Anomalous Hydrodynamics in One Dimensional Electronic Fluid.

We construct multi-mode viscous hydrodynamics for one dimensional spinless electrons. Depending on the scale, the fluid has six (shortest lengths), four (intermediate, exponentially broad regime), or three (asymptotically long scales) hydrodynamic modes. Interaction between hydrodynamic modes leads to anomalous scaling of physical observables and waves propagating in the fluid. In a four-mode regime, all modes are ballistic and acquire KPZ-like broadening with asymmetric power-law tails. "Heads" and "tails" of the waves contribute equally to thermal conductivity, leading to $\omega^{-1/3}$ scaling of its real part. In a three-mode regime, the system is in the universality class of classical viscous fluid[9,24]. Self-interaction of the sound modes results in KPZ-like shape, while the interaction with the heat mode results in asymmetric tails. The heat mode is governed by Levy flight distribution, whose power-law tails give rise to $\omega^{-1/3}$ scaling of heat conductivity.