Coexistence of coarsening and mean field relaxation in the long-range Ising chain
Federico Corberi, Alessandro Iannone, Manoj Kumar, Eugenio Lippiello, Paolo Politi
Abstract
We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance r <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>r</mml:mi> </mml:math> decaying as r^{-\alpha} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:math> . For \alpha =0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with \alpha >1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. 0<\alpha <1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we show that the system shows both features, with probability P_\alpha (N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> of having the latter one, with the different limiting behaviours \lim _{N\to \infty}P_\alpha (N)=0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mo>lim</mml:mo> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (at fixed \alpha<1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ) and \lim _{\alpha \to 1}P_\alpha (N)=1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mo>lim</mml:mo> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> (at fixed finite N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> ). 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^\alpha <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> </mml:math> .