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.

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

2000 ◽  
Vol 37 (3) ◽  
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 θc 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.

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

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.

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

scholarly journals Non-perturbative effects in spin glasses

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Michele Castellana ◽  
Giorgio Parisi
Keyword(s):  
Fixed Point ◽  
Ising Model ◽  
Spin Glasses ◽  
Numerical Study ◽  
Mean Field ◽  
Critical Dimension ◽  
Upper Critical Dimension ◽  
Ising Spin ◽  
Ferromagnetic Ising Model ◽  
The Mean

Abstract We present a numerical study of an Ising spin glass with hierarchical interactions—the hierarchical Edwards-Anderson model with an external magnetic field (HEA). We study the model with Monte Carlo (MC) simulations in the mean-field (MF) and non-mean-field (NMF) regions corresponding to d ≥ 4 and d < 4 for the d-dimensional ferromagnetic Ising model respectively. We compare the MC results with those of a renormalization-group (RG) study where the critical fixed point is treated as a perturbation of the MF one, along the same lines as in the "Equation missing"-expansion for the Ising model. The MC and the RG method agree in the MF region, predicting the existence of a transition and compatible values of the critical exponents. Conversely, the two approaches markedly disagree in the NMF case, where the MC data indicates a transition, while the RG analysis predicts that no perturbative critical fixed point exists. Also, the MC estimate of the critical exponent ν in the NMF region is about twice as large as its classical value, even if the analog of the system dimension is within only ~2% from its upper-critical-dimension value. Taken together, these results indicate that the transition in the NMF region is governed by strong non-perturbative effects.

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