  In parallel to the condensed-matter realization of quantum Hall (Chern
insulators), quantum spin Hall (topological insulators), and fractional quantum
Hall (fractional Chern insulators) effects, we propose that bilayer flat band
(FB) lattices with one FB in each layer constitute solid-state analogues of
bilayer quantum Hall (BQH) system, leading to anomalous BQH (ABQH) effect,
without magnetic field. By exact diagonalization of a bilayer Kagome lattice
Hamiltonian, as a prototypical example, we demonstrate the stabilization of
excitonic condensate Halperin's (1,1,1) state at the total filling $v_T=1$ of
the two Fbs. Furthermore, by tuning the inter-layer tunneling and distance
between the Kagome layers at $v_T=2/3$, we show phase transitions among
Halperin's (3,3,0), spin-singlet (1,1,2), and particle-hole conjugate of
Laughlin's 1/3 states, as previously observed in BQH systems. Our work opens a
new research direction in the field of FB physics by demonstrating bilayer FB
materials as an attractive avenue for realizing exotic ABQH states including
non-Abelian anyons.
