  We show that a class of exactly solvable quantum Ising models, including the
transverse-field Ising model and anisotropic XY model, can be characterized as
the loops in a two-dimensional auxiliary space. The transverse-field Ising
model corresponds to a circle and the XY model corresponds to an ellipse, while
other models yield cardioid, limacon, hypocycloid, and Lissajous curves etc. It
is shown that the variation of the ground state energy density, which is a
function of the loop, experiences a nonanalytical point when the winding number
of the corresponding loop changes. The winding number can serve as a
topological quantum number of the quantum phases in the extended quantum Ising
model, which sheds some light upon the relation between quantum phase
transition and the geometrical order parameter characterizing the phase
diagram.
