AN EXACTLY SOLVABLE TWO-FOLD CAYLEY TREE MODEL

1989 ◽  
Vol 03 (10) ◽  
pp. 1523-1537 ◽  
Author(s):  
CAN F. DELALE
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Cayley Tree ◽  
Spontaneous Magnetization ◽  
Zero Field ◽  
Tree Graph ◽  
Tree Model ◽  
Exactly Solvable ◽  
Ferromagnetic Ising Model

A two-fold Cayley tree graph with fully q-coordinated sites is constructed and the ferromagnetic Ising model on the constructed graph is solved exactly. It is shown that a phase transition results in zero field at the critical Bethe temperature with spontaneous magnetization below the critical Bethe temperature.

LOW-TEMPERATURE SERIES EXPANSIONS FOR SQUARE-LATTICE ISING MODEL WITH FIRST AND SECOND NEIGHBOUR INTERACTIONS

2002 ◽  
Vol 16 (32) ◽  
pp. 4911-4917
Author(s):  
YEE MOU KAO ◽  
MALL CHEN ◽  
KEH YING LIN
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Low Temperature ◽  
Spontaneous Magnetization ◽  
Temperature Series ◽  
Square Lattice ◽  
Series Expansions ◽  
Zero Field ◽  
Ferromagnetic Ising Model ◽  
Neighbour Interaction

We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.

scholarly journals Glassy behaviour in the ferromagnetic Ising model on a Cayley tree

1996 ◽  
Vol 29 (18) ◽  
pp. 5773-5804 ◽  
Author(s):  
R Mélin ◽  
J C Anglès d'Auriac ◽  
P Chandra ◽  
B Douçot
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Ferromagnetic Ising Model

scholarly journals Phase Transition in Ferromagnetic Ising Model with a Cell-Board External Field

2015 ◽  
Vol 162 (1) ◽  
pp. 139-161
Author(s):  
Manuel González-Navarrete ◽  
Eugene Pechersky ◽  
Anatoly Yambartsev
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
External Field ◽  
A Cell ◽  
Ferromagnetic Ising Model

The ferromagnetic Ising model on a Cayley tree: a damage spreading analysis

2000 ◽  
Vol 267 (2-3) ◽  
pp. 127-131 ◽  
Author(s):  
M.M. Xavier Jr. ◽  
F.D. Nobre ◽  
A.M. Mariz ◽  
F.A. da Costa
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Damage Spreading ◽  
Ferromagnetic Ising Model

scholarly journals The critical temperature of the 2D-Ising model through deep learning autoencoders

2020 ◽  
Vol 93 (12) ◽  
Author(s):  
Constantia Alexandrou ◽  
Andreas Athenodorou ◽  
Charalambos Chrysostomou ◽  
Srijit Paul
Keyword(s):  
Phase Transition ◽  
Deep Learning ◽  
Critical Temperature ◽  
Ising Model ◽  
Latent Variable ◽  
Scaling Effects ◽  
Physical Systems ◽  
Ferromagnetic Ising Model ◽  
Quasi Order ◽  
Two Phases

Abstract We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We focus our investigation on the 2-dimensional ferromagnetic Ising model and then test the application of the autoencoder on the anti-ferromagnetic Ising model. We use spin configurations produced for the 2-dimensional ferromagnetic and anti-ferromagnetic Ising model in zero external magnetic field. For the ferromagnetic Ising model, we study numerically the relation between one latent variable extracted from the autoencoder to the critical temperature Tc. The proposed autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the ℤ2 symmetry as the temperature increases. We provide a finite volume analysis for a sequence of increasing lattice sizes. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent variable z̃, which enables us to predict the critical temperature. One can define a latent susceptibility and use it to quantify the value of the critical temperature Tc(L) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that Tc(L) extrapolates to the known theoretical value as L →∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision. Subsequently, we test the application of the autoencoder on the anti-ferromagnetic Ising model, demonstrating that the proposed network can detect the phase transition successfully in a similar way. Graphical abstract

Zero-temperature ising spin dynamics on the homogeneous tree of degree three

2000 ◽  
Vol 37 (03) ◽  
pp. 736-747 ◽  
Author(s):  
C. Douglas Howard
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Spin Dynamics ◽  
Spin Chains ◽  
Homogeneous Tree ◽  
Zero Temperature ◽  
Spin Configuration ◽  
Ising Spin ◽  
Temperature Dynamics ◽  
Ferromagnetic Ising Model

We investigate zero-temperature dynamics for a homogeneous ferromagnetic Ising model on the homogeneous tree of degree three (𝕋) with random (i.i.d. Bernoulli) spin configuration at time 0. Letting θ denote the probability that any particular vertex has a +1 initial spin, for infinite spin clusters do not exist at time 0 but we show that infinite ‘spin chains’ (doubly infinite paths of vertices with a common spin) exist in abundance at any time ϵ > 0. We study the structure of the subgraph of 𝕋 generated by the vertices in time-ϵ spin chains. We show the existence of a phase transition in the sense that, for some critical θ c with spin chains almost surely never form for θ < θc but almost surely do form in finite time for θ > θc . We relate these results to certain quantities of physical interest, such as the t → ∞ asymptotics of the probability that any particular vertex changes spin after time t.

Exact zero-field susceptibility of the Ising model on a Cayley tree

1998 ◽  
Vol 177-181 ◽  
pp. 185-187 ◽  
Author(s):  
Tatijana Stosˇić ◽  
Borko D. Stosˇić ◽  
Ivon P. Fittipaldi
Keyword(s):  
Ising Model ◽  
Cayley Tree ◽  
Zero Field
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