Continuum of Bound States in a Non-Hermitian Model

Qiang Wang, Changyan Zhu, Xu Zheng, Haoran Xue, Baile Zhang, and Y. D. Chong

Phys. Rev. Lett. 130, 103602 – Published 8 March 2023

Abstract

In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call “continuum Landau modes” (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.

Received 18 October 2022
Accepted 14 February 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.103602

Physics Subject Headings (PhySH)

Topological phases of matter

Non-Hermitian systems Topological materials

Atomic, Molecular & Optical

Authors & Affiliations

Qiang Wang ¹, Changyan Zhu¹, Xu Zheng¹, Haoran Xue ¹, Baile Zhang ^1,2,*, and Y. D. Chong ^1,2,†

¹Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
²Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore

^*blzhang@ntu.edu.sg
^†yidong@ntu.edu.sg

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Vol. 130, Iss. 10 — 10 March 2023

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Images

Figure 1
Effects of a uniform magnetic field on the spectra of 2D models. (a) For a nonrelativistic particle with quadratic dispersion, the spectrum collapses into discrete Landau levels. (b) For a Dirac particle, the spectrum forms an unbounded sequence of Landau levels. (c) For the non-Hermitian Hamiltonian (1), the complex linear dispersion relation turns into a continuum of bound states filling the complex energy plane.
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Figure 2
Continuum Landau modes (CLMs) in a 2D lattice. (a) Schematic of a square lattice with reciprocal hoppings along $x$ (gray lines), nonreciprocal hoppings along $y$ (black arrows), and on-site mass $m_{x, y} = B (y - i x)$ (size and darkness of the circles indicate the real and imaginary parts). (b),(c) Complex energy spectra for finite lattices with (b) $B = 0.03$ and (c) $B = 0.3$ . The color of each dot corresponds to the participation ratio (PR) of the eigenstate; a more localized state has lower PR. The arrows on the color bar indicate the highest PR for the CLM ansatz, for each $B$ . The dashed boxes are the bounds on CLM eigenenergies derived from Eq. (7). (d) Plot of $Re (E)$ versus $⟨ y ⟩$ (left panel) and $Im (E)$ versus $⟨ x ⟩$ (right panel) for $B = 0.3$ . The black dashes and gray dotted lines, respectively, indicate the theoretical central trend line (corresponding to $E_{k + q}^{0} \to 0$ ) and bounding lines derived from Eq. (7). (e),(f) Wave function amplitude $| ψ_{r} |$ for the eigenstates marked by yellow stars in (b) and (c), respectively. Hollow and filled circles, respectively, indicate the variation with $x$ and $y$ , along lines passing through the center of each Gaussian; solid curves show the CLM predictions. Insets show the distribution in the 2D plane. In (b)–(f), we use $t_{x} = t_{y} = 1$ and a lattice size of $60 \times 60$ , with open boundary conditions.
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Figure 3
Rainbow trapping and wave funneling in 1D lattices. (a) 1D lattice with nonreciprocal nearest neighbor hoppings of $t$ (solid arrows) and $- t$ (dashed arrows). Each site $j$ has real mass $m_{j} = B (j - j_{0})$ , where $j \in [1, N]$ and $j_{0} = (N + 1) / 2$ . (b) Complex energy spectrum for the lattice in (a) with $N = 2000$ , $t = 1$ , and $B = 0.01$ . The color of each dot indicates the eigenstate’s position expectation value $⟨ x ⟩$ . The dashed box shows the CLM energy bounds described in the text. (c) Site-dependent amplitudes under steady state excitation at frequency $ω$ for the lattice in (b), with an additional per-site damping term $Δ m = 1.9 i$ to avoid blowup. On each site, the excitation has uniform amplitude but a random phase drawn uniformly from $[0, 2 π)$ . The two dashes show the working band $[- B N / 2, B N / 2]$ . The peak position found to be proportional to $ω$ . (d) 1D lattice with reciprocal nearest neighbor hopping $t$ , and on-site mass $m_{j} = i B (j - j_{0})$ . (e) Complex energy spectrum for the lattice in (d) with $N = 2000$ , $t = 1$ , and $B = 0.01$ . (f) Site-dependent amplitudes under the same excitation as in (c), using the lattice in (e) with additional per-site damping $Δ m = 9.9 i$ . Dashes show the working band $[- 2 t, 2 t]$ . Funneling toward the large- $j$ boundary is observed. In (c) and (f), the coupling of the excitation to each site is $κ = 0.2$ .
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