# Models for Edge States in the Electronic Spectra of Dimer Chains

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This Demonstration shows two simple toy models for the appearance of zero-energy states in a one-dimensional (1D) system such as a simple finite chain of dimers and a two-dimensional (2D) system built from coupled finite chains of dimers. The electronic Hamiltonian matrix is solved in both cases within the tight-binding formalism, where is the hopping matrix element between -type and -type sites, whereas is the hopping matrix element between -type and -type sites within each finite chain. In the 2D case, the inter-chain coupling parameter affects the dispersivity of the electronic bands: if , the bands are almost flat. It was shown in [1] that tuning the ratio introduces anisotropy into the system, therefore inducing a topological transition governing the localization properties at the system edges. Hence, if , there will be no zero energy states, whereas if , edge states appear in the electronic spectra. [2] rationalizes this topological transition in terms of a geometrical phase, the so-called Zak phase, which is equal to in the presence of edge states; otherwise it is zero. The Zak phase is related to another relevant geometric phase (the Berry phase) by contour integration over the first Brillouin zone (see [1] and references therein).

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Contributed by: Jessica Alfonsi (September 2014)

(Padova, Italy)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: absence of edge states in a chain of dimers

Snapshot 2: appearance of edge states in a chain of dimers

Snapshot 3: absence of edge states in a 2D system made of coupled chains of dimers

Snapshot 4: turning point of the topological transition for the onset of edge states in a 2D system made of coupled chains of dimers

Snapshot 5: edge states in a 2D system made of coupled chains of dimers and enhanced dispersion of energy bands

References

[1] P. Delplace, "États de bord et cônes de Dirac dans des cristaux bidimensionnels," Ph.D. thesis, Université Paris-Sud XI, France, 2010. tel.archives-ouvertes.fr/tel-00607781/fr.

[2] P. Delplace, D. Ullmo, and G. Montambaux, "Zak Phase and the Existence of Edge States in Graphene," *Physical Review B*, 84(19), 2011 pp. 195452–195464. journals.aps.org/prb/abstract/10.1103/PhysRevB.84.195452.

## Permanent Citation