  We present a new class of quantum phase transitions that refer neither to
local order parameter and critical fluctuations nor to continuous symmetry
breaking but are assigned by the step-wise change in topology of the
multi-particle system expressed in homotopy terms. The energy for homotopy
phases changes in a step-wise manner and the different homotopies induce
specific correlations in planar interacting multi-particle system representing
different commesurability patterns of quantum incompressible states. We
illustrate the concept of the homotopy phase transition in the simplest quantum
multi-particle system of two repulsing electrons on a 2D finite jellium exposed
to a strong perpendicular magnetic field. The homotopy phases related to
fractional quantum Hall states are described and compared with their
experimental manifestation.
