Localization of Chiral Edge States by the Non-Hermitian Skin Effect

Quantum Hall systems host chiral edge states extending along the one-dimensional boundary of any two-dimensional sample. In solid state materials, the edge states serve as perfectly robust transport channels that produce a quantized Hall conductance; due to their chirality, and the topological protection by the Chern number of the bulk band structure, they cannot be spatially localized by defects or disorder. Here, we show experimentally that the chiral edge states of a lossy quantum Hall system can be localized. In a gyromagnetic photonic crystal exhibiting the quantum Hall topological phase, an appropriately structured loss configuration imparts the edge states’ complex energy spectrum with a feature known as point-gap winding. This intrinsically non-Hermitian topological invariant is distinct from the Chern number invariant of the bulk (which remains intact) and induces mode localization via the “non-Hermitian skin effect.” The interplay of the two topological phenomena—the Chern number and point-gap winding—gives rise to a non-Hermitian generalization of the paradigmatic Chern-type bulk-boundary correspondence principle. Compared to previous realizations of the non-Hermitian skin effect, the skin modes in this system have superior robustness against local defects and disorders.

Figure 1

Chern-type bulk-boundary correspondence and its non-Hermitian generalization. (a) Chiral edge states. (b) Edge- or corner-localized chiral edge states. The determination of boundary modes requires a hybrid topological invariant ($C$; ${\nu }_x$, ${\nu }_y$).

Figure 2

First-principles study of edge-localized chiral edge states. (a) Unit cell of the PhC. $a=10\mathrm{mm}$ and $d=3\mathrm{mm}$. (b) Eigenfrequencies for double-PBCs (blue region) and double-OBCs (rhombic dots). (c) Finite structure with double-OBCs. (d) Eigenfunctions for states labeled “1” and “2” in (b). (e) Supercell with $x\text{−}\mathrm{PBC}/y\text{−}\mathrm{OBC}$. (f),(g) Eigenfrequencies and eigenfunctions for the supercell in (e). The black arrows in (f) denote the $k_x$ increasing directions. The red arrows in (g) indicate the group velocities of the edge states. (h)–(j) Similar to (e)–(g) but for $y\text{−}\mathrm{PBC}/x\text{−}\mathrm{OBC}$.

Figure 3

Observation of edge-localized chiral edge states. (a),(b) Photograph of the fabricated PhC with top copper plate removed (a) and shifted (b) for clear visualization. (c),(e) Measured intensity distributions excited by a line source near the right or left boundary in (a), operating at frequencies of 15.3 GHz and 17.2 GHz, respectively. (d),(f) Measured ratios of edge-localized energy. The arrows point to 15.3 GHz in (d) and 17.2 GHz in (f).

Figure 4

Observation of corner-localized chiral edge states. (a) Unit cell of the PhC. (b) Colored lines: eigenfrequencies for the supercell in (c) under $x\text{−}\mathrm{PBC}/y\text{−}\mathrm{OBC}$ and (d) under $y\text{−}\mathrm{PBC}/x\text{−}\mathrm{OBC}$. Rhombic dots: Eigenfrequencies for the sample under double-OBCs in (g). (e),(f) Eigenfunctions for the sample in (g). (h),(i) Measured intensity distributions excited by the corner source in (g). (j) Measured ratios of corner-localized energy. The arrows represent 15.3 GHz and 17.2 GHz.