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Certain holographic states of matter with a global U(1) symmetry support asound mode at zero temperature, caused neither by spontaneous symmetry breakingof the global U(1) nor by the emergence of a Fermi surface in the infrared. Inthis work, we show that such a mode is also found in zero density holographicquantum critical states. We demonstrate that in these states, the appearance ofa zero temperature sound mode is the consequence of a mixed `t Hooft anomalybetween the global U(1) symmetry and an emergent higher-form symmetry. Atnon-zero temperatures, the presence of a black hole horizon weakly breaks theemergent symmetry and gaps the collective mode, giving rise to a sharpDrude-like peak in the electric conductivity. A similar gapped mode arises atlow temperatures for non-zero densities when the state has an emergent Lorentzsymmetry, also originating from an approximate anomalous higher-form symmetry.However, in this case the collective excitation does not survive at zerotemperature where, instead, it dissolves into a branch cut due to strongbackreaction from the infrared, critical degrees of freedom. We comment on therelation between our results and the application of the Luttinger theorem tocompressible holographic states of matter.

Zero sound and higher-form symmetries in compressible holographic phases