Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence

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Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence

Elisabet Edvardsson, Flore K. Kunst, and Emil J. Bergholtz

Phys. Rev. B 99, 081302(R) – Published 27 February 2019

Abstract

Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulk-boundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface, and bulk degrees of freedom we establish that the non-Hermitian extensions of higher-order topological phases exhibit an even richer phenomenology than their Hermitian counterparts and that this can be understood in a unifying way within our biorthogonal framework. Saliently this works in the presence of the non-Hermitian skin effect, and also naturally encompasses genuinely non-Hermitian phenomena in the absence thereof.

Received 22 December 2018

DOI:https://doi.org/10.1103/PhysRevB.99.081302

Physics Subject Headings (PhySH)

Quantum optics Topological insulators Topological phases of matter

Topological materials

Interdisciplinary PhysicsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Elisabet Edvardsson^*, Flore K. Kunst, and Emil J. Bergholtz

Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden

^*Corresponding author: elisabet.edvardsson@fysik.su.se

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Issue

Vol. 99, Iss. 8 — 15 February 2019

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Images

Figure 1
(a) Schematic depiction of the lattice with $A$ , $B$ , $B^{'}$ , and $B^{″}$ sublattices in red, blue, green, and black, respectively, and the unit cells labeled by $m_{1}$ and $m_{2}$ . The red-blue and green-black chains are coupled to each other via nearest-neighbor hopping, $s$ , and the Hamiltonian on each of these two types of chains has the on-site term $(-) \sin (t)$ , and hopping terms $a_{\pm} (t) = - t_{1} + δ \cos (t) \pm γ / 2$ and $b (t) = - t_{1} - δ \cos (t)$ , where $t_{1}$ , $γ$ , and $δ$ are nearest-neighbor hopping parameters. (b) The absolute value of the energy with $γ = 2.5$ , $t_{1} = 1.5$ , $δ = 1$ , $s = 0.25$ , and $M_{1} = M_{2} = 11$ as a function of $t$ is shown in blue (gray) for the open (periodic) system. The orange (green) dashed lines correspond to $| r_{R, 1} | = 1$ ( $| r_{L, 1} | = 1$ ), while the black solid lines correspond to $| r_{L, 1} r_{R, 1} | = 1$ . (c) The localization of the hinge state [in red in (b)] on a lattice with $M_{1} = M_{2} = 11$ computed using the biorthogonal (top row), right (middle), and left (bottom) expectation values of the projection operator.
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Figure 2
Distribution of the (a) bulk and (b) surface band with energies (a) $| E | = 2.0264$ and (b) $| E | = 0.5345$ for the same parameters as in Fig. 1 for the cut $t = - 0.5$ . The left and middle columns show the localization of the right and left wave function individually, and the right column shows the biorthogonal localization.
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Figure 3
(a) Schematic depiction of the lattice with $A$ , $B$ , and $B^{'}$ sublattices in red, blue, and green, respectively, and the unit cells labeled by $m_{1}$ and $m_{2}$ . In the Hamiltonian the sites in the “up” triangles are connected via a nearest-neighbor hopping $t_{1}$ , while in the “down” triangles, the nearest-neighbor hopping in a clockwise fashion is $t_{+} = t_{2} + γ / 2$ , and $t_{-} = t_{2} - γ / 2$ in the counterclockwise motion. (b) The absolute value of the energy with $t_{2} = 1$ , $γ = 1.5$ , and $M_{1} = M_{2} = 20$ as a function of $t_{1}$ is shown in blue (gray) for the open (periodic) system. The orange (green) dashed lines correspond to $| r_{R, 1} | = | r_{L, 2} | = 1$ ( $| r_{R, 2} | = | r_{L, 1} | = 1$ ), while the black solid lines correspond to $| r_{L, 1} r_{R, 1} | = | r_{L, 2} r_{R, 2} | = 1$ . (c) The localization of the corner state [in red in (b)] on a lattice with $M_{1} = M_{2} = 6$ computed using the biorthogonal (top row), right (middle), and left (bottom) expectation values of the projection operator.
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Figure 4
(a) Schematic depiction of the lattice with $A$ , $B$ , and $B^{'}$ sublattices in red, blue, and green, respectively, and the unit cells labeled by $m_{1}$ and $m_{2}$ . The Hamiltonian is explicitly depicted, and the same as in Fig. 3. (b) The absolute value of the energy with $t_{2} = 1$ , $γ = 1.5$ , and $M_{1} = M_{2} = 20$ as a function of $t_{1}$ is shown in blue (gray) for the open (periodic) system. The black dashed lines correspond to $| r_{R, 2} | = | r_{L, 1} | = 1$ . (c) The localization of the lowest energy band on a lattice with $M_{1} = M_{2} = 6$ for different cuts in $t_{1}$ using the biorthogonal (top row), right (middle), and left (bottom) expectation values of the projection operator.
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