Cavity-Mediated Electron-Electron Interactions: Renormalizing Dirac States in Graphene

TL;DR

The paper develops a non-perturbative photon-free QED-HF approach to model extended crystals in optical cavities, deriving a downfolded effective Hamiltonian $\hat{H}_{\text{eff}} = \hat{H}_{\text{e}} + \frac{\hbar \tilde{\omega}}{2} + \hat{H}_{\text{l}} + \hat{H}_{\text{nl}}$ that captures local and nonlocal cavity-mediated interactions. By solving self-consistent HF equations for graphene, it reveals that quantum vacuum fluctuations of cavity photons induce sizeable, polarization-dependent renormalizations of Dirac states: a circularly polarized mode opens a topological Dirac gap (with $\Delta \propto A_0^2/\omega$ and Berry curvature indicating $C=1$), while a linearly polarized mode produces a topologically trivial but flat, anisotropic gap due to long-range nonlocal interactions. When two symmetric linear modes are present, intrinsic graphene symmetries are restored and Dirac cones reappear with a modified Fermi velocity; introducing anisotropy in the two-mode setup allows continuous control over gap size and topology. The framework generalizes to multi-mode cavities and suggests avenues for ab initio cavity quantum electrodynamics in solids, enabling non-perturbative discovery of cavity-induced quantum phenomena.

Abstract

Embedding materials in optical cavities has emerged as a strategy for tuning material properties. Accurate simulations of electrons in materials interacting with quantum photon fluctuations of a cavity are crucial for understanding and predicting cavity-induced phenomena. In this article, we develop a non-perturbative quantum electrodynamical approach based on a photon-free self-consistent Hartree-Fock framework to model the coupling between electrons and cavity photons in crystalline materials. We apply this theoretical approach to investigate graphene coupled to the vacuum field fluctuations of cavity photon modes with different types of polarizations. The cavity photons introduce nonlocal electron-electron interactions, originating from the quantum nature of light, that lead to significant renormalization of the Dirac bands. In contrast to the case of graphene coupled to a classical circularly polarized light field, where a topological Dirac gap emerges, the nonlocal interactions induced by a quantum linearly polarized photon mode give rise to the formation of flat bands and the opening of a topologically trivial Dirac gap. When two symmetric cavity photon modes are introduced, Dirac cones remain gapless, but a Fermi velocity renormalization yet indicates the relevant role of nonlocal interactions. These effects disappear in the classical limit for coherent photon modes. This new self-consistent theoretical framework paves the way for the simulation of non-perturbative quantum effects in strongly coupled light-matter systems, and allows for a more comprehensive discovery of novel cavity-induced quantum phenomena.

Figures (10)

• Figure 1: Illustration of the renormalized Dirac cones of monolayer graphene coupled to cavity photon modes of different polarizations. Due to cavity-mediated electron interactions, a circularly polarized photon induces an isotropic Dirac gap with nontrivial band topology, while a linearly polarized photon induces a flat and anisotropic Dirac gap with trivial topology. In contrast, two isotropic linearly polarized photons, with the same frequency and amplitude and the perpendicular polarization directions, do not induce the Dirac gap but modify the Dirac Fermi velocity. In the setup, graphene is on the $xy$ plane with $z=0$, and $x$ and $y$ are along the zigzag and armchair directions of the graphene structure, respectively.
• Figure 2: Dirac states in graphene coupled with a circularly polarized photon mode with $\hbar\omega=0.3$ eV, $A_0=2\times10^{-8}\frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$, and $\mathbf{e}=\mathbf{e}_x+i\mathbf{e}_y$. (a) HF bands of the $+\mathbf{K}$ valley (inset) along $\mathbf{\Gamma} \leftarrow +\mathbf{K} \rightarrow \mathbf{M}$ path. (b) Representation of the blue band in (a) in the two-dimensional reciprocal zone $\{k_x , k_y\} \in [-1,1]$$10^{-3}$Å$^{-1}$ centered at the crystal momentum $+\mathbf{K}$. (c) Component $|c_c|^2$ of the conduction basis state $\varphi_{c\mathbf{k}}^0$ for the lower valence band ($\varphi_{v\mathbf{k}} = c_v \varphi_{v\mathbf{k}}^0 + c_c \varphi_{c\mathbf{k}}^0$) in (b). (d) Variation of the electron density, $\Delta \rho (\mathbf{r}) = \sum_\mathbf{k} |\varphi_{v \mathbf{k}}|^2 - |\varphi_{v \mathbf{k}}^0|^2$, at the specific $z = 0.33~\text{\AA}$ plane, where the $2p_z$ atomic orbital of carbon has its maximum. The left (right) panel shows the contribution to density from the $+\mathbf{K}$ ($-\mathbf{K}$) valley. The cross signs mark the position of the $A$ and $B$ sites. (e,f) Evolution of the Dirac band gap in panel (a) as a function of the $A_0$ (with fixed $\hbar\omega=0.3$ eV) and photon energy $\hbar\omega$ (with fixed $A_0 = 2 \times 10^{-8} \frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$), respectively.
• Figure 3: Dirac states in graphene coupled with a linearly polarized photon mode with $\hbar\omega=0.3~\text{eV}$, $A_0 = 2 \times 10^{-8} \frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$, and $\mathbf{e} = \mathbf{e}_x$. (a) HF bands of the $+\mathbf{K}$ valley with and without nonlocal interaction. The former are also shown in the presence of a tiny sublattice potential value $V_{AB} = \pm 2 \times 10^{-5} t_0$. (b) Representation of the blue band in (a) in $\{k_x , k_y\} \in [-1,1]$$10^{-3}$Å$^{-1}$ centered at $+\mathbf{K}$. (c) Component $|c_c|^2$ of the conduction basis state $\varphi_{c\mathbf{k}}^0$ for the lower valence band ($\varphi_{v\mathbf{k}} = c_v \varphi_{v\mathbf{k}}^0 + c_c \varphi_{c\mathbf{k}}^0$) in (b). (d) Variation of electron density, $\Delta \rho (\mathbf{r}) = \sum_\mathbf{k} |\varphi_{v \mathbf{k}}|^2 - |\varphi_{v \mathbf{k}}^0|^2$, at $z = 0.33~\text{\AA}$ plane for $V_{AB} = 0$. The left (right) panel shows the contribution from the $+\mathbf{K}$ ($-\mathbf{K}$) valley. (e,f) The difference of wavefunction amplitude $\delta_{AB}$ (phase $\theta_{AB}$) between the $A$ and $B$ sites for the valence states across the plane $k_x = k_0 = -2 \times 10^{-4}~\text{\AA}^{-1}$ (the gray plane in (b)). Blue (Orange) lines are for graphene with $V_{AB} = 0$ ($\pm 2 \times 10^{-5} t_0$). (g,h) Evolution of the Dirac band gap in panel (a) as a function of $A_0$ (with fixed $\hbar\omega=0.3$ eV) and $\hbar\omega$ (with fixed $A_0 = 2 \times 10^{-8} \frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$) for $V_{AB} = 0$, respectively.
• Figure 4: Evolution of cavity-renormalized Dirac states and band topology. (a) Using a single photon mode with $\hbar\omega=0.3$ eV, $A_0 = 2 \times 10^{-8} \frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$, and $\mathbf{e} = \mathbf{e}_x \cos{\eta} + i \mathbf{e}_y \sin{\eta}$, the Dirac band gap changes with mode ellipticity degree $\tan{\eta}$. (b) HF bands of the $\pm \mathbf{K}$ valleys, from local and nonlocal interactions, for ellipticity degrees $\tan{\eta} = 1, 0.41~\text{and}~0.02$, corresponding to ellipticity angles $\eta = \frac{\pi}{4}$, $\frac{\pi}{8}$ and $\frac{\pi}{180}$, respectively. (c) Berry curvature for the valence band in (b). (d) Change of the Dirac band gap with mode amplitude ratio $\frac{A_{0y}}{A_{0x}}$ in the presence of two linearly polarized photon modes with $\mathbf{e} = \mathbf{e}_x$ and $\mathbf{e}_y$. The photon energy of the two modes is $\hbar\omega=0.3$ eV, and the amplitude of $x$-polarized mode is fixed as $A_{0x} = 2 \times10^{-8} \frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$. (e) HF bands of $\pm \mathbf{K}$ velleys, from local and nonlocal interactions, for the amplitude ratio $\frac{A_{0y}}{A_{0x}} = 0.0, 0.5,~\text{and}~1.0~$. (f) Berry curvature for the valence band in (e), where the cross signs indicate that the Berry curvature is singular for the valence band at the crystal momenta $\pm \mathbf{K}$ .
• Figure 5: Dirac states in graphene coupled with two isotropic linearly polarized photon modes. The two modes have the same $\hbar\omega=0.3$ eV and $A_0 = 2 \times 10^{-8} \frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$, but the perpendicular polarization vectors $\mathbf{e} = \mathbf{e}_x$ and $\mathbf{e}_y$. (a) HF bands for the $+\mathbf{K}$ valley. The lower panel shows the variation of the bands due to addition of the nonlocal interaction. (b) Representation of the blue band in (a) in $\{k_x , k_y\} \in [-1,1]$$10^{-3}$Å$^{-1}$ centered at $+\mathbf{K}$. (c) Component $|c_c|^2$ of the conduction basis state $\varphi_{c\mathbf{k}}^0$ for the lower valence band ($\varphi_{v\mathbf{k}} = c_v \varphi_{v\mathbf{k}}^0 + c_c \varphi_{c\mathbf{k}}^0$) in (b). (d) Variation of the electron density, $\Delta \rho (\mathbf{r}) = \sum_\mathbf{k} |\varphi_{v \mathbf{k}}|^2 - |\varphi_{v \mathbf{k}}^0|^2$, at $z = 0.33~\text{\AA}$ plane. The four panels show the contribution from the valence states with crystal momenta $-\mathbf{k}_0$, $+\mathbf{k}_0$, $+\mathbf{k}_0^\prime$ and $+\mathbf{k}_0^{\prime\prime}$. $-\mathbf{k}_0$ is inversely symmetric to $+\mathbf{k}_0$, and the $+\mathbf{k}_0^\prime$ and $+\mathbf{k}_0^{\prime\prime}$ are rotated by $120^{\circ}$ and $240^{\circ}$ with respect to $+\mathbf{k}_0$. (e,f) Evolution of the Fermi velocity along $+\mathbf{K} \rightarrow \mathbf{M}$ as a function of $A_0$ (with fixed $\hbar\omega=0.3$ eV) and $\hbar \omega$ (with fixed $A_0 = 2 \times 10^{-8} \frac{\text{kg} \cdot \text{m}}{\text{C} \cdot \text{s}}$), respectively.