Ground state and excitations of quasiperiodic 1D narrow-band moiré systems: a mean field approach

TL;DR

This work addresses how quasiperiodicity and interactions shape ground states and excitations in a 1D narrow-band moiré model. Using a variational mean-field (Hartree–Fock) approach, the authors construct a phase diagram and compare with exact DMRG results to show accurate reproduction of the critical, multifractal CDW phase, including a quasifractal CDW characterized by many wavevectors $K_n = \pi+2\pi\tau n$. They also develop a real-space time-dependent Hartree-Fock (tdHF) / RPA framework to access the excitation spectrum and generalized susceptibilities, uncovering sub-gap collective modes in the extended regime and the absence of a zero-energy phason in the critical regime. The results support the validity of mean-field treatments for systems hosting multifractal critical states and suggest that such approaches can inform understanding of correlated phases in higher-dimensional moiré materials such as twisted bilayer graphene.

Abstract

We demonstrate that a mean field approximation can be confidently employed in quasiperiodic moiré systems to treat interactions and quasiperiodicity on equal footing. We obtain the mean field phase diagram for an illustrative one-dimensional moiré system that exhibits narrow bands and a regime with non-interacting multifractal critical states. By systematically comparing our findings with existing exact results, we identify the regimes where the mean field approximation provides an accurate description. Interestingly, in the critical regime, we obtain a quasifractal charge density wave, consistent with the exact results. To complement this study, we employ a real-space implementation of the time-dependent Hartree-Fock, enabling the computation of the excitation spectrum and response functions at the RPA level. These findings indicate that a mean field approximation to treat systems hosting multifractal critical states, as found in two-dimensional quasiperiodic moiré systems, is an appropriate methodology.

Figures (10)

• Figure 1: Mean field phase diagram for the quasiperiodic case. The white line marks the points where the Fourier transform of the charge density fluctuations change from extend to localized behavior. Each color maps the magnitude of the order parameter, $\mathcal{O}_{CDW}=\max\boldsymbol{n}-\min\boldsymbol{n},$ where $\boldsymbol{n}$ is the vector of the charge density. $\pi$-CDW corresponds to the Charge Density Wave with order only at $k=\pi.$ QPM-CDW to the Quasiperiodic moiré charge density wave, where a finite number of wave vectors are present. Quasi-fractal corresponds to the regime of the charge density wave where an extremely large number of wave-vectors are present in the fluctuations. A system with size $N=504$ and modulation period $\tau=\frac{293}{504}$ was used.
• Figure 2: a) Mean field phase diagram for the periodic case. Each color maps the magnitude of the order parameter, $\mathcal{O}_{CDW}$. The horizontal dashed lines mark the value of the modulation strength where the bands become narrower, with a higher density of states, thus reducing enhancing the order paramter. $\pi$-CDW corresponds to the Charge Density Wave with order only at $k=\pi.$ PM-CDW to the Periodic moiré Charge Density Wave, where a finite number of possible wavevectors (in this case 12) are present in the charge density fluctuations. b) Density of states of the non-interacting Hamiltonian, at the Fermi level as function of the periodic modulation strength, $V_{2}$. A system size $N=480$ and modulation period $\tau=\frac{7}{12}$ was used.
• Figure 3: Mean field results of the CDW phase for a system size of $N=112$, for different quasiperiodic modulation strengths. a) Real space charge density modulation for $V_{2}=0.1$ and $U=1$. b) Fourier transform of the fluctuations of the charge density for $V_{2}=0.1$ and $U=1$. The vertical dashed lines correspond to the position of the peaks, $K_{n}=\pi+2\pi\tau n$. d)Real space charge density modulation for $V_{2}=2.0$ and $U=0.1$. e) Fourier transform of the fluctuations of the charge density for $V_{2}=2.0$ and $U=0.1$. Vertical dashed lines correspond to the position of the peaks, $K_{n}=\pi+2\pi\tau n$. Panels c) and f) correspond to the finite size scaling analysis for selected peaks of panels b) and e), respectively. The selected peaks correspond to $K_{m}=\pi+2\pi\tau m,$ with $m=0,5,10,20,40$.
• Figure 4: IPR of the charge density fluctuations as a function of the interaction strength for quasiperiodic modulation a) $V_{2}=0.5$ and b) $V_{2}=2.0$, for different chain sizes. The inset shows a detailed finite size scaling analysis for a selected value of the interaction, marked as a dashed line in panel a).
• Figure 5: a) Charge distribution for a periodic system, with $\tau_{c}=\frac{7}{12},$for $U=1$ and $V_{2}=1$, where the charge distribution has a periodicity of $12$ atoms. b) Fourier transform of the charge fluctuations. The dashed lines correspond to $K_{n}=\pi+2\pi\tau_{c}n$ with $n=-5,\cdots,6$.