The nature of three dimensions: Non-local behavior in the three-dimensional (3D) Ising model

Zhang, Z. D.

doi:10.1088/1742-6596/827/1/012001

Cited by 7 publications

(9 citation statements)

References 40 publications

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“…If this idea is the case, then the acceleration of (3+1)D universe has relevance only with properties of some 3D equilibrium statistic models or Euclidean field theories, while the vacuum energy of (3+1)D quantum field theories could affect dynamics only of some (3+2) or (4+1)D universes. This is consistent with our results for an emergent time axis in the 3D Ising model [10,11,16,17,20,59], and also in TQSM and TQFTs.…”

Section: Topological Quantum Field Theoriessupporting

confidence: 93%

“…, where σj (j = 1, 2, 3) are Pauli matrices. The partition function of the ferromagnetic 3D Ising model at the zero magnetic field can be given as follows [8][9][10][11]16,17]:…”

Section: Exact Solution Of the Ferromagnetic 3d Ising Modelmentioning

confidence: 99%

“…. For the ferromagnetic 3D Ising model, the Clifford algebraic representation plays an important role in solving analytically its exact solution [10,11,16,17]. The difficulties arisen by the transfer matrix V3 (as seen from Equation ( 5)) for exactly solving the ferromagnetic 3D Ising model at the zero magnetic field consist of nonlocal behavior, non-Gaussian, and noncommutative operators [11].…”

Section: Exact Solution Of the Ferromagnetic 3d Ising Modelmentioning

confidence: 99%

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Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

Zhang

2022

Symmetry

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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.

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Section: Topological Quantum Field Theoriessupporting

confidence: 93%

“…, where σj (j = 1, 2, 3) are Pauli matrices. The partition function of the ferromagnetic 3D Ising model at the zero magnetic field can be given as follows [8][9][10][11]16,17]:…”

Section: Exact Solution Of the Ferromagnetic 3d Ising Modelmentioning

confidence: 99%

Section: Exact Solution Of the Ferromagnetic 3d Ising Modelmentioning

confidence: 99%

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Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

Zhang

2022

Symmetry

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“…There have been several rounds of comments/responses between some statistical physicists and Zhang [30][31][32]39,40,46,47]. We do not want to repeat Zhang's responses published already [45][46][47][48][49][50]. Just mention briefly: The error for the Jordan-Wigner transform in Zhang's original paper [44] is not a problem, which has been corrected in ref.…”

Section: Introduction To 3d Ising Modelmentioning

confidence: 99%

Clifford algebra approach of 3D Ising model

Zhang

Сузукі

March

2018

Adv. Appl. Clifford Algebras

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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 somein Jordan algebras is applied instead of the usual matrix multiplication AB (Theorem IV: 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. ..

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“…Recent studies [44][45][46] reveal, that non-local effects also play an important role in the Ising model. Although the Ising model only captures nearest neighbor spin interaction, in 3D a term with an effective long range interaction appears in the partition function [45]. Although the results of these studies cannot be directly applied to our model, they represent a further hint at the presence and influence of non-local effects.…”

Section: Eigenmodes At T =mentioning

confidence: 99%

Eigenmodes of a disordered FeCo magnonic crystal at finite temperatures

Paischer

Buczek

Buczek³

et al. 2021

J. Phys.: Condens. Matter

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In this report we present a systematic study of the magnonic modes in the disordered Fe 0.5 Co 0.5 alloy based on the Heisenberg Hamiltonian using two complementary approaches. In order to account for substitutional disorder, on the one hand we directly average the transverse magnetic susceptibility in real space over different disorder configurations and on the other hand we use the coherent potential approximation (CPA). While the method of direct averaging is numerically exact, it is computationally expensive and limited by the maximal size of the supercell which can be simulated on a computer. On the contrary the CPA does not suffer from this drawback and yields a cheap numerical scheme. Therefore, we additionally compare the results of these two approaches and show that the CPA gives very good results for most of the magnetic properties considered in this report, including the magnon energies and the spatial shape of the eigenmodes. However, it turns out that while reproducing the general trend, the CPA systematically underestimates the disorder induced damping of the magnons. This provides evidence that the physics of impurity scattering in this system is governed by non-local effects missing in the CPA. Finally, we study the real space eigenmodes of the system, including their spatial shapes, and analyze their temperature dependence within the random phase approximation.

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The nature of three dimensions: Non-local behavior in the three-dimensional (3D) Ising model

Cited by 7 publications

References 40 publications

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

Clifford algebra approach of 3D Ising model

Eigenmodes of a disordered FeCo magnonic crystal at finite temperatures

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