Zak phase and the existence of edge states in graphene

Abstract

We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase Z(k ), which is a Berry phase across an appropriately chosen one dimensional Brillouin zone, and the existence of a localized state of momentum k at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one dimensional toy model as well as for graphene ribbons with zigzag edges. The range of k for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, it confirms and generalizes the results of several previous works.

Edge-state blockade of transport in quantum dot arrays

“…For the monomer chain realized at the transition point δτ = 0 (for finite systems it is rather a crossover at δτ ≈ τ 0 /N [14]), the eigenstates read φ (n) ∝ sin[π n/(N + 1)], where = 1, . .…”

“…It is characterized by a topological invariant, the Zak phase [12], which depends on the ratio between the inter-and intradimer coupling and has been measured recently [13]. For finite chains in the topologically nontrivial phase, a pair of exponentially decaying edge states emerges [14]. Moreover, Coulomb interaction may lead to long-range tunneling of doublons between edge states [15].…”

Kibble-Zurek Mechanism in Topologically Nontrivial Zigzag Chains of Polariton Micropillars

“…Indeed, depending on the difference of the tunneling coefficients within and between the dimers, such chains form topologically different conduction bands, characterized by a π difference in the Zak phase [19]. As it was shown recently [20], the number of states in the conduction band depends on this phase, and the states which are not included in the bulk are localized on the edges. These edge states do not rely on inter-particle interactions but are topologically protected: they are robust against disorder and perturbations.…”

“…The pair "ab" forms the unit cell. Following the definitions established in the previous works [20], the tunneling constant in the first link (within the cell) is called t , while the tunneling in the second link (between cells) is t. The corresponding tight-binding Hamiltonian reads (m is the cell number):…”

“…The Zak phase is determined by the ratio of the tunneling coefficients within and between the dimers [20]: ζ n = 0 if t /t > 1, and ζ n = π, if t /t < 1. The topological transition ζ n = 0 ↔ ζ n = π occurs at t = t .…”