  This work brings forward an alternative experimental approach to infer the
topological character of phase transitions in insulators. This method relies on
subjecting the target system to a set of external fields, each of which
consists of two parts, i.e., a weak spatiotemporally slowly-varying component
on top of a constant offset. The fields are chosen in such a way, so that they
respectively induce slow variations in the wave vector describing the bulk band
structure, as well as a parameter which allows tuning the bulk gap. Such a
process maps the Berry singularities of the base space to a synthetic space
spanned by the parameters related to the external fields. By measuring the
response of the system to the weak part of the perturbations, when these are
additionally chosen to form spacetime textures, one can construct a quantity
that is here-termed Berry singularity marker (BSM). The BSM enables the Berry
singularity detection as it becomes nonzero only in the close vicinity of a
Berry singularity and is equal to its charge. The calculation of the BSM
requires the measurement of the susceptibility tensor for the applied external
fields. Near the Berry singularities, the BSM is dominated by a universal
value, which is determined by the quantum metric tensor of the system. While in
this work I restrict to 1D AIII insulators, the proposed approach is general.
Notably, a key feature of the present method is that it can be implemented in a
lock-in fashion, that is, one can "filter out" from the BSM any possible
contribution from disorder by performing more measurements. Hence, the present
construction paves the way for a disorder resilient diagnosis of topological
phase transitions, that appears particularly relevant for disordered
topological insulator and hybrid Majorana platforms, while it can be readily
implemented using topo-electric circuits.
