  We develop the density matrix renormalization group approach to
systematically identify the topological order of the quantum spin liquid (QSL)
through adiabatically obtaining different topological sectors of the QSL on an
infinite cylinder. As an application, we study the anisotropic kagome
Heisenberg model known for hosting a Z$_2$ QSL, however no numerical
simulations have been able to access all four sectors before. We obtain the
complete set of four topological degenerate ground states distinguished by the
presence or absence of the spinon and vison quasiparticle line, which fully
characterizes the topological nature of the quantum phase. We have also studied
the kagome Heisenberg model, which has attracted a lot of recent attention. We
find two topological sectors accurately and also estimate various properties of
the other topological sectors, where the larger correlation length is found
indicating the possible proximity to another phase.
