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We study energy transport in the integrable $\mathbb Z_3$ parafermionic chainusing the partitioning protocol. By exploiting the Bethe-ansatz solution forthe thermodynamics of the system, we develop a generalized hydrodynamicdescription of the non-equilibrium steady states, which we benchmark usingnumerical simulations based on matrix product states. The model features alow-energy conformal limit with central charge $c=4/5$, which affects thelow-temperature energy current, as we explicitly show. Moreover, we exploitthat, for energies close to the maximally excited state, the system is alsocritical and described by a conformal field theory with $c=1$. By consideringthe two halves prepared at two temperatures both low in value but opposite insign, we are able to investigate in an exact and controlled way the junctionbetween two conformal field theories with different central charges.Notwithstanding the absence of global conformal invariance, we find resultsthat approximate to a high degree those of out-of-equilibrium conformal fieldtheories. Our study extends the generalized hydrodynamics to a novel framework,where it can be profitably used for exploring new physical phenomena.

Energy transport in an integrable parafermionic chain via generalized hydrodynamics