Topological protection in non-Hermitian Haldane honeycomb lattices

Open Access

Topological protection in non-Hermitian Haldane honeycomb lattices

Pablo Reséndiz-Vázquez, Konrad Tschernig, Armando Perez-Leija, Kurt Busch, and Roberto de J. León-Montiel

Phys. Rev. Research 2, 013387 – Published 31 March 2020

Abstract

Topological phenomena in non-Hermitian systems have recently become a subject of great interest in the photonics and condensed-matter communities. In particular, the possibility of observing topologically protected edge states in non-Hermitian lattices has sparked an intensive search for systems where this kind of states are sustained. Here we present a study of the emergence of topological edge states in two-dimensional Haldane lattices exhibiting balanced gain and loss. In line with recent studies on other Chern insulator models, we show that edge states can be observed in the so-called broken $P T$ -symmetric phase, that is, when the spectrum of the gain-loss-balanced system's Hamiltonian is not entirely real. More importantly, we find that such topologically protected edge states emerge irrespective of the lattice boundaries, namely, zigzag, bearded, or armchair.

Received 11 December 2019
Accepted 4 March 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013387

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Edge states Symmetry protected topological states Topological insulators

Honeycomb lattice

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Pablo Reséndiz-Vázquez ^1,*, Konrad Tschernig ^2,3, Armando Perez-Leija^2,3, Kurt Busch^2,3, and Roberto de J. León-Montiel ^1,†

¹Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 Ciudad de México, Mexico
²Max-Born-Institut, Max-Born-Straße 2A, 12489 Berlin, Germany
³Humboldt-Universität zu Berlin, Institut für Physik, AG Theoretische Optik & Photonik, D-12489 Berlin, Germany

^*pablo.resendiz@correo.nucleares.unam.mx
^†roberto.leon@nucleares.unam.mx

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 2, Iss. 1 — March - May 2020

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Schematic representation of the non-Hermitian honeycomb Haldane lattice with zigzag, bearded, and armchair edges. The blue (orange) dots represent the amplifying (lossy) sites in the lattice. Note that the Haldane model includes two types of interactions between sites, a nearest-neighbor interaction $t_{1}$ , shown as green arrows, and second-nearest-neighbor couplings $t_{2}$ , with a constant phase $e^{\pm i ϕ}$ for clockwise (anticlockwise) coupling direction, depicted by the red arrows. In order to implement balanced gain and loss, we set the on-site energies of the blue sites to $i Γ$ and those of the orange sites to $- i Γ$ . (b) and (c) The time evolution of an edge state over a $60 \times 31$ sites Haldane-ribbon lattice for $t_{1} = 1.0 s^{- 1}$ , $t_{2} = 0.3 s^{- 1}$ , and $ϕ = π / 2$ . Panel (b) shows the free evolution of the edge state, whereas panel (c) shows its propagation in the presence of a rectangular defect on the right edge of the lattice. (d), (e), and (f) The real (blue line) and imaginary (orange line) energy eigenvalues per lattice mode for $Γ = 0 s^{- 1}$ , $Γ = 0.1 s^{- 1}$ , and $Γ = 1.0 s^{- 1}$ , respectively. The eigenvalues related to the topologically protected edge states are shown in the region encircled by the black ellipse. Note that in panel (f) $Γ$ exceeds the critical gain-loss ratio $Γ_{c} = 0.7 s^{- 1}$ , so no purely real eigenvalues are observed.
Reuse & Permissions
Figure 2
Critical value of the gain-loss rate $Γ_{c}$ (defined as the maximum value of gain-loss for which at least 20 edge states remain within the dissipation- or amplification-free region) as a function of the Haldane flux, $ϕ$ for bearded-armchair (blue), armchair-zigzag (purple), bearded-zigzag type 1 (red), and bearded-zigzag type 2 (green). As discussed in the text, the topological phase transition occurs in a flux range defined by $0 < ϕ < π$ . Note that for the bearded-armchair termination, $Γ_{c} = 0.70 s^{- 1}$ , while for armchair-zigzag termination, $Γ_{c} = 0.61 s^{- 1}$ . For bearded-zigzag type 1 and 2, we obtain $Γ_{c} = 0.38 s^{- 1}$ and $Γ_{c} = 0.42 s^{- 1}$ , respectively.
Reuse & Permissions
Figure 3
(a)–(c) Schematic representation of the three geometries considered in this section. In (a.1)–(c.1) we show the free time evolution of an edge state over a $61 \times 30$ site Haldane-ribbon lattice with armchair-zigzag, bearded type 1-zigzag, and bearded type 2-zigzag terminations for $t_{1} = 1.0 s^{- 1}$ , $t_{2} = 0.3 s^{- 1}$ , and $ϕ = π / 2$ ; whereas (a.2)–(c.2) show the propagation of the edge state in the presence of a finite triangular defect.
Reuse & Permissions
Figure 4
(a)–(i) Real (blue line) and imaginary (orange line) parts of the lattice eigenmodes for (a), (d), (g) $Γ = 0 s^{- 1}$ , (b), (e), (h) $Γ = 0.1 s^{- 1}$ , and (c), (f), (i) $Γ = 1.0 s^{- 1}$ . Panels (a)–(c) correspond to armchair-zigzag terminations, whereas (d)–(f) and (g)–(i) correspond to bearded-zigzag type 1 and bearded-zigzag type 2 terminations, respectively. The eigenvalues related to the topologically protected edge states are shown in the region encircled by the black ellipse. Note that in (c), (f), and (i) $Γ$ exceeds the critical gain-loss ratio for all the geometries, so purely real eigenvalues are completely absent.
Reuse & Permissions

Physical Review Research