Sound modes in one-dimensional quantum liquids*

We study sound in one-dimensional Galilean invariant quantum liquids. In contrast to classical fluids, instead of a single sound mode we find a broad range of frequencies in which the system supports two sound modes. The nature of these modes depends on the type of particles forming the quantum liquid. In systems of spin-1/2 fermions, we find the first and second sound modes propagating at two different speeds. Similarly to superfluid ⁴He, the first sound is a wave of density, whereas the second one is a wave of entropy. In spinless quantum liquids we find two hybrid sound modes, which are simultaneous oscillations of both density and entropy, with relative phase shifts 0 or π. Their speeds are nearly equal, with the difference scaling linearly with the temperature.