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

TL;DR

This paper introduces bilayer flat-band Hamiltonians exhibiting full localization and symmetry-protected topological phases, constructed from cube operators, with potential applications in generalized SYK models.

Contribution

The work constructs exact projective Hamiltonians on bilayer lattices with topological classification and gapless edge modes, extending the Creutz ladder model.

Findings

All bulk states are localized with extensive LIOMs.

Hamiltonians exhibit symmetry-protected topological phases with edge modes.

A generalized SYK model is derived from the edge states.

Abstract

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