Quantized valley Hall response from local bulk density variations

The application of a mechanical strain to a 2D material can create pseudo-magnetic fields and lead to a quantized valley Hall effect. However, measuring valley-resolved effects remains a challenging task due to their inherent fragility and dependence on the sample's proper design. Additionally, non-local transport probes based on multiterminal devices have often proven to be inadequate in yielding conclusive evidence of the valley Hall signal. Here, we introduce an alternative way of detecting the quantized valley Hall effect, which entirely relies on local density measurements, performed deep in the bulk of the sample. The resulting quantized signal is a genuine Fermi sea response, independent of the edge physics, and reflects the underlying valley Hall effect through the Widom-St\v{r}eda formula. Specifically, our approach is based on measuring the variation of the particle density, locally in the bulk, upon varying the strength of the applied strain. This approach to the quantized valley Hall effect is particularly well suited for experiments based on synthetic lattices, where the particle density (or integrated density of states) can be spatially resolved.