Quantum Hydrodynamics of Fractonic Superfluids with Lineon Condensate: from Navier-Stokes-like Equations to Landau-like Criterion

Fractonic superfluids are exotic states of matter with spontaneously broken higher-rank $U(1)$ symmetry. The latter is associated with conserved quantities that include not only particle number (i.e. charge) but also higher moments, such as dipoles, quadrupoles, and angular moments. Due to the presence of such conserved quantities, the mobility of particles is restricted either completely or partially. In this work, we systematically study hydrodynamical properties of fractonic superfluids, especially focusing on the fractonic superfluids with conserved angular moments. The constituent bosons are called "lineons" with $d$-components in $d$-dimensional space. From Euler-Lagrange equation, we derive the continuity equation and Navier-Stokes-like equations, in which the angular moment conservation introduces extra terms. Furthermore, we discuss the current configurations that are related to the defects. Like the conventional superfluid, we study the critical values of velocity fields and density currents, which gives rise to a Landau-like criterion. At the end of this work, several future directions are discussed.