Low frequency propagating shear waves in holographic liquids

Recently, it has been realized that liquids are able to support solid-like transverse modes with an interesting gap in momentum space developing in the dispersion relation. We show that this gap is also present in simple holographic bottom-up models, and it is strikingly similar to the gap in liquids in several respects. Firstly, the appropriately defined relaxation time in the holographic models decreases with temperature in the same way. More importantly, the holographic $k$-gap increases with temperature and with the inverse of the relaxation time. Our results suggest that the Maxwell-Frenkel approach to liquids, involving the additivity of liquid hydrodynamic and solid-like elastic responses, can be applicable to a much wider class of physical systems and effects than thought previously, including relativistic models and strongly-coupled quantum field theories. More precisely, the dispersion relation of the propagating shear waves is in perfect agreement with the Maxwell-Frenkel approach. On the contrary the relaxation time appearing in the holographic models considered does not match the Maxwell prediction in terms of the shear viscosity and the instantaneous elastic modulus but it shares the same temperature dependence.