Haldane phase in the sawtooth lattice: Edge states, entanglement spectrum and the flat band

TL;DR

This study uses numerical methods to explore the phase diagram of a frustrated Bose-Hubbard model on a sawtooth lattice, revealing a topological Haldane insulator phase characterized by edge states and unique entanglement properties.

Contribution

It identifies a topological Haldane phase in a flat-band Bose-Hubbard system, highlighting its stability and entanglement features despite non-degenerate spectra.

Findings

Discovery of the Haldane insulator phase with edge states

Entanglement spectrum matches AKLT state without degeneracy

Haldane phase stability depends on band flatness

Abstract

Using density matrix renormalization group numerical calculations, we study the phase diagram of the half filled Bose-Hubbard system in the sawtooth lattice with strong frustration in the kinetic energy term. We focus in particular on values of the hopping terms which produce a flat band and show that, in the presence of contact and near neighbor repulsion, three phases exist: Mott insulator (MI), charge density wave (CDW), and the topological Haldane insulating (HI) phase which displays edge states and particle imbalance between the two ends of the system. We find that, even though the entanglement spectrum in the Haldane phase is not doubly degenerate, it is in excellent agreement with the entanglement spectrum of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state built in the Wannier basis associated with the flat band. This emphasizes that the absence of degeneracy in the entanglement spectrum is not necessarily a signature of a non-topological phase, but rather that the (hidden) protecting symmetry involves non-local states. Finally, we also show that the HI phase is stable against small departure from flatness of the band but is destroyed for larger ones.