Topological phases are quantum
phases that can not be described by any local order parameter.
Interestingly, topological phases
in interacting fermion systems can be much richer than boson/
spin systems due to the Fock space structure of many body fermion Hilbert
space. Unfortunately, the well-known mathematical tools -- tensor category theory cannot be defined in
the Fock
space and we do not have a precise mathematical language to describe
topological phases for
fermion systems. Thus generalizing the tensor category theory into the Fock
space will be of
great importance and highly complementary to physics in the future. To illustrate
the basic
idea of describing/classifying topological phases in interacting fermion
systems, I will focus
on an exactly solvable model that describes a new kind of topological phase
which cannot be
realized in any interacting boson system. This model has the same ground state
degeneracy on
torus as the toric code model, but with different braiding S and T matrix. I
will also mention
some other recent progresses along this direction.
