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).

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 GROUND STATES OF ONE-DIMENSIONAL LONG-RANGE FERROMAGNETIC ISING MODEL WITH EXTERNAL FIELD

2012 ◽  
Vol 26 (03) ◽  
pp. 1150014 ◽  
Author(s):  
AZER KERIMOV
Keyword(s):  
Phase Diagram ◽  
Ising Model ◽  
External Field ◽  
Lattice Points ◽  
Temperature Phase ◽  
Zero Temperature ◽  
One Dimensional ◽  
Ferromagnetic Ising Model ◽  
The One ◽  
Temperature Phase Diagram

A zero-temperature phase-diagram of the one-dimensional ferromagnetic Ising model is investigated. It is shown that at zero temperature spins of any compact collection of lattice points with identically oriented external field are identically oriented.

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 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

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.

scholarly journals Критический индекс восприимчивости 1D-изинговского ферромагнетика, замкнутого в кольцо

2018 ◽  
Vol 60 (7) ◽  
pp. 1318
Author(s):  
Ж.В. Дзюба ◽  
В.Н. Удодов
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Particle Interaction ◽  
Three Dimensional ◽  
One Dimensional ◽  
Dimensional Scaling ◽  
Finite Dimensional ◽  
The Monte Carlo Method ◽  
Ferromagnetic Ising Model ◽  
The One

AbstractUsing the Monte Carlo method, critical behavior of the one-dimensional ferromagnetic Ising model has been investigated with allowance for the interaction of the second and third neighbors and four-particle interaction. The obtained results on the critical temperature were compared with the critical temperature of the quasi-one-dimensional Ising magnetic [(СН_3)_3NH] · FeCl_3 · 2H_2O and with the magnitude of the exchange interaction J/k _B = 17.4 K. Within the scope of the finite-dimensional scaling theory, the critical susceptibility exponent has been calculated. It has been shown that values of the susceptibility exponent for the one-dimensional Ising model with periodic boundary conditions are considerably less than the known values of the exponents for three-dimensional systems. The critical susceptibility exponent strongly depends on energy parameters; namely, it decreases with an increase in them.

scholarly journals LOCAL MAGNETIZATION IN CRITICAL ISING MODEL WITH BOUNDARY MAGNETIC FIELD

1994 ◽  
Vol 09 (24) ◽  
pp. 2227-2234 ◽  
Author(s):  
R. CHATTERJEE ◽  
A. ZAMOLODCHIKOV
Keyword(s):  
Magnetic Field ◽  
Critical Temperature ◽  
Ising Model ◽  
Half Plane ◽  
Correlation Functions ◽  
Spin Correlation ◽  
Point Function ◽  
Local Magnetization

We describe a simple way to derive spin correlation functions in 2-D Ising model at critical temperature but with nonzero magnetic field at the boundary. Local magnetization (i.e. one-point function) is computed explicitly for half-plane and disk geometries.

MAGNETIZATION DECAY IN THE DILUTED ISING MODEL

1996 ◽  
Vol 07 (01) ◽  
pp. 89-97 ◽  
Author(s):  
PETER GRASSBERGER
Keyword(s):  
Asymptotic Behavior ◽  
Critical Temperature ◽  
Ising Model ◽  
Exponential Decay ◽  
Stretched Exponential ◽  
Damage Spreading ◽  
Monte Carlo Techniques ◽  
Magnetization Decay ◽  
Damage Healing ◽  
Stretched Exponential Decay

Using Monte Carlo techniques, we study the decay of magnetization in diluted two-dimensional Ising models at and below the critical temperature Tc of the undiluted Ising model, but above the critical temperature of the diluted system. Using damage spreading (or rather damage "healing"), we are able to measure down to much lower final magnetizations (10–9) and to much larger times than previous authors. Nevertheless, we do not yet find the predicted asymptotic behavior in the Griffiths phase T < Tc. But we can at least exclude a stretched exponential decay as found in previous papers, for T < Tc. Finally, we discuss the case T = Tc where a stretched exponential decay can be proven to hold, at least for p < pc. We indeed do see a stretched exponential for p = pc (and T = Tc), but we show that it cannot describe the asymptotic behavior either.

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