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

<jats:p>We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance <jats:inline-formula><jats:alternatives><jats:tex-math>r</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>r</mml:mi></mml:math></jats:alternatives></jats:inline-formula> decaying as <jats:inline-formula><jats:alternatives><jats:tex-math>r^{-\alpha}</jats:tex-math><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></jats:alternatives></jats:inline-formula>. For <jats:inline-formula><jats:alternatives><jats:tex-math>\alpha =0</jats:tex-math><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></jats:alternatives></jats:inline-formula>, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with <jats:inline-formula><jats:alternatives><jats:tex-math>\alpha >1</jats:tex-math><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></jats:alternatives></jats:inline-formula> there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. <jats:inline-formula><jats:alternatives><jats:tex-math>0<\alpha <1</jats:tex-math><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></jats:alternatives></jats:inline-formula>, we show that the system shows both features, with probability <jats:inline-formula><jats:alternatives><jats:tex-math>P_\alpha (N)</jats:tex-math><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></jats:alternatives></jats:inline-formula> of having the latter one, with the different limiting behaviours <jats:inline-formula><jats:alternatives><jats:tex-math>\lim _{N\to \infty}P_\alpha (N)=0</jats:tex-math><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></jats:alternatives></jats:inline-formula> (at fixed <jats:inline-formula><jats:alternatives><jats:tex-math>\alpha<1</jats:tex-math><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></jats:alternatives></jats:inline-formula>) and <jats:inline-formula><jats:alternatives><jats:tex-math>\lim _{\alpha \to 1}P_\alpha (N)=1</jats:tex-math><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></jats:alternatives></jats:inline-formula> (at fixed finite <jats:inline-formula><jats:alternatives><jats:tex-math>N</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math></jats:alternatives></jats:inline-formula>). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time <jats:inline-formula><jats:alternatives><jats:tex-math>\tau _\alpha (N)\sim N^\alpha</jats:tex-math><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></jats:alternatives></jats:inline-formula>. </jats:p>