Non-Hermitian dislocation modes: Stability and melting across exceptional points

Letter

Non-Hermitian dislocation modes: Stability and melting across exceptional points

Archisman Panigrahi, Roderich Moessner, and Bitan Roy

Phys. Rev. B 106, L041302 – Published 11 July 2022

Abstract

The traditional bulk-boundary correspondence assuring robust gapless modes at the edges and surfaces of insulating and nodal topological materials gets masked in non-Hermitian (NH) systems by the skin effect, manifesting an accumulation of a macroscopic number of states near such interfaces. Here we show that dislocation lattice defects are immune to such skin effect or at most display a weak skin effect (depending on its relative orientation with the Burgers vector), and as such they support robust topological modes in the bulk of a NH system, specifically when the parent Hermitian phase features band inversion at a finite momentum. However, the dislocation modes gradually lose their support at their core when the system approaches an exceptional point, and finally melt into the boundary of the system across the NH band gap closing. We explicitly demonstrate these findings for a two-dimensional NH Chern insulator, thereby establishing that dislocation lattice defects can be instrumental to experimentally probe pristine NH topology.

Received 24 May 2021
Revised 8 June 2022
Accepted 29 June 2022

DOI:https://doi.org/10.1103/PhysRevB.106.L041302

Physics Subject Headings (PhySH)

Disclinations & dislocations Topological phase transition Topological phases of matter

Lattice models in condensed matter Symmetries in condensed matter Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Archisman Panigrahi ¹, Roderich Moessner², and Bitan Roy ^3,*

¹Indian Institute of Science, Bangalore 560012, India
²Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany
³Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, USA

^*Corresponding author: bitan.roy@lehigh.edu

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Issue

Vol. 106, Iss. 4 — 15 July 2022

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Images

Figure 1
(a) Construction of an edge dislocation with Burgers vector $b = a e_{x}$ through the Volterra cut-and-paste procedure on a square lattice by removing a line of atoms (cyan circles), ending at its center or core (orange circle) and subsequently reconnecting (dashed lines) the sites (blue and red dots) living on the edges (blue and red lines) across it. Thus, the dislocation core does not host naked edges, unlike the perimeter of the system. Phase diagrams in the presence of the NH perturbation (b) $h_{x}$ (and $h_{y}$ ) and (c) $h_{z}$ for $t = t_{0} = 1$ [Eq. (1)]. The eigenvalue spectra are gapped only in the red shaded region, where the NH Chern number $C_{NH} = - 1$ [Eq. (2)] and system supports topologically protected dislocation modes [Figs. 2 and 3]. EPs first appear in the BZ along the black lines (defining $h_{x}^{c}$ ) through the $X = (π, 0) / a$ and $Y = (0, π) / a$ points (dashed line) or the $M = (π, π) / a$ point (solid line) in (b). In (c), EPs enter the BZ at momentum satisfying $cos (k_{x}) = cos (k_{y}) = m_{0} / 2$ for $h_{z} = h_{z}^{c}$ (black solid line). Above the black lines EPs move in the BZ before they collide and disappear (not shown explicitly).
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Figure 2
Eigenvalues of the NH operator $H_{NH} = σ \cdot d (k) + i σ \cdot h$ in a periodic system with an edge dislocation-antidislocation pair for (a) $h_{x} = 0.2$ , (b) $h_{x} = 0.6$ , (c) $h_{y} = 0.2$ , (d) $h_{y} = 0.6$ , (e) $h_{z} = 0.2$ , and (f) $h_{z} = 0.6$ , when $t = t_{0} = - m_{0} = 1$ . The amplitude of the right eigenvectors corresponding to the dislocation modes [red dots in (a)–(f)] are, respectively, displayed in (g), (h), (i), (j), (k), and (l). Any site with its value less than $0.5 \times 10^{- 4}$ is left empty. With increasing NH perturbations, the spectral gap to the rest of bulk states (black dots) decreases, and the dislocation modes gradually lose their support at their cores. For $h_{x, y, z} \geq h_{x, y, z}^{c}$ [white shaded region in Figs. 1 and 1], with the appearance of EPs in the BZ, the dislocation modes melt into rest of the bulk states (as there is no skin effect in periodic systems) and disappear.
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Figure 3
Amplitude of right eigenvectors of the dislocation modes for (a) $h_{x} = 0.2$ , (b) $h_{z} = 0.4$ , (c) $h_{y} = 0.05$ , (d) $h_{x} = 0.55$ , (e) $h_{z} = 0.7$ and (f) $h_{y} = 0.3$ in systems with open boundary condition. Any site with its value less than $0.5 \times 10^{- 4}$ is left empty. With increasing $h_{x} (h_{y})$ , dislocation modes gradually lose support at dislocation cores and their weight shifts toward the right (top) edge or skin of the system, due to the broken $C_{4}$ symmetry. Such shift with increasing $h_{z}$ is, however, predominantly toward the four corners, as this NH perturbation preserves the $C_{4}$ symmetry.
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Figure 4
Amplitude of all the right eigenvectors except for the dislocation modes (shown in Fig. 3) for $h_{x} = 0.4$ [(a) and (c)] and $h_{y} = 0.4$ [(b) and (d)] in systems with open [(a) and (b)] and periodic [(c) and (d)] boundary conditions. Any site with its value less than $0.3 \times 10^{- 3}$ is left empty in (a) and (b). Therefore, in open boundary system the dislocation core is always devoid of skin effect. By contrast in periodic system dislocation core is does not show skin effect for $h_{x}$ , while it features a weak skin effect for $h_{y}$ , whose amplitude is at least an order magnitude smaller than that in open boundary system and the dislocation modes. This is so because the associated Burgers vector $b = a e_{x}$ is parallel (orthogonal) to the direction of the skin effect for $h_{x} (h_{y})$ [see text].
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