Energy diffusion in the long-range interacting spin systems

TL;DR

This paper studies how energy spreads in long-range interacting spin systems, revealing conditions for normal and anomalous diffusion across different dimensions and interaction decay rates.

Contribution

It establishes rigorous theorems linking the decay exponent to diffusion behavior, providing a comprehensive understanding of energy diffusion in long-range spin models.

Findings

Normal diffusion occurs when lpha > D in D dimensions.

Anomalous diffusion is linked to lpha < 3/2 in 1D.

Theoretical conditions for diffusion are proven to be optimal.

Abstract

We investigate energy diffusion in long-range interacting spin systems, where the interaction decays algebraically as $V(r) \propto r^{-\alpha}$ with the distance $r$ between the sites. We consider prototypical spin systems, the transverse Ising model, and the XYZ model in the $D$-dimensional lattice with finite $\alpha >D$ which guarantees the thermodynamic extensivity. In one dimension, both normal and anomalous diffusion are observed, where the anomalous diffusion is attributed to anomalous enhancement of the amplitude of the equilibrium current correlation. We prove the power-law clustering property of arbitrary orders of joint cumulants in general dimensions. Applying this theorem to equal-time current correlations, we further prove several theorems leading to the statement that the sufficient condition for normal diffusion in one dimension is $\alpha > 3/2$ regardless of the models. The fluctuating hydrodynamics approach consistently explains L\'{e}vy diffusion for $\alpha < 3/2$, which implies the condition is optimal. In higher dimensions of $D \geq 2$, normal diffusion is indicated as long as $\alpha > D$.