  Motivated by the very recent experimental realization of electronic Lieb
lattices and research interest on topological states of matter, we study the
topological phase transitions driven by Anderson disorder on spin-orbit coupled
Lieb lattices in the presence of spin-independent and dependent potentials. By
combining the numerical transport and self-consistent Born approximation
methods, we found that both time-reversal invariant and broken Lieb lattices
can host disorder-induced gapful topological phases, including the quantum spin
Hall insulator (QSHI) and quantum anomalous Hall insulator (QAHI) phases. For
the time-reversal invariant case, this disorder can induce a topological phase
transition directly from normal insulator (NI) to the QSHI. While for the
time-reversal broken case, the disorder can induce either a QAHI-QSHI phase
transition or a NI-QAHI-QSHI phase transition. Remarkably, the time-reversal
broken QSHI phase can be induced by Anderson disorder on the spin-orbit coupled
Lieb lattices without time-reversal symmetry.
