Probing the collective excitations of excitonic insulators in an optical cavity

TL;DR

This work investigates how an excitonic insulator encoded in a one-dimensional two-orbital lattice couples to quantum cavity photons. By deriving a perturbative light–matter interaction and computing the cavity photon Green's function within an RPA framework, the authors show that EI collective phase modes hybridize with cavity photons, producing an avoided crossing and a softening of the lower mode at a critical coupling. The study reveals a clear diagnostic: a sharp below-gap splitting of the cavity mode signals Coulomb-driven excitonic condensation, while its absence points to phonon-dominated pairing; trivial and topological insulators show minimal, non-hybridizing photon responses. The proposed heterodyne detection scheme provides a concrete route to experimentally map these signatures and distinguish the microscopic origin of excitonic order, with implications for cavity QED control of correlated materials.

Abstract

The light--matter interaction in optical cavities offers a promising ground to create hybrid states and manipulate material properties. In this work, we examine the effect of light-matter coupling in the excitonic insulator phase using a quasi one-dimensional lattice model with two opposite parity orbitals at each site. We show that the model allows for a coupling between the collective phase mode and cavity photons. Our findings reveal that the collective mode of the excitonic state significantly impacts the dispersion of the cavity mode, giving rise to an avoiding band crossing in the photon dispersion. This phenomenon is absent in trivial and topological insulator phases and also in phonon-mediated excitonic insulators, underscoring the unique characteristics of collective excitations in excitonic insulators. Our results demonstrate the significant impact of light-matter interaction on photon propagation in the presence of excitonic collective excitations.

Figures (7)

• Figure 1: A one-dimensional electronic system consisting of two orbitals of opposite parities ($s$ and $p_x$) at each site is centrally positioned in an optical cavity, which is shown by two large mirrors perpendicular to $y$ axis. The intra-orbital $J_\alpha$ ($\alpha=s, p_x$) and the nearest-neighbor inter-orbital $J_{sp_x}$ parameters describe the hoppings between the orbitals. The cavity's electromagnetic field is polarized along the lattice and propagates in the $y$-direction.
• Figure 2: Phase diagram of the s–p chain model adapted (data is digitized) from khatibi2020excitonic with permission. The real part of exciton order parameter $\phi$ in the (D/J, V/J) plane and winding number $\nu$ are used to distinguish three phases. The colored area marks the excitonic insulator, while white regions denote topological ($\nu=-1$) and trivial ($\nu=0$) insulating phases, respectively.
• Figure 3: The heat map of $\mathop{\mathrm{Re}}\nolimits[\mathcal{D}(q,\omega)]$ is shown. We adopt natural units such that $\hbar=6.6\times10^{-16} \mathrm{eV.s}$ and $c=3\times10^8 \mathrm{ m/s}$ and set the cavity photon fundamental frequency to $\hbar\omega_c/J=0.1$. The other parameters are fixed as $J=0.1\mathrm{eV}$ and lattice constant $a=4 \text{\AA}$. Panels (a,b,c) correspond to the excitonic insulator phase (parameters: $V/J=2.5, D/J=1.1, J_{sp}/J=0.5, \phi=0.116$), (d,e,f) to the trivial insulator phase (parameters: $V/J=2.5, D/J=2, J_{sp}/J=0.5, \phi=0$), and (g,h,i) to the topological insulator phase (parameters: $V/J=1, D/J=1.1, J_{sp}/J=0.5, \phi=0$). In each row, from left to right, panels correspond to the strength of the light-matter interaction $g=0.2$, $g=0.3$ and $g=0.4$. In panels (a–c), the dashed black line marks the excitonic phase mode energy, approximately $\omega_{\mathrm{phase}}/J=0.23$.
• Figure 4: (a) Density plot of the excitonic order parameter $\phi$ in the $\lambda/J-V/J$ plan. Heat maps of $\mathop{\mathrm{Re}}\nolimits[\mathcal{D}(q,\omega)]$ in (b) Coulomb-driven (parameters: $V/J=2.8$, $D/J=0.5$, $J_{sp}/J=0.5$, $\phi=0.1755$, $\lambda/J=0.1$ ) and (c) phonon-driven (parameters: $V/J=0.1$, $D/J=0.5$, $J_{sp}/J=0.5$, $\lambda/J=1.4$, $\phi=0.075$) excitonic-insulator phases. In both regimes the single-particle gap is set to $E_g/J\approx 2$, the light-matter interaction to $g=0.2$, phonon frequency to $\omega_{\mathrm{pn}}/J=0.1$ and the cavity fundamental frequency to $\hbar\omega_c/J=0.3$ ($J=0.1$ and $a=4 \text{\AA}$).
• Figure 5: (a) Dyson equation for the photon Green’s function. (b) Equation for the photon self-energy $\Pi(q,\omega)$. (c) Screened coulomb interaction, where $\chi^0(q,\omega)$ represents the electronic polarization of the matter.