Valley polarization of Landau levels driven by residual strain in the ZrSiS surface band

In a multi-valley electronic band structure, lifting of the valley degeneracy is associated with rotational symmetry breaking in the electronic fluid, and may emerge through spontaneous symmetry breaking order, or through a large response to a small external perturbation such as strain. In this work we use scanning tunneling microscopy to investigate an unexpected rotational symmetry breaking in Landau levels formed in the unusual floating surface band of ZrSiS. We visualize a ubiquitous splitting of Landau levels into valley-polarized sub-levels. We demonstrate methods to measure valley-selective Landau level spectroscopy, to infer unknown Landau level indices, and to precisely measure each valley's Berry phase in a way that is agnostic to the band structure and topology of the system. These techniques allow us to obtain each valley's dispersion curve and infer a rigid valley-dependent contribution to the band energies. Ruling out spontaneous symmetry breaking by establishing the sample-dependence of this valley splitting, we explain the effect in terms of residual strain. A quantitative estimate indicates that uniaxial strain can be measured to a precision of $ \lt 0.025 \% $. The extreme valley-polarization of the Landau levels results from as little as $ \sim 0.1 \% $ strain, and this suggests avenues for manipulation using deliberate strain engineering.