Clifford algebra approach of 3D Ising model

Zhang, Zhidong; Сузукі, Осаму; March, N. H.

doi:10.1007/s00006-018-0923-2

Cited by 68 publications

(54 citation statements)

References 45 publications

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“…Two conjectures were proposed by the author in [10] for solving the exact solution of the ferromagnetic 3D Ising model at the zero external magnetic field. Zhang, Suzuki, and March proved four theorems (Trace Invariance Theorem, Linearization Theorem, Local Transformation Theorem, Commutation Theorem) in [11], which rigorously proved Zhang's two conjectures. Furthermore, Suzuki and Zhang [12,13] rigorously proved the two conjectures by the method of the Riemann-Hilbert problem, the monoidal transformation, and the Gauss-Bonnet-Chern formula.…”

Section: Introductionmentioning

confidence: 90%

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: Introductionmentioning

confidence: 90%

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

Zhang

2022

Symmetry

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“…The state of the spin system in the Ising model can be described analytically for a two-dimensional square lattice 17 . There are solutions for some 3D systems 18 , 19 . For a spin-free system, an accurate analytical solution is not possible.…”

Section: Modelmentioning

confidence: 99%

Composition of the Frenkel–Kontorova and Ising models for investigation the magnetic properties of a ferromagnetic monolayer on a stretching substrate

Белим

Tikhomirov

2021

Sci Rep

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In the article, computer simulation on the behavior of a ferromagnetic thin film on a non-magnetic substrate by computer simulation is performed. The substrate is described by the two-dimensional Frenkel–Kontorova potential. The Ising model is used to describe the magnetic properties of a two-dimensional ferromagnetic film. The Wolf cluster algorithm is used to model the magnetic behavior of the film. A square lattice is considered for an unperturbed ferromagnetic film. Computer simulations show that mismatch of film and substrate periods results in film splitting into regions with different atomic structures. Magnetic properties for the obtained structure have been investigated. The hysteresis loop is calculated using the Metropolis algorithm. Deformations of the substrate lead to a decrease in the phase transition temperature. The Curie temperature decreases both when the substrate is compressed and when stretched. The change in phase transition temperature depends on the decreasing rate of exchange interaction with distance and the amplitude of interaction with the substrate. When the substrate is compressed, an increase in the amplitude of the interaction between the film and the substrate results in an increase in the phase transition temperature. The opposite effect occurs when the substrate is stretched. The hysteresis loop changes its shape and parameters when the substrate is deformed. Compression and stretching of the substrate results in a decrease in coercive force. The reduction in coercive force when compressing the substrate is greater than when stretching. The magnetization of the film is reduced by deformations at a fixed temperature.

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“…7. Apart from the theoretical models mentioned in Table 4, Z. Zhang et al 60,61 has studied different 3D-Ising models for a simple orthorhombic system. Among them, the exact solution (3D-Ising-Exact, with β = 0.375, γ = 1.25), renormalization group calculation (3D-Ising-RG, with β = 0.340, γ = 1.244) and high-temperature series expansion (3D-Ising-SE, with β = 0.312, γ = 1.25) are also compared in Fig.…”

Section: Scaling Analysismentioning

confidence: 99%

Field induced crossover in critical behaviour and direct measurement of the magnetocaloric properties of La0.4Pr0.3Ca0.1Sr0.2MnO3

Ghorai

Skini

Hedlund

et al. 2020

Sci Rep

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La0.4Pr0.3Ca0.1Sr0.2MnO3 has been investigated as a potential candidate for room temperature magnetic refrigeration. Results from X-ray powder diffraction reveal an orthorhombic structure with Pnma space group. The electronic and chemical properties have been confirmed by X-ray photoelectron spectroscopy and ion-beam analysis. A second-order paramagnetic to ferromagnetic transition was observed near room temperature (289 K), with a mean-field like critical behaviour at low field and a tricritical mean-field like behaviour at high field. The field induced crossover in critical behaviour is a consequence of the system being close to a first-order magnetic transition in combination with a magnetic field induced suppression of local lattice distortions. The lattice distortions consist of interconnected and weakly distorted pairs of Mn-ions, where each pair shares an electron and a hole, dispersed by large Jahn–Teller distortions at Mn3+ lattice sites. A comparatively high value of the isothermal entropy-change (3.08 J/kg-K at 2 T) is observed and the direct measurements of the adiabatic temperature change reveal a temperature change of 1.5 K for a magnetic field change of 1.9 T.

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Clifford algebra approach of 3D Ising model

Cited by 68 publications

References 45 publications

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

Composition of the Frenkel–Kontorova and Ising models for investigation the magnetic properties of a ferromagnetic monolayer on a stretching substrate

Field induced crossover in critical behaviour and direct measurement of the magnetocaloric properties of La0.4Pr0.3Ca0.1Sr0.2MnO3

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