The light-matter correlation energy functional of the cavity-coupled two-dimensional electron gas via quantum Monte Carlo simulations

TL;DR

The paper tackles the lack of ab initio methods for describing cavity-induced light-matter coupling in bulk systems by applying a quantum electrodynamical auxiliary-field quantum Monte Carlo (QED-AFQMC) method to a minimal, yet representative, cavity-coupled two-dimensional electron gas in a modulating potential. It introduces a finite-size mitigation strategy using twist-averaged boundary conditions, enabling reliable extraction of the light-matter correlation energy $E_{c,el-ph}$ in the thermodynamic limit and providing a numerical foundation for QED density functional theory (QEDFT) functionals. The authors demonstrate that a modified weak-coupling perturbation theory with a constant renormalized frequency $\tilde{\Omega}$ describes the data well over a broad range, while the infinite-coupling limit can be captured by the Lang-Firsov transformed Hamiltonian $H_{\infty}$, yielding a finite correlation energy despite strong coupling. They further propose a simple parameterization of $E_{c,el-ph}$ as a functional of the cavity parameters and electronic density, $E_{c,el-ph} \approx - \frac{\mathcal{Q}^2}{c_1 + c_2 \rho^{2/D} \frac{V_c \tilde{\Omega}}{N \boldsymbol{\epsilon}^2}}$ with fitted $c_1=4.672$ and $c_2=63.73$, offering a practical ingredient for QEDFT and enabling extensions to 3D, multi-mode cavities, and inclusion of Coulomb interactions in future work. Overall, the work provides key benchmarks and a concrete energy functional pathway for quantitatively modeling light-matter effects in quasi-2D materials.

Abstract

We perform extensive simulations of the two-dimensional cavity-coupled electron gas in a modulating potential as a minimal model for cavity quantum materials. These simulations are enabled by a newly developed quantum-electrodynamical (QED) auxiliary-field quantum Monte Carlo method. We present a procedure to greatly reduce finite-size effects in such calculations. Based on our results, we show that a modified version of weak-coupling perturbation theory is remarkably accurate for a large parameter region. We further provide a simple parameterization of the light-matter correlation energy as a functional of the cavity parameters and the electronic density. These results provide a numerical foundation for the development of the QED density functional theory, which was previously reliant on analytical approximations, to allow quantitative modeling of a wide range of systems with light-matter coupling.

Figures (3)

• Figure 1: Electron-photon correlation energy of the cavity-coupled electron gas in the modulated potential. (a) Sketch of the model with electron gas in a soft external potential $v_\text{ext}$. The polarization of the cavity mode is in one of the lattice directions. (b) The unreasonable effectiveness of weak-coupling perturbation theory with $\tilde{\Omega}=\text{const}$ (PT$_{\tilde{\Omega}}$) compared to leaving $\Omega=\text{const}$ (PT$_{\Omega}$) and QMC data for $v/\rho = 2.7$ and $\Omega/\rho=0.9$. (c-d) Fit (solid lines) of the light-matter correlation energy $E_\text{c,\,el-ph}$ to slices of the parameter space, spanned by the average electron density $\rho=1/a^2$, the potential depth $v$, the frequency $\Omega$, and the light matter coupling $|\boldsymbol{\epsilon}|$. Black dashed lines correspond to the weak-coupling perturbation theory (PT$_{\tilde{\Omega}}$), while black circles are AFQMC results of the infinitely light-matter coupled asymptotic model, which was not included in the fit. The black solid line shows the extrapolation of the fit to infinite coupling. The errorbars are smaller than the symbol size.
• Figure 2: Toroidal magnetic finite-size effects. In both panels, $a = 3\,a_0$, $\Omega = 0.05\,\text{Ha}$, and $N=N_\text{uc}$ is the number of electrons and unit cells. (a) The low-energy spectrum of the one-dimensional cavity-coupled electron gas computed with exact diagonalization. At finite light-matter coupling, the ground state, $E_\text{ED}$, is a doublet belonging to total crystal momentum $K = \pm 4\pi/L = \pm \pi/a$ (yellow circles) and correspondingly finite $\braket{\mathbf A}$. The first excited state, $E_\text{ED}^{\braket{\vb{A}}=0}$, is a doublet with $K=0$. The Hilbert space was constrained to 21 momenta and $n_\text{ph} < 20$, which is converged to the complete basis set limit. (b) A comparison of the finite-size scaling of the two-dimensional light-matter correlation energy derived from different subtraction schemes, obtained from AFQMC. $\overline{E^\kappa - E^\kappa_\text{CS}}$ denotes the average over 60 twisted boundary conditions.
• Figure 3: Average density gradient fluctuations in the polarization direction, $\mathcal{Q}^2$, of the modulated electron gas, as a function of the squared potential depth $v^2$. Obtained from a noninteracting calculation. The solid line is a fit as described in the main text.