Study of the frustrated Ising model on a square lattice based on the exact density of states

Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. Furthermore, we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit. Finally, we extend our results to the antiferromagnetic Ising model.

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Low-temperature antiferromagnetism of Ising model with competing interactions

ВЕСТНИК ПЕРМСКОГО УНИВЕРСИТЕТА ФИЗИКА ◽

10.17072/1994-3598-2021-2-64-71 ◽

2021 ◽

pp. 64-71

Author(s):

Kirill Tsiberkin ◽

Keyword(s):

Ising Model ◽

Low Temperature ◽

Temperature Shift ◽

Second Order ◽

Square Lattice ◽

Antiferromagnetic Coupling ◽

Competing Interactions ◽

Saturation State ◽

Spin Configuration ◽

Competing Interaction

The paper presents a numerical analysis of equilibrium state and spin configuration of square lattice Ising model with competing interaction. The most detailed description is given for case of ferromagnetic interaction of the first-order neighbours and antiferromagnetic coupling of the second-order neighbours. The numerical method is based on Metropolis algorithm. It uses 128×128 lattice with periodic boundary conditions. At first, the simulation results show that the system is in saturation state at low temperatures, and it turns into paramagnetic state at the Curie point. The competing second-order interaction makes possible the domain structure realization. This state is metastable, because its energy is higher than saturation energy. The domains are small at low temperature, and their size increases when temperature is growing until the single domain occupies the whole simulation area. In addition, the antiferromagnetic coupling of the second-order neighbours reduces the Curie temperature of the system. If it is large enough, the lattice has no saturation state. It turns directly from the domain state into paramagnetic phase. There are no extra phases when the system is antiferromagnetic in main order, and only the Neel temperature shift realizes here.

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Neural Network for Prediction of Curie Temperature of Two-Dimensional Ising Model

Dal'nevostochnyi Matematicheskii Zhurnal ◽

10.47910/femj202105 ◽

2021 ◽

Vol 21 (1) ◽

pp. 51-60

Author(s):

A.O. Korol ◽

◽

V.Yu. Kapitan ◽

◽

...

Keyword(s):

Neural Network ◽

Ising Model ◽

Curie Temperature ◽

High Temperature Phase ◽

Square Lattice ◽

Temperature Phase ◽

The Neural Network ◽

Different Temperatures ◽

Low Temperature Phase ◽

Ising Spins

The authors describe a method for determining the critical point of a second-order phase transitions using a convolutional neural network based on the Ising model on a square lattice. Data for training were obtained using Metropolis algorithm for different temperatures. The neural network was trained on the data corresponding to the low-temperature phase, that is a ferromagnetic one and high-temperature phase, that is a paramagnetic one, respectively. After training, the neural network analyzed input data from the entire temperature range: from 0.1 to 5.0 (in dimensionless units) and determined (the Curie temperature T_c). The accuracy of the obtained results was estimated relative to the Onsager solution for a flat lattice of Ising spins.

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THE SIMULATION OF 2D SPIN-1 ISING MODEL WITH POSITIVE BIQUADRATIC INTERACTION ON A CELLULAR AUTOMATON

International Journal of Modern Physics C ◽

10.1142/s0129183103005443 ◽

2003 ◽

Vol 14 (10) ◽

pp. 1305-1320 ◽

Cited By ~ 14

Author(s):

BÜLENT KUTLU

Keyword(s):

Phase Transition ◽

Ising Model ◽

Cellular Automaton ◽

Intermediate Phase ◽

Finite Size ◽

Square Lattice ◽

Two Dimensional ◽

Finite Size Scaling ◽

Creutz Cellular Automaton ◽

Antiferromagnetic Spin

The two-dimensional antiferromagnetic spin-1 Ising model with positive biquadratic interaction is simulated on a cellular automaton which based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transition of the model are presented for a comparison with those obtained from other calculations. We confirm the existence of the intermediate phase observed in previous works for some values of J/K and D/K. The values of the static critical exponents (β, γ and ν) are estimated within the framework of the finite-size scaling theory for D/K<2J/K. Although the results are compatible with the universal Ising critical behavior in the region of D/K<2J/K-4, the model does not exhibit any universal behavior in the interval 2J/K-4<D/K<2J/K.

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THE PHONON SPECTRUM OF THE OCTAGONAL TILING

International Journal of Modern Physics B ◽

10.1142/s0217979293002468 ◽

1993 ◽

Vol 07 (06n07) ◽

pp. 1505-1525 ◽

Cited By ~ 11

Author(s):

J. LOS ◽

T. JANSSEN ◽

F. GÄHLER

Keyword(s):

Specific Heat ◽

Density Of States ◽

Phonon Spectrum ◽

Low Frequency ◽

Debye Model ◽

Square Lattice ◽

Total Density ◽

Unit Cells ◽

Linear Behaviour ◽

Total Density Of States

A study of the phonon spectrum of the octagonal tiling is presented, by calculating and analysing the properties of the spectrum of perfect and randomized commensurate approximants with unit cells containing up to 8119 vertices. The total density of states, obtained by numerical integration over the Brillouin zone, exhibits much structure, and in the low frequency range of the spectrum there is deviation from the normal linear behaviour in the form of pseudogaps. For randomized approximants these pseudogaps disappear and the density of states is globally smoothened. It turns out that the widths of the gaps in the dispersion vanish in the low frequency limit. Therefore the scaling behaviour of the lowest branches tends to the behaviour of an absolutely continuous spectrum, which is not the case at higher frequencies. As an application, the vibrational specific heat of the different tiling models is calculated and compared to the specific heat of a square lattice and of a Debye model.

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