  Charge-density waves (CDWs) in Weyl semimetals (WSMs) have been shown to
induce an exotic axionic insulating phase in which the sliding mode (phason) of
the CDW acts as a dynamical axion field, giving rise to a large positive
magneto-conductance [Wang et al., Phys. Rev. B 87, 161107(R) (2013); Roy et
al., 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 orbital
magnetization 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 occurs
in 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-energy
effective theory and a minimal tight-binding (TB) model, we find that the DPME
originates from an effective valley axion field that couples the
electromagnetic gauge field with a strain-induced pseudo-gauge field. In
particular, whereas the piezoelectric effects studied in previous works are
characterized by 2D Berry curvature, the DPME represents the first example of a
fundamentally 3D strain effect originating from the Chern-Simons 3-form. We
further find that the DPME has a discontinuous change at a critical value of
the phase of the CDW order parameter. We demonstrate that, when there is a jump
in the DPME, the surface of the system undergoes a topological quantum phase
transition (TQPT), while the bulk remains gapped. Hence, the DPME provides a
bulk signature of the boundary TQPT in a TR-invariant Weyl-CDW.
