Quasi-additive estimates on the Hamiltonian for the one-dimensional long range Ising model

2017 ◽  
Vol 58 (7) ◽  
pp. 073301 ◽  
Author(s):  
Jorge Littin ◽  
Pierre Picco
Keyword(s):  
Ising Model ◽  
Long Range ◽  
One Dimensional ◽  
The One

scholarly journals Coexistence of coarsening and mean field relaxation in the long-range Ising chain

2021 ◽  
Vol 10 (5) ◽  
Author(s):  
Federico Corberi ◽  
Alessandro Iannone ◽  
Manoj Kumar ◽  
Eugenio Lippiello ◽  
Paolo Politi
Keyword(s):  
Ising Model ◽  
Long Range ◽  
Characteristic Time ◽  
Opposite Sign ◽  
Mean Field ◽  
Ising Chain ◽  
One Dimensional ◽  
Dynamical Scaling ◽  
Long Range Interactions ◽  
The One

We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance rr decaying as r^{-\alpha}r−α. For \alpha =0α=0, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with \alpha >1α>1 there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. 0<\alpha <10<α<1, we show that the system shows both features, with probability P_\alpha (N)Pα(N) of having the latter one, with the different limiting behaviours \lim _{N\to \infty}P_\alpha (N)=0limN→∞Pα(N)=0 (at fixed \alpha<1α<1) and \lim _{\alpha \to 1}P_\alpha (N)=1limα→1Pα(N)=1 (at fixed finite NN). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time \tau _\alpha (N)\sim N^\alphaτα(N)∼Nα.

scholarly journals ONE-DIMENSIONAL LONG-RANGE FERROMAGNETIC ISING MODEL UNDER WEAK AND SPARSE EXTERNAL FIELD

2009 ◽  
Vol 23 (32) ◽  
pp. 5899-5906
Author(s):  
AZER KERIMOV
Keyword(s):  
Ising Model ◽  
External Field ◽  
Long Range ◽  
Low Temperatures ◽  
Gibbs State ◽  
Range Interaction ◽  
One Dimensional ◽  
Long Range Interaction ◽  
Ferromagnetic Ising Model ◽  
The One

We consider the one-dimensional ferromagnetic Ising model with very long range interaction under weak and sparse biased external field and prove that at sufficiently low temperatures, the model has a unique limiting Gibbs state.

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