  The geometric response of quantum Hall liquids is an important aspect to
understand their topological characteristics in addition to the electromagnetic
response. According to the Wen-Zee theory, the topological spin is coupled to
the curvature of the space in which the electrons reside. The presence of
conical geometry provides a local isolated geometric singularity, making it
suitable for exploring the geometric response. In the context of
two-dimensional electrons in a perpendicular magnetic field, each Landau orbit
occupies the same area. The cone geometry naturally provides a structure in
which the distances between two adjacent orbits gradually change and can be
easily adjusted by altering the tip angle. The presence of a cone tip
introduces a geometric singularity that affects the electron density and
interacts with the motion of electrons, which has been extensively studied.
Furthermore, this type of geometry can automatically create a smooth interface
or crossover between the crystalline charge-density-wave state and the
liquid-like fractional quantum Hall state. In this work, the properties of this
interface are studied from multiple perspectives, shedding light on the
behavior of quantum Hall liquids in such geometric configurations.
