Non-Hermitian Bethe-Salpeter Equation for Open Systems: Emergence of Exceptional Points in Excitonic Spectra from First Principles

TL;DR

This work develops a rigorous, first-principles generalization of the Bethe-Salpeter equation to open quantum systems by embedding dissipation in a nonequilibrium Green's function framework on the Keldysh contour with Lindblad dynamics. It yields a microscopic non-Hermitian excitonic Hamiltonian and a steady-state, conserving Bethe-Salpeter formulation, including a static HSEX kernel and a dissipation-induced contribution to the two-particle kernel. When applied to valley excitons in transition metal dichalcogenides coupled to engineered photon baths, the approach reveals exceptional points in momentum space, ranging from discrete points to rings, which drive nonanalytic valley polarization, complex photoluminescence patterns, and nontrivial Berry curvature and conductivities. The results demonstrate a principled pathway to predict and control non-Hermitian and topological excitonic properties via environmental engineering, and they are compatible with ab initio BSE implementations for material-specific predictions.

Abstract

In open quantum systems hosting excitons, dissipation mechanisms critically shape the excitonic dynamics, band-structure and topological properties. A microscopic understanding of excitons in such non-Hermitian settings demands a first-principles generalization of the Bethe-Salpeter equation (BSE). Building on a recently introduced nonequilibrium Green's function formalism compatible with Lindbladian dynamics, we derive a non-Hermitian BSE from diagrammatic perturbation theory on the Keldysh contour, and obtain a microscopic excitonic Hamiltonian that incorporates dissipation while preserving causality. We apply the formalism to valley excitons in transition metal dichalcogenides coupled to structured photon baths. We uncover a rich landscape of exceptional points in momentum space, forming either discrete sets or continuous manifolds, depending on bath structure. The exceptional points give rise to nonanalytic valley-polarization, unusual polarization pattern in photoluminescence, and nontrivial topological signatures. Our results establish a first-principles framework for predicting and controlling excitonic behavior in open quantum materials, showing how engineered environments can be leveraged to induce and manipulate non-Hermitian and topological properties.

Figures (5)

• Figure 1: Excitonic spectrum for bath B1. Real (left) and imaginary (right) part of the eigenvalues $E^\pm_{\mathbf q}$ as a function of the exciton momentum $q$ for different symmetry-breaking parameters $\theta_{1}$. The upper and lower branches are shown in solid-blue and dashed-red, respectively. At $\theta_{1}=\pi/4$, a ring of EP emerges inside the light cone. The insets show the three-dimensional dispersion at $\theta_{1}=\pi/4$, highlighting the cylindrical symmetry.
• Figure 2: Excitonic spectrum for bath B0. Real (left) and imaginary (right) part of the eigenvalues $E^\pm_{\mathbf q}$ as a function of the exciton momentum $q$ for different angles $\theta_{0}-\varphi$. The upper and lower branches are shown in solid-blue and dashed-red, respectively. At $\varphi=\theta_{0}\pm \pi/4$, an EP emerges. The insets show the three-dimensional dispersion, highlighting the anistropic structure in momentum space.
• Figure 3: Valley polarization $P_{{\mathbf q}}$ as a function of momentum $q$ for bath B1 (top) and B0 (bottom). For B1 different values of $\theta_{1}$ are considered. For $\theta_{1}=0$ the polarization vanishes at all momenta, reflecting perfect balance between the valleys. At finite $\theta_{1}$ the polarization reaches extremal values at the exceptional momentum. In particular, at $\theta_{1}=\pm\pi/4$ the valley polarization saturates, $P_{{\mathbf q}}=\pm1$, indicating complete localization in either the $K$ or $K'$ valley. For B0 different values of the angle $\varphi$ are considered. The valley polarization saturates at the exceptional momentum only for $\varphi=\theta_{0}\pm\pi/4$.
• Figure 4: Elliptical polarization patterns of the electric field ${\mathbf E}(t)={\rm Re}[{\mathbf J} e^{{\rm i}\omega t}]$ for the $\pm$ branches and for bath B1 (top) and B0 (bottom). The orientation and shape of the ellipses encode the polarization state, while the color scale represents the ellipticity. At the EPs the electric field is circularly polarized, whereas for $q \to 0$ and $q \to \infty$ the polarization reduces to linear. The relative orientation of the polarization axes exhibits a $\pi/2$ difference between the two branches at large $q$.
• Figure 5: (Left) Berry curvature $\Omega_{q}$ at $\theta_{1}=\pi/4$. The curve shows a divergence for $q \to 0$ and a discontinuous jump of magnitude $2\lambda^{2}\ln(D^{(1)}/U)$ across the exceptional momentum $q_e$, consistent with the analytic expressions in Eq. (\ref{['eq:berry_curv']}). Middle-Right. Hall conductivity (middle), $\sigma_{xy}$, and Nernst conductivity (right), $\alpha_{xy}$, against temperature for several values of the ratio $D^{(1)}/U$. In both cases, the emergence of the exceptional ring at $D^{(1)}/U=1$ inverts the trend in how these quantities vary with $D^{(1)}/U$.