One-dimensional extended Hubbard model coupled with an optical cavity

TL;DR

This work addresses how vacuum fluctuations in a single-mode optical cavity influence the quantum phase transition between spin-density-wave and charge-density-wave states in the one-dimensional extended Hubbard model. It employs numerically exact tensor-network methods (DMRG/TEBD on matrix-product states) to compute ground-state and excited-state properties, revealing that the ground-state photon number $N_{\rm ph}$ and photon squeezing encode the SDW-CDW transition, with distinct behavior depending on the relative size of the coupling $G$ and the cavity frequency $\Omega$. Vacuum Rabi splitting manifests in the optical conductivity for small $G$, while the photon spectral function splits only when nearest-neighbor interactions $V$ are nonzero, explained by a Hopfield-like polariton framework with an effective coupling $G_{\rm eff}=G\sqrt{\langle \Delta J^2 \rangle_{GS}/N}$. The results demonstrate that quantum light can diagnose and control phase transitions in strongly correlated materials and point to experimental routes for observing exciton-polariton physics in cavity-embedded quantum systems.

Abstract

We study the one-dimensional extended Hubbard model coupled with an optical cavity, which describes an interplay of the effect of vacuum fluctuation of light and the quantum phase transition between the charge- and spin-density-wave phases. The ground state and excitation spectrum of the model are calculated by numerically exact tensor-network methods. We find that the photon number of the ground state is enhanced (suppressed) along the quantum phase transition line when the light-matter coupling is comparable to (much smaller than) the cavity frequency. We also show that the exciton peak in the optical conductivity and photon spectrum that exists without the cavity exhibits the vacuum Rabi splitting at resonance due to the light-matter interaction. This behavior is in contrast to the case without excitons, where the photon spectrum is merely broadened without splitting due to the lack of a sharp resonance.

Figures (9)

• Figure 1: Schematic picture of the one-dimensional extended Hubbard system confined in a cavity, where the quantized electromagnetic field with frequency $\Omega$ is coupled to electrons with the coupling constant $G$. $W$, $U$, and $V$ are the hopping amplitude, on-site interaction, and nearest-neighbor interaction, respectively. The SDW phase appears for $V \lesssim U/2$, while the CDW phase appears for $V \gtrsim U/2$.
• Figure 2: [(a), (b)] Photon number $N_{\rm ph}$ as a function of $U$ and $V$ for the one-dimensional extended Hubbard model coupled to an optical cavity with the coupling constant (a) $G=0.5$ and (b) $G=6$. [(c), (d)] Comparison of the photon number calculated from the DMRG (blue marks) and the perturbation combined with the squeeze transformation (red lines) as a function of $U/W$ with $V/W=2.5$ for (c) $G=0.5$ and (d) $G=6$. (e) The current fluctuation (top panel) and the kinetic energy (bottom panel) for the one-dimensional extended Hubbard model without a cavity. (f) Dependence of the photon number on the coupling constant $G$ plotted in the log scale. The results are obtained for the system size $N=80$ and the cavity frequency $\Omega/W = 10$.
• Figure 3: (a) $\Delta P/\Delta X$ and (b) $\Delta P\Delta X$ as a function of the on-site and nearest-neighbor interactions $U$ and $V$ for the one-dimensional extended Hubbard model coupled to an optical cavity with $N=80$, $\Omega/W=10$, and $G=6.0$.
• Figure 4: (a) Wigner function and (b) the photon distribution function calculated by DMRG for the one-dimensional extended Hubbard coupled to an optical cavity with $N=80$, $\Omega/W=10$, $G=6$, $U/W=10$, and $V/W=3$. [(c), (d)] As a comparison, we show the corresponding results given by the squeezed vacuum state with the squeeze factor $\zeta$ [Eq. \ref{['eq:zeta']}].
• Figure 5: Real part of the optical conductivity $\mathrm{Re}(\sigma(\omega))$ ((a), (b), (c), (d)) and photon spectral function $S(\omega)$ ((e), (f), (g), (h)) for the one-dimensional extended Hubbard model coupled with the light-matter interaction (blue curves, $G=0.5$) with different nearest-neighbor interactions $V/W=0,~3.5,~5.0$, and $7.5$. The cavity frequencies are set to $\Omega/W=10,~5.6,~2.1$ and $11.8$, respectively, which are chosen from the peak center position of the optical conductivity without the light-matter interaction (red curves, $G=0$). The results are obtained for the system with $N=40$, $G=0.5$ and $U/W=10$, and the classical field parameters are $A_p=0.01,~t_p=1,~\sigma_p=0.02$, and $\omega_p=10$. The damping parameter is $\eta=0.2$