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Charge-density waves (CDWs) in Weyl semimetals (WSMs) have been shown toinduce an exotic axionic insulating phase in which the sliding mode (phason) ofthe CDW acts as a dynamical axion field, giving rise to a large positivemagneto-conductance [Wang et al., Phys. Rev. B 87, 161107(R) (2013); Roy etal., Phys. Rev. B 92, 125141 (2015); J. Gooth et al., Nature 575, 315 (2019)].In this work, we predict that dynamical strain can induce a bulk orbitalmagnetization in time-reversal- (TR-) invariant WSMs that are gapped by a CDW.We term this effect the "dynamical piezomagnetic effect" (DPME). Unlike in [J.Gooth et al., Nature 575, 315 (2019)], the DPME introduced in this work occursin a bulk-constant (i.e., static and spatially homogeneous in the bulk) CDW,and does not rely on fluctuations, such as a phason. By studying the low-energyeffective theory and a minimal tight-binding (TB) model, we find that the DPMEoriginates from an effective valley axion field that couples theelectromagnetic gauge field with a strain-induced pseudo-gauge field. Inparticular, whereas the piezoelectric effects studied in previous works arecharacterized by 2D Berry curvature, the DPME represents the first example of afundamentally 3D strain effect originating from the Chern-Simons 3-form. Wefurther find that the DPME has a discontinuous change at a critical value ofthe phase of the CDW order parameter. We demonstrate that, when there is a jumpin the DPME, the surface of the system undergoes a topological quantum phasetransition (TQPT), while the bulk remains gapped. Hence, the DPME provides abulk signature of the boundary TQPT in a TR-invariant Weyl-CDW.

Dynamical Piezomagnetic Effect in Time-Reversal Invariant Weyl Semimetals with Axionic Charge-Density Waves