Energy diffusion in the long-range interacting spin systems

TL;DR

This work analyzes energy diffusion in long-range interacting spin lattices with interactions $V(r)\propto r^{-α}$ across dimensions. By proving a cumulant power-law clustering theorem and applying Green–Kubo-type analyses to equal-time current correlations, it establishes a universal 1D threshold $α> frac{3}{2}$ for normal diffusion (optimal, with Lévy diffusion for $α< frac{3}{2}$) and shows that in $D≥2$ dimensions diffusion is normal whenever $α>D$, consistent with thermodynamic extensivity. The authors connect equal-time current amplitudes to diffusion via rigorous bounds and reveal Lévy diffusion through fluctuating hydrodynamics, yielding the exponent relation $η=2α-2$. Numerical simulations for transverse Ising, XY, and XYZ models corroborate the theoretical bounds and scaling, highlighting the framework’s applicability to a broad class of LR diffusive systems.

Abstract

We investigate energy diffusion in long-range interacting spin systems, where the interaction decays algebraically as $V(r) \propto r^{-α}$ 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 $α>D$ 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 $α> 3/2$ regardless of the models. The fluctuating hydrodynamics approach consistently explains Lévy diffusion for $α< 3/2$, which implies the condition is optimal. In higher dimensions of $D \geq 2$, normal diffusion is indicated as long as $α> D$.

Figures (33)

• Figure 1: The thermal conductivity for the long-range transverse Ising model with the parameters $J=h=1$, and $T=10$ ($k_{\rm B}\equiv 1$). (a): $\kappa_N$ as a function of size $N$. The inset is the time dependence as well as the integration of the total current correlation for $\alpha=1.3$ and $N=1024$. (b): $C_N(0)$ as a function of $N$.
• Figure 2: The correlation function $\langle {\cal J}_n {\cal J}_0\rangle$ for the transverse Ising model with the parameter $\alpha=1.3$ and $T=10~(k_{\rm B}\equiv 1)$ in the main panel. The inset shows the Green-Kubo like formula. The solid line is the theoretical line (\ref{['upp-main1']}), i.e., $n^{-(2\alpha - 2)}$. The equal-time correlation $\langle {\cal J}_n {\cal J}_0\rangle$ explains the power law exponent, $\eta=2\alpha-2$.
• Figure 3: Space-time energy correlation function $C(n,t):=\langle \delta \hat{h}_{n} (t) \delta \hat{h}_0\rangle$ for the transverse Ising model with the parameters $J=1$ and $h=1$ ((a) and (b)) and the XYZ model with the parameters $J_x=1,J_y=0.8,J_z=0.5$ ((c) and (d)). $T=10~(k_{\rm B}\equiv 1)$. System size is $N=512$. In (a) and (c), we use the parameter $\alpha=1.3$, while (b) and (d) are for $\alpha=1.6$. It is clear that in both models, $\alpha=1.3$ case obeys the Lévy scaling, while $\alpha=1.6$ shows the diffusive scaling.
• Figure 4: Rapid decay of total current correlation $C_N(t):= \sum_n \left \langle {\cal J}_n (t) {\cal J}_0 (0) \right \rangle$ and the convergence of Green-Kubo integral $\int^\infty_0 dt C_N(t)$. (a): Transverse Ising case for $\alpha=1.6$ and $J=1, h=-0.5, N=1024$, (b): XYZ case for $\alpha=1.1$ and $J_x=1, J_y=0.8, J_z=0.5, N=256$ and (c): XYZ case for $\alpha=1.8$ and $J_x=1, J_y=0.8, J_z=0.5, N=256$.
• Figure 5: The schematic diagrams for (\ref{['ising_proof']}).

Theorems & Definitions (21)

• Theorem 1: Cumulant power-law clustering theorem
• Definition 1
• Proposition 1
• Lemma 2
• proof : Proof of Lemma \ref{['chain']}
• proof
• proof
• Theorem 2
• proof
• Theorem 3