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In this work, we present bilayer flat-band Hamiltonians, in which all bulkstates are localized and specified by extensive local integrals of motion(LIOMs). The present systems are bilayer extension of Creutz ladder, which isstudied previously. In order to construct models, we employ building blocks,cube operators, which are linear combinations of fermions defined in each cubeof the bilayer lattice. There are eight cubic operators, and the Hamiltoniansare composed of the number operators of them, the LIOMs. A suitable arrangementof locations of the cube operators is needed to have exact projectiveHamiltonians. The projective Hamiltonians belong to a topologicalclassification class, BDI class. With the open boundary condition, theconstructed Hamiltonians have gapless edge modes, which commute with each otheras well as the Hamiltonian. This result comes from a symmetry analogous to theone-dimensional chiral symmetry of the BDI class. These results indicate thatthe projective Hamiltonians describe a kind of symmetry protected topologicalphase matter. Careful investigation of topological indexes, such as Berryphase, string operator, is given. We also show that by using the gapless edgemodes, a generalized Sachdev-Ye-Kitaev (SYK) model is constructed.

Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems