High-$T_{\mathrm{c}}$ cuprate superconductors host spin, charge, and lattice instabilities. In particular, in the antiferromagnetic glass phase, over a large doping range, lanthanum-based cuprates display a glass-like spin freezing with antiferromagnetic correlations. Previously, sound velocity anomalies in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (LSCO) for hole doping $p=x\ge 0.145$ were reported and interpreted as arising from a coupling of the lattice to the magnetic glass [M. Frachet, I. Vinograd et al., Nat. Phys. 16, 1064 (2020)]. Here we report both sound velocity and attenuation in LSCO $p=0.12$, i.e., at a doping level for which the spin freezing temperature is the highest. Using high magnetic fields and comparing with nuclear magnetic resonance measurements, we confirm that the anomalies in the low temperature ultrasound properties of LSCO are produced by a coupling between the lattice and the spin glass. Moreover, we show that both sound velocity and attenuation can be simultaneously accounted for by a simple phenomenological model originally developed for canonical spin glasses. Our results point towards a strong competition between superconductivity and spin freezing, tuned by the magnetic field. A comparison of different acoustic modes suggests that the slow spin fluctuations have a nematic character.

• Received 29 October 2020
• Revised 1 February 2021
• Accepted 26 February 2021

DOI:https://doi.org/10.1103/PhysRevB.103.115133