Fourier's law breakdown for the planar-rotor chain with long-range coupling

TL;DR

This study investigates heat conduction in a long-range interacting planar-rotor chain, revealing a breakdown of Fourier's law for certain interaction ranges and proposing a universal stretched exponential form for conductance.

Contribution

It introduces a novel universal stretched exponential model describing thermal conductance in long-range interacting chains, highlighting the transition point at $ ext{alpha} = 2$ where Fourier's law breaks down.

Findings

Fourier's law is violated for $ ext{alpha} < 2$.

A universal $q$-stretched exponential form describes conductance.

Transition at $ ext{alpha} = 2$ between anomalous and normal heat conduction.

Abstract

The thermal conductivity, $\kappa$, of a homogeneous chain of generically-ranged interacting planar rotors, more precisely the inertial $\alpha-XY$ model, is numerically studied with the coupling constant decaying as $r^{-\alpha}$. The conductance ($\sigma\equiv \kappa/L$) is calculated for typical values of $\alpha$, temperature $T$ and lattice size $L$. For large $L$, the results can be collapsed into an universal $q$-stretched exponential form, namely $L^{\delta_{\alpha}} \sigma \propto e_{q_{\alpha}}^{-B_{\alpha}(L^{\gamma_{\alpha}}T)^{\eta_{\alpha}}}$, where $e_q^z\equiv [1+ (1-q)z]^{1/(1-q)}$. The parameters $(\gamma_{\alpha},\delta_{\alpha},B_{\alpha}, \eta_{\alpha})$ are $\alpha$-dependent, and $q_{\alpha}$ is the index of the $q$-stretched exponential. This form is achievable due to the ratio $\eta_{\alpha}/(q-1)$ being almost constant with respect to the lattice size $L$. Two distinct regimes are identified: for $0 \le \alpha < 2$, the Fourier law is violated, whereas for $\alpha\ge 2$, it is satisfied. These findings provide significant insights into heat conduction mechanisms in systems with long-range interactions.