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Anomalous transport in low dimensional spin chains is an intriguing topicthat can offer key insights into the interplay of integrability and symmetry inmany-body dynamics. Recent studies have shown that spin-spin correlations inspin chains, where integrability is either perfectly preserved or broken bysymmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ)universality class. Similarly, energy transport can show ballistic ordiffusive-like behaviour. Although such behaviour has been studied underequilibrium conditions, no results on nonequilibrium spin transport inclassical spin chains has been reported so far. In this work, we investigateboth spin and energy transport in classical spin chains (integrable andnon-integrable) when coupled to two reservoirs at two differenttemperatures/magnetization. In both the integrable case andbroken-integrability (but spin-symmetry preserving), we report anomalousscaling of spin current with system size ($\mathbb{J}^s \propto L^{-\mu}$) withan exponent, $\mu \approx 2/3$, falling under the KPZ universality class. Onthe other hand, it is noteworthy that energy current remains ballistic($\mathbb{J}^e \propto L^{-\eta}$ with $\eta \approx 0$) in the purelyintegrable case and there is departure from ballistic behaviour ($\eta > 0$)when integrability is broken regardless of spin-symmetry. Under nonequilibriumconditions, we have thoroughly investigated spatial profiles of localmagnetization and energy. We find interesting nonlinear spatial profiles whichare hallmarks of anomalous transport. We also unravel subtle strikingdifferences between the equilibrium and nonequilibrium steady state through thelens of spatial spin-spin correlations.

Nonequilibrium spin transport in integrable and non-integrable classical spin chains