Topological Acoustic Diode

We show that certain three-dimensional topological phases can act as acoustic diodes realizing nonlinear odd acoustoelastic effects. Beyond uncovering topologically-induced anomalous acoustic second-harmonic generation and rectification, we demonstrate how such nonlinear responses are uniquely captured by the momentum-space nonmetricity tensor in the quantum state Hilbert-space geometry. In addition to completing the classification of quantum geometric observables in the quadratic response regime, our findings reveal unexplored avenues for experimental realizations of acoustic diodes using effective $θ$ vacua of axion insulators adaptable for topological engineering applications.