Exceptional Excitons

Abstract

Non-Hermitian physics is reshaping our understanding of quantum systems by revealing states and phenomena without Hermitian counterparts. While non-Hermiticity is typically associated with gain-loss processes in open systems, we uncover a fundamentally different route to non-Hermitian behavior emerging from non-equilibrium correlations. In photoexcited semiconductors, the effective interaction between electrons and holes gives rise to a pseudo-Hermitian Bethe-Salpeter Hamiltonian (PH-BSH) that governs excitonic states in the presence of excited populations. Within this framework, we identify a previously unknown class of excitonic quasiparticles - exceptional excitons - corresponding to exceptional points embedded inside the electron-hole continuum. Exceptional excitons emerge at the onset of population inversion, and represent the strongly renormalized counterparts of the system's equilibrium excitons. They are spatially localized, protected against hybridization with the continuum, and remain long-lived even in regimes where conventional excitons undergo a Mott transition. Crucially, exceptional excitons appear only when the PH-BSH is evaluated with non-thermal, resonantly generated carrier populations that support an excitonic superfluid. Ab initio results for monolayer WS_2 explicitly demonstrate this scenario and show that exceptional excitons can be realized with existing ultrafast pumping techniques. We also identify distinctive optical and photoemission signatures that enable their unambiguous detection.

Figures (9)

• Figure 1: Excitons in photoexcited materials. Panel a: Schematic illustration of exciton evaporation above the Mott density in the presence of thermal carrier distribution. Panel b: Illustration of exciton survival in the presence of high carrier population in the superfluid state. Panel c: (upper part) Evolution of the real part of the A-exciton energy and of the bandgap (i.e. the onset of the e-h continuum) for WS$_{2}$, evaluated from Eq. (\ref{['phbsh']}) with thermal carrier populations $f^{{\rm th}}_{\nu {\mathbf k}}$ at 150 K, by varying the excitation density $n$; (lower part) imaginary part of A-exciton energy. Panel d: Same as panel c, but with PH-BSH evaluated from Eq. (\ref{['phbsh']}) with superfluid populations $f^{{\rm sf}}_{\nu {\mathbf k}}$ obtained by solving self-consistently Eq. (\ref{['eq:exciton_sf']}). Here and in the rest of the paper the excitation density $n$ is the sum of densities at the $K$ and $K'$ valleys of WS$_{2}$, see Methods sections.
• Figure 2: Real-time generation of superfluid population. Panel a: (upper part) Real-time evolution of the conduction-band occupations $f_{c {\mathbf k}}(t)$ in the $K$-valley of WS$_{2}$ following photoexcitation, obtained using time-dependent HSEX simulations (see Methods section). The temporal profile of the pump pulse (shown in the lower part) has a duration of $\sim 100$ fs, central frequency $\omega_{{\rm P}}=2.1$ eV and a fluence $F=25~\mu{\rm Jcm}^{-2}$. Occupations $f_{c {\mathbf k}}(t)$ are calculated within a square plaquette of side $0.4~\mathring{\mathrm{A}}^{-1}$ centered at the $K$-point; the corresponding colormap is shown in the inset. The total excited-carrier density is $n=10^{13}{\rm cm}^{-2}$. Panel b: Three-dimensional visualization of the steady-state distridution $f_{c {\mathbf k}}$ across all momenta in the plaquette. Panel c: momentum-dependent discrepancy $|f^{{\rm sf}}_{c {\mathbf k}}-f_{c {\mathbf k}}|$ between the superfluid distribution evaluated according to Eq. (\ref{['eq:exciton_sf']}) and the steady-state values evaluated with the real-time simulation. The relative error is less than 1%. Panel d: steady-state excitation density $n$ as a function of the pump frequency $\omega_{{\rm P}}$ for different values of the fluence. Light-blue shading denotes the parameter regime where population inversion occurs and the exceptional points basin is reached with the error in the carrier distribution of the same order as in panel c. The equilibrium value of the A-exciton energy $E^{{\rm eq}}_{{\rm x}}=2.06$ eV is indicated with a vertical green line.
• Figure 3: Excitonic eigenmode coalescence. Panel a: Evolution of the conduction-band carrier distribution $f_{c {\mathbf k}}$ in WS$_{2}$ corresponding to a total excitation density $n=7.5 \times^{12}{\rm cm}^{-2}$, as the control parameter $\alpha$ is tuned from the thermal limit $f_{c {\mathbf k}}= f^{{\rm th}}_{c {\mathbf k}}$ at $T=150~$K ($\alpha=0$) to the superfluid limit $f_{c {\mathbf k}}= f^{{\rm sf}}_{c {\mathbf k}}$ ($\alpha=1$). Panel b: Real and imaginary parts of the complex-conjugate excitonic eigenvalues $E_{\pm}$ as a function of $\alpha$. Panel c: Eigenvector similarity $|\Delta \Psi_{{\mathbf k}}|$ for different representative values of $\alpha$. Panel d: Normalized overlap $|\Psi^{\ast}_{-}\Psi_{+}|$ (blue curve) and determinant of the eigenvector matrix $|{\rm Det} \Psi|$ (red curved) as a function of $\alpha$. In panels c and d, the eigenvectors $\Psi_{\pm}$ are normalized to 1.
• Figure 4: Spectral properties of excitonic modes. Panel a: Spectral function $A_{{\rm x}}(\omega)$ defined in Eq. (\ref{['spec']}) for WS$_{2}$ with an excitation density $n=10^{13}{\rm cm}^{-2}$ in the presence of thermal carriers at $T=150$ K (black curve). For comparison, the corresponding equilibrium spectral function obtained by setting $H=H^{{\rm eq}}$ (i.e. $\delta H=0$) is also shown (green curve). The inset displays the real (blue curve) and imaginary (red curve) parts of the embedding self-energy $\Sigma(\omega)$, together with the line $\omega -H_{{\rm xx}}$ (dashed). Panel b: Same quantities as in panel a, but evaluated for superfluid carriers at the same excitation density $n=10^{13}{\rm cm}^{-2}$. In both panels the equilibrium spectral function and the self-energy are computed using $\eta=40$ meV.
• Figure 5: Optical properties. Panel a: Calculated absorption spectrum of WS$_{2}$ in the presence of thermal carriers at $T=150$ K for different excitation densities. The red-shaded region marks the onset and progression of the Mott regime, where bound excitons disappear. Panel b: Same as panel a, but for superfluid carrier populations. The blue-shaded region denotes the regime in which exceptional excitons emerge (BCS regime).