Topological Order and Entanglement Spectrum of Zigzag Ribbon

F. D. M. Haldane

Young Heon Kim, Hye Jeong Lee, S.-R. Eric Yang

Interacting disordered zigzag graphene nanoribbons have fractional charges, are quasi-one-dimensional, and display an exponentially small gap. Our numerical computations showed that the topological entanglement entropy of these systems has a small finite but universal value, independent of the strength of the interaction and the disorder. The result that was obtained for the topological entanglement entropy shows that the disorder-free phase is critical and becomes unstable in the presence of disorder. Our result for the entanglement spectrum in the presence of disorder is also consistent with the presence of a topologically ordered phase.

In-Hwan Lee, Hoang-Anh Le, S.-R. Eric Yang

Graphene zigzag nanoribbons, initially in a topologically ordered state, undergo a topological phase transition into crossover phases distinguished by quasi-topological order. We computed mutual information for both the topologically ordered phase and its crossover phases, revealing the following results: (i) In the topologically ordered phase, A-chirality carbon lines strongly entangle with B-chirality carbon lines on the opposite side of the zigzag ribbon. This entanglement persists but weakens in crossover phases. (ii) The upper zigzag edge entangles with non-edge lines of different chirality on the opposite side of the ribbon. (iii) Entanglement increases as more carbon lines are grouped together, regardless of the lines' chirality. No long-range entanglement was found in the symmetry-protected phase in the absence of disorder.

Bogdan Andrei Bernevig

Xiao-Gang Wen

Guanyu Zhu, Qing‐Feng Sun, Y.-Y. Zhang +3 more

We investigate an ultracold Fermi gas in a hexagonal lattice subjected to a strong Rashba spin-orbit coupling (SOC). We focus on the marginal Abelian limit with the strength of SOC $\ensuremath{\kappa}=\ensuremath{\pi}/2$, where the derived Dirac fermions are spin-1/2 spinors in contrast to the pure hexagonal lattice. We find a double-degeneracy at the center of the Brillouin zone, which is a spin-3/2 Dirac-Weyl fermion that can be represented as two spin-1/2 fermions. While at the corner of the Brillouin zone, there occur another two spin-1/2 fermions. These Dirac spinors display a verity of spin textures beyond the usual topological properties. Furthermore, we investigate the energy spectrum of one-dimensional (1D) zigzag and armchair ribbons, and show that the appearance of a new cone at the center of the Brillouin zone can induce two flat bands in a 1D zigzag ribbon. These unique properties allow us to transport a single spin-1/2 Dirac spinor through the 1D zigzag ribbon with two quasi-two-dimensional leads, where only the spin-up or spin-down freedom of the Dirac spinors can transit through the ribbon.