A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry.

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.

Exact solution of two-dimensional (2D) Ising model with a transverse field: A low-dimensional quantum spin system

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 2n·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.

The common feature for a nontrivial hard problem is the existence of nontrivial topological structures, non-planarity graphs, nonlocalities, or long-range spin entanglements in a model system with randomness. For instance, the Boolean satisfiability (K-SAT) problems for K ≥ 3 MSATK≥3 are nontrivial, due to the existence of non-planarity graphs, nonlocalities, and the randomness. In this work, the relation between a spin-glass three-dimensional (3D) Ising model MSGI3D with the lattice size N = mnl and the K-SAT problems is investigated in detail. With the Clifford algebra representation, it is easy to reveal the existence of the long-range entanglements between Ising spins in the spin-glass 3D Ising lattice. The internal factors in the transfer matrices of the spin-glass 3D Ising model lead to the nontrivial topological structures and the nonlocalities. At first, we prove that the absolute minimum core (AMC) model MAMC3D exists in the spin-glass 3D Ising model, which is defined as a spin-glass 2D Ising model interacting with its nearest neighboring plane. Any algorithms, which use any approximations and/or break the long-range spin entanglements of the AMC model, cannot result in the exact solution of the spin-glass 3D Ising model. Second, we prove that the dual transformation between the spin-glass 3D Ising model and the spin-glass 3D Z2 lattice gauge model shows that it can be mapped to a K-SAT problem for K ≥ 4 also in the consideration of random interactions and frustrations. Third, we prove that the AMC model is equivalent to the K-SAT problem for K = 3. Because the lower bound of the computational complexity of the spin-glass 3D Ising model CLMSGI3D is the computational complexity by brute force search of the AMC model CUMAMC3D, the lower bound of the computational complexity of the K-SAT problem for K ≥ 4 CLMSATK≥4 is the computational complexity by brute force search of the K-SAT problem for K = 3 CUMSATK=3. Namely, CLMSATK≥4=CLMSGI3D≥CUMAMC3D=CUMSATK=3. All of them are in subexponential and superpolynomial. Therefore, the computational complexity of the K-SAT problem for K ≥ 4 cannot be reduced to that of the K-SAT problem for K < 3.

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.

In this work, we analyze in detail the transfer matrices as well as the partition function of the three-dimensional (3D) Ising model, in order to understand the topological effects in three dimensions. We attempt to extract the non-local contributions to the physical properties of the 3D Ising system, by comparing the conjectured exact solution with the results of the conventional high-temperature and low-temperature expansions. This is because the conjectured solution takes into account both the local and non-local behaviors, while these conventional expansions consider only the local environments. Then, we summarize the results of our previous work, most of them being joint work with Professor N. H. March, on the critical exponents of the 3D Ising model and some related topics. This work provides a deep understanding on the topological contributions and the non-local behavior in the 3D Ising model and their effects on the critical exponents in the 3D Ising universality.

Studying percolation phase transitions offers valuable insights into the characteristics of phase transitions, shedding light on the underlying mechanisms that govern the formation of global connectivity within a system. We explore the percolation phase transition in the 3D cubic Ising model by employing two machine learning techniques. Our results demonstrate that machine learning methods can distinguish different phases during the percolation transition. Through the finite-size scaling analysis on the output of the neural networks, the percolation temperature and a correlation length exponent in the geometrical percolation transition are extracted and compared to those in the thermal magnetization phase transition within the 3D Ising model. These findings provide a valuable method for enhancing our understanding of the properties of the QCD critical point, which belongs to the same universality class as the 3D Ising model.

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.

OpenACC programs of the Swendsen–Wang multi-cluster spin flip algorithm

The classical nucleation theory (CNT) is tested systematically by computer simulations of the two-dimensional (2D) and three-dimensional (3D) Ising models with a Glauber-type spin flip dynamics. While previous studies suggested potential problems with CNT, our numerical results show that the fundamental assumption of CNT is correct. In particular, the Becker-Döring theory accurately predicts the nucleation rate if the correct droplet free energy function is provided as input. This validates the coarse graining of the system into a one dimensional Markov chain with the largest droplet size as the reaction coordinate. Furthermore, in the 2D Ising model, the droplet free energy predicted by CNT matches numerical results very well, after a logarithmic correction term from Langer's field theory and a constant correction term are added. But significant discrepancies are found between the numerical results and existing theories on the magnitude of the logarithmic correction term in the 3D Ising model. Our analysis underscores the importance of correctly accounting for the temperature dependence of surface energy when comparing numerical results and nucleation theories.

GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model

We present results from simulations of the hysteresis loops in the two-dimensional (2D) Ising model and a cell-dynamical system (CDS) in a linearly, rather than sinusoidally, varying external field. We find in the CDS a disorder-induced transition, which has behavior similar to the critical point in the 2D Ising model. Below the critical point, the area of the hysteresis loops, representing the dissipation per cycle, scales with the rate of the driving field H\ifmmode \dot{}\else \.{}\fi{} as A=${\mathit{A}}_{0}$+aH${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{\mathrm{\ensuremath{\alpha}}}$, with a nearly constant \ensuremath{\alpha}\ensuremath{\sim}0.36\ifmmode\pm\else\textpm\fi{}0.08 for the Ising model and 0.66\ifmmode\pm\else\textpm\fi{}0.02 for the CDS. Thus, the CDS belongs to the class of mean-field models, which is different from that of the Ising model. Above the critical point, both the Ising model and the CDS give ${\mathit{A}}_{0}$=0 and an \ensuremath{\alpha} that increases with temperature and disorder.

With the rapid development of the graphics processing unit (GPU), a recent GPU offers incredible resources for general purpose computing. First, we apply this new technology to Monte Carlo simulations of the 2D lattice Ising models. By implementing the checkerboard algorithm, results are obtained up to 54, 62 and 68 times faster on the GPU than on a current CPU core for the honeycomb, square and triangular lattice respectively. An implementation of the 3D cubic lattice Ising model on a GPU is able to generate results up to 141 times faster than on a current CPU core. The critical point of the 2D and 3D Ising model obtained by finite size scaling techniques is consistent with the theoretical results for the 2D Ising model and previous simulation results for the 3D Ising model.

In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent ν∼1.3 is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent α∼−1.9 . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.