Direct measurement of a non-Hermitian topological invariant in a hybrid light-matter system

Abstract

Topology is central to understanding and engineering materials that display robust physical phenomena immune to imperfections. Different topological phases of matter are characterized by topological invariants. In energy-conserving (Hermitian) systems, these invariants are determined by the winding of eigenstates in momentum space. In non-Hermitian systems, a topological invariant is predicted to emerge from the winding of the complex eigenenergies. Here, we directly measure the non-Hermitian topological invariant arising from exceptional points in the momentum-resolved spectrum of exciton polaritons. These are hybrid light-matter quasiparticles formed by photons strongly coupled to electron-hole pairs (excitons) in a halide perovskite semiconductor at room temperature. We experimentally map out both the real (energy) and imaginary (linewidth) parts of the spectrum near the exceptional points and extract the novel topological invariant—fractional spectral winding. Our work represents an essential step toward realization of non-Hermitian topological phases in a condensed matter system.

Fig. 1.. Complex spectral structure near pairs of EPs in momentum space.

(A) Energy (real part of the complex spectrum) of the exciton-polariton modes in a microcavity with linear birefringence, calculated using the model Eq. 1. The mean energy is subtracted for clarity. Energy crossings occur at two opposite regions in the 2D momentum space (k_x, k_y). (B) Enlarged view of the dashed region in (A) in the Hermitian limit, showing a Dirac point. (C) Energy of the dashed region in (A) in the non-Hermitian case, with nonzero iσ_{x, y} components, showing the Dirac point splitting into a pair of EPs (pink dots) connected by the nodal line, bulk Fermi arc (green), where the energies cross. (D) Imaginary part of the complex spectrum corresponding to the linewidth for the dashed region in (A), showing the imaginary Fermi arc (orange), where the linewidths cross, emanating from the EPs (pink dots). (E) Energy of the system with a weak, real-valued σ_z term perturbation. (F) Same as (E) but with a strong perturbation leading to the annihilation of the EPs and opening of the gap. (G) Simplified complex energy structure of the two eigenstates, showing the bulk (green) and imaginary (orange) Fermi arcs connecting at the EPs and forming two closed contours. A single contour can also form (dashed orange) for the different sign of the parameters in Eq. 3. (H to J) In-plane pseudospin angle in momentum space of the upper eigenstate corresponding to (from left to right) (C), (E), and (F), respectively. (K and L) Spectral phase $\text{Arg}(\mathrm{\Delta }\tilde{E})$ in momentum space corresponding to (C) and (E), respectively. In (H) and (I), pink dots correspond to the EP, dashed lines correspond to the bulk Fermi arc, and white arrowed contours correspond to the half-charge (H, K, and L) and integer (I and J) windings around the singularities.

Fig. 2.. Experimental investigation of the complex exciton-polariton eigenenergies.

(A) Schematics of the planar microcavity made of SiO₂/Ta₂O₅ DBRs with an embedded CsPbBr₃ perovskite crystal. (B) Schematics of the laboratory (x, y, z) axis and the polarization measurement axis (H, V). The exciton-polariton in-plane momentum depends on the angles (θ,ϕ) of the PL emission. (C) Linearly polarized PL intensity (I_V-I_H) measured along (k_x, k_y = 0) and (k_x = 0, k_y). Dashed lines are the extracted peak energies of the two polarized modes. The dispersion is approximately symmetric for k → − k. Inset: Schematics of the measurements in momentum space with respect to the Fermi arcs. (D) Linewidths of the modes in (C) with the mean subtracted. Inset: Enlarged region near k = 0.

Fig. 3.. Mapping out complex energies near the EP pair.

(A) Schematics of the EP pair (pink dots) connected by the bulk (green) and imaginary (orange) Fermi arcs. Dashed lines (b to f) represent the lines (directions) in k-space, along which the measurements in (B) to (F) are performed. (B to F) Measured energies and linewidths (mean-subtracted) of the two modes: (B) Parallel to and very near the bulk Fermi arc; (C) parallel to the bulk Fermi arc intersecting the imaginary Fermi arc twice, which corresponds to two linewidth crossings and no crossing in energy; (D) perpendicular to the bulk Fermi arc very near the top EP, showing crossing in both energy and linewidth; (E) along the center of the real Fermi arc, showing crossing in energy and anticrossing in linewidth; (F) near the EP but outside the real Fermi arc showing no crossing in energy but crossing in linewidth. The complex eigenvalues are sorted so that a smooth crossing (D and E) or anticrossing (B, C, and F) in the real part is ensured. The values for k are as follows: (B) k_x = −5.19 μm⁻¹, (C) k_x = −5.07 μm⁻¹, (D) k_y = 0.40 μm⁻¹, (E) k_y = 0.21 μm⁻¹, and (F) k_y = 0.09 μm⁻¹. Error bars represent the 95% confidence interval fitting results.

Fig. 4.. Chirality and topology of the EPs.

(A) Poincaré sphere with arrows representing the pseudospin of exciton polaritons away from the EP (thin red and blue), near the EP (thick red and blue), and at the EP (thick purple). Dashed vertical arrows are the effective out-of-plane field arising from the imaginary component of the complex in-plane artificial magnetic field. (B) Theoretical texture of circular polarization (S₃) arising from the inclusion of non-Hermiticity into the model of Eq. 1. (C) Measured energy-integrated circular polarization (S₃) showing the same spin structure as in (B) but with a weak S₃ background coming from the bare perovskite (see section: Pseudospin texture in the complex artificial gauge field). Right: Enlarged images of the marked regions showing the position of EPs (black points). (D) Theoretical values of $\text{arg}({\tilde{E}}_{+}-{\tilde{E}}_{-})$for one EP pair with the arrows schematically showing the fractional winding number. Parameters are the same as in Fig. 1 (C and D). (E and F) Measured values of $\text{arg}({\tilde{E}}_{+}-{\tilde{E}}_{-})$near the two pairs of EPs demonstrating the half-integer spectral winding around each EP.