Title: Magnetostriction, piezomagnetism and domain nucleation in a kagome antiferromagnet

Abstract: Whenever the elastic energy of a solid depends on magnetic field, there is a magnetostrictive response. Field-linear magnetostriction implies piezomagnetism and vice versa. Here, we show that Mn$_3$Sn, a non-collinear antiferromanget with Weyl nodes, hosts a large and almost perfectly linear magnetostriction even at room temperature. The longitudinal and transverse magnetostriction, with opposite signs and similar amplitude are restricted to the kagome planes and the out-of-plane response is negligibly small. By studying four different samples with different Mn:Sn ratios, we find a clear correlation between the linear magnetostriction, the spontaneous magnetization and the concentration of Sn vacancies. The recently reported piezomagnetic data fits in our picture. We show that linear magnetostriction and piezomagnetism are both driven by the field-induced in-plane twist of spins. A quantitative account of the experimental data requires the distortion of the spin texture by Sn vacancies. We find that the field-induced domain nucleation within the hysteresis loop corresponds to a phase transition. Within the hysteresis loop, a concomitant mesoscopic modulation of local strain and spin twist angles, leading to twisto-magnetic stripes, arises as a result of the competition between elastic and magnetic energies.