structure of the three-dimensional(3D) Ising model is given from the points of view of topology,algebra,and geometry.By analyzing the relationships among transfer matrices of the 3D Ising model,Reidemeister moves in the knot theory,Yang-Baxter and tetrahedron equations,the following facts are illustrated for the 3D Ising model.1) The complex quaternion basis constructed for the 3D Ising model naturally represents the rotation in a(3+1)-dimensional space-time as a relativistic quantum statistical mechanics model,which is consistent with the 4-fold integrand of the partition function obtained by taking the time average.2) A unitary transformation with a matrix that is a spin representation in 2 n·l·o-space corresponds to a rotation in 2n·l·o-space,which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model.3) A tetrahedron relationship would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model,and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures.4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φx,φy,and φz.The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail.The conjectured exact solution is compared with numerical results,and the singularities at/near infinite temperature are inspected.The analyticity in β=1/(kBT) of both the hard-core and the Ising models has been proved only for β>0,not for β=0.Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.
An overview of the mathematical structure of the three-dimensional(3D) Ising model is given from the points of view of topology,algebra,and geometry.By analyzing the relationships among transfer matrices of the 3D Ising model,Reidemeister moves in the knot theory,Yang-Baxter and tetrahedron equations,the following facts are illustrated for the 3D Ising model.1) The complex quaternion basis constructed for the 3D Ising model naturally represents the rotation in a(3+1)-dimensional space-time as a relativistic quantum statistical mechanics model,which is consistent with the 4-fold integrand of the partition function obtained by taking the time average.2) A unitary transformation with a matrix that is a spin representation in 2 n·l·o-space corresponds to a rotation in 2n·l·o-space,which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model.3) A tetrahedron relationship would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model,and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures.4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φx,φy,and φz.The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail.The conjectured exact solution is compared with numerical results,and the singularities at/near infinite temperature are inspected.The analyticity in β=1/(kBT) of both the hard-core and the Ising models has been proved only for β>0,not for β=0.Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.
An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1) - dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2^(nlo)-space corresponds to a rotation in 2nlo-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases Φx, Φy, and Φz. The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected. The analyticity in β = 1/(kB T) of both the hard-core and Ising models has been proved for β > 0, not for β = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.
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.
The three-dimensional (3D) Ising model is mapped into a 3D spinless fermionic model by the Jordan-Wigner transformation. The exact solution of the 3D model for spinless fermions is derived analytically by performing a diagonalization process consisting of the Clifford algebraic approach, the Fourier transformation and the Bogoliubov transformation. The Clifford algebraic approach is the same as that developed for the 3D Ising model, using a time average within the Jordan-von Neumann-Wigner framework, a linearization procedure and a local gauge transformation. The formulas for eigenvalues, partition function, subsequent thermodynamic properties and critical behaviors are presented. The dimensionality and the topological phases are investigated. The present results for many spinless fermions in a 3D lattice are applicable for studying the mechanisms of magnetism, superfluid, superconductors and topological materials.
The use of a transfer matrix method to solve the 3D Ising model is straightforwardly generalized from the 2D case. We follow B.Kaufman's approach. No approximation is made, however the largest eigenvalue cannot be identified. This problem comes from the fact that we follow the choice of directions of 2-dimensional rotations in the direct product space of the 2D Ising model such that all eigenvalue equations reduce miraculously to only one equation. Other choices of directions of 2-dimensional rotations for finding the largest eigenvalue may lose this fascinating feature. Comparing the series expansion of internal energy per site at the high temperature limit with the series obtained from the computer graphic method, we find these two series have very similar structures. A possible correct via a factor Φ(x) is suggested to fit the result of the graphic method.
Abstract The gauge invariance is an important phenomenon widely occurring in various matters with many-body interactions. The gauge theories are very important for understanding the evolution of fundamental physical processes in particle physics, high-energy physics, and condensed matter physics. In this work, the origin of nonlocal effects is inspected, and the contributions of nontrivial topological structures to physical properties are investigated in detail for both the three-dimensional (3D) Ising model and the 3D Z 2 lattice gauge model. Then, the exact solution for the 3D Z 2 lattice gauge theory is derived by the duality between the two models. The partition function, the spontaneous magnetization, the true range of correlation, and the critical point are rigorously derived for the 3D Z 2 lattice gauge theory. The critical exponents of the 3D Z 2 lattice gauge theory are in the same universality class as the 3D Ising model, which are α = 0, β = 3/8, γ = 5/4, δ = 13/3, η = 1/8, and ν = 2/3. Several fundamental issues, such as dimensionality, duality, symmetry, manifold and degenerate states, are investigated for these many-body interacting spin systems. The connections with superfluid, superconductors, particle physics, etc. are evaluated. Furthermore, physical significances and mathematical aspects of the 3D Z 2 lattice gauge theory are discussed with respect to topology, geometry, and algebra. The present results for the 3D Z 2 lattice gauge theory are helpful for figuring out the basic features of 3D lattice gauge theories with other symmetries, such as U(1), SU(2), and SU(3). This work shines a light on the interdisciplinarity between many-body interacting systems, algebra, topology, and geometry.
In this work, we first reveal the nature of three dimensions: the existence of non-local behavior. Taking the three-dimensional (3D) Ising model as an example, we analyze the transfer matrices as well as the partition function to discuss in details the origin of non-local behavior in three dimensions. We then point out that fail to deal with non-local behavior is one of the main disadvantages of several theoretical approaches (such as conventional low-temperature expansions, conventional high-temperature expansions, perturbative renormalization group and Monte Carlo methods, etc), all of which take local environments into account only. This work provides a deep understanding on the natural difference between 3D bulk materials and 2D materials, and shows that how to treat the non-local behavior and the interaction between spins along the third dimension is a key point for solving exactly the problems of the 3D systems with the nearest neighboring interactions. Finally, we will briefly review recent advance in the 3D Ising model and related topics.