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

PHASE TRANSITION IN THE ISING MODEL ON LOCAL-WORLD EVOLVING NETWORKS

2008 ◽  
Vol 19 (11) ◽  
pp. 1717-1726
Author(s):  
WEI QIANG ◽  
GUANGDAO HU ◽  
PENGDA ZHAO
Keyword(s):  
Phase Transition ◽  
Critical Temperature ◽  
Ising Model ◽  
Scale Free Network ◽  
Scale Free ◽  
Evolving Networks ◽  
Free Network ◽  
Effective Phase ◽  
Limiting Case ◽  
Ising Spins

We study the critical behavior of the Ising model on the local-world evolving network. Monte Carlo simulations with the standard Metropolis local update algorithms are performed extensively on the network with different parameters. Ising spins put onto network vertices exhibit an effective phase transition from ferromagnetism to paramagnetism upon heating. The critical temperature has been demonstrated to increase linearly with the average degree of the network as TC ~ 〈k〉. Simulation results on local-world evolving networks with various parameters show logarithmical relationships of the critical temperature with the size of the local world as TC ~ ln (ml), and with the size of the network as TC ~ ln (N), respectively. The latter is the generalization of the conclusion for the Ising model on the Barabási–Albert scale-free network, a limiting case of the local-world evolving network.

scholarly journals Plastic Yielding as a Phase Transition

1999 ◽  
Vol 66 (2) ◽  
pp. 289-298 ◽  
Author(s):  
M. Ortiz
Keyword(s):  
Phase Transition ◽  
Shear Stress ◽  
Critical Temperature ◽  
Ising Model ◽  
Yield Point ◽  
Slip Plane ◽  
Dislocation Loop ◽  
Lattice Gas ◽  
Two Dimensional ◽  
Statistical Mechanical

A statistical mechanical theory of forest hardening is developed in which yielding arises as a phase transition. For simplicity, we consider the case of a single dislocation loop moving on a slip plane through randomly distributed forest dislocations, which we treat as point obstacles. The occurrence of slip at the sites occupied by these obstacles is assumed to require the expenditure of a certain amount of work commensurate with the strength of the obstacle. The case of obstacles of infinite strength is treated in detail. We show that the behavior of the dislocation loop as it sweeps the slip plane under the action of a resolved shear stress is identical to that of a lattice gas, or, equivalently, to that of the two-dimensional spin-1/2 Ising model. In particular, there exists a critical temperature Tc below which the system exhibits a yield point, i.e., the slip strain increases sharply when the applied resolved shear stress attains a critical value. Above the critical temperature the yield point disappears and the slip strain depends continuously on the applied stress. The critical exponents, which describe the behavior of the system near the critical temperature, coincide with those of the two-dimensional spin-1/2 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

Phase Transition in 2D Anisotropic Ising Model with Next-Nearest Neighbor Interactions in a Magnetic Field

SPIN ◽  
2018 ◽  
Vol 08 (03) ◽  
pp. 1850010
Author(s):  
D. Farsal ◽  
M. Badia ◽  
M. Bennai
Keyword(s):  
Magnetic Field ◽  
Phase Transition ◽  
Phase Diagram ◽  
Monte Carlo ◽  
Magnetic Susceptibility ◽  
Critical Temperature ◽  
Ising Model ◽  
Nearest Neighbor ◽  
Two Dimensional ◽  
The Magnetic Field

The critical behavior at the phase transition of the ferromagnetic two-dimensional anisotropic Ising model with next-nearest neighbor (NNN) couplings in the presence of the field is determined using mainly Monte Carlo (MC) method. This method is used to investigate the phase diagram of the model and to verify the existence of a divergence at null temperature which often appears in two-dimensional systems. We analyze also the influence of the report of the NNN interactions [Formula: see text] and the magnetic field [Formula: see text] on the critical temperature of the system, and we show that the critical temperature depends on the magnetic field for positive values of the interaction. Finally, we have investigated other thermodynamical qualities such as the magnetic susceptibility [Formula: see text]. It has been shown that their thermal behavior depends qualitatively and quantitatively on the strength of NNN interactions and the magnetic field.

scholarly journals Exponential Decay of Truncated Correlations for the Ising Model in any Dimension for all but the Critical Temperature

2019 ◽  
Vol 374 (2) ◽  
pp. 891-921 ◽  
Author(s):  
Hugo Duminil-Copin ◽  
Subhajit Goswami ◽  
Aran Raoufi
Keyword(s):  
Phase Diagram ◽  
Critical Temperature ◽  
Ising Model ◽  
Critical Point ◽  
Exponential Decay ◽  
Point Function ◽  
Ferromagnetic Ising Model ◽  
Pure Phases ◽  
Clustering Property

AbstractThe truncated two-point function of the ferromagnetic Ising model on $${\mathbb {Z}}^d$$Zd ($$d\ge 3$$d≥3) in its pure phases is proven to decay exponentially fast throughout the ordered regime ($$\beta >\beta _c$$β>βc and $$h=0$$h=0). Together with the previously known results, this implies that the exponential clustering property holds throughout the model’s phase diagram except for the critical point: $$(\beta ,h) = (\beta _c,0)$$(β,h)=(βc,0).

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.

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.

FRUSTRATION EFFECTS ON SMALL-WORLD NETWORKS

2004 ◽  
Vol 15 (09) ◽  
pp. 1269-1277 ◽  
Author(s):  
PAULO R. A. CAMPOS ◽  
VIVIANE M. DE OLIVEIRA ◽  
F. G. BRADY MOREIRA
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Critical Behavior ◽  
Magnetic Ordering ◽  
Small World ◽  
Two Dimensions ◽  
Small World Networks ◽  
Ferromagnetic Ising Model ◽  
Antiferromagnetic Ising Model ◽  
Frustration Effects

We investigate the frustration effects on small-world networks by studying antiferromagnetic Ising model in two dimensions. When the rewiring is constrained to those sites such that the interaction still occurs between spins in distinct sublattices and frustration does not take place, we observe that the system behaves as in previous investigations of ferromagnetic Ising model. However, when the rewiring procedure does not only produce interactions between spins in distinct sublattices, small-world configurations can effectively produce geometrical frustration and we attain a different critical behavior. In the frustrated case, the critical temperature decreases with the augment of the rewiring probability and the magnetic ordering presents two different regimes for low and high p.

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