In the quantum Hall effect, a two-dimensional gas of electrons is subjected to an out-of-plane magnetic field and electron transport quantises: the Hall conductivity plateaus at values proportional to integers and rational numbers in units of fundamental constants, with remarkably low error. Shortly after experimental discovery of the quantum Hall effect in 1980, theorists developed a framework explaining this quantization as a consequence of topological phases of matter, or those phases with signatures unaffected by sufficiently small perturbations. In particular, a theory in terms of point charges coupling to external fields beautifully described this physics.

The quantum Hall effect is now seen as one of the pillars of modern condensed matter physics leading to considerable other discoveries. A vast zoo of topological phases are now being studied, descending from the theory of the quantum Hall effect. This on-going search for topological phases of matter has recently led to discovery of topological skyrmion phases of matter hidden from established classification schemes, however, which contradict key assumptions of these classification schemes. 

These assumptions stem from the classification schemes, like the original theory of the quantum Hall effect, only applying to topological phases of point charges, with some more recent theories considering higher-dimensional charged objects. Charges, however, can be generalized to extended topological textures in vector fields of myriad observables (e.g. polarisation or magnetisation) known as skyrmions. A paradigm shift is therefore required, by generalising the framework of the quantum Hall effect to the quantum skyrmion Hall effect corresponding to quantized transport of topological textures in vector fields, which coarse-grains to quantized charge transport in special cases.