Effective band-projected description of interacting quasiperiodic systems

Abstract

We study the interplay between electronic interactions and quasiperiodicity in a one-dimensional narrow-band system, focusing on ground-state and low-energy excitation properties. Using band projection as low-energy effective approach, we show that a projection restricted to first order in the interaction strength fails to reproduce the correlated phase diagram. This contrasts with the standard success of first-order band projection in translationally invariant flatband systems and highlights the essential role of virtual processes involving remote bands in quasiperiodic settings. By incorporating second-order interband contributions perturbatively, we obtain an effective Hamiltonian that quantitatively reproduces the exact phase iagram previously obtained using density matrix renormalization group calculations, including the transition between a Luttinger liquid and a charge-density-wave phase and the crossover to a quasifractal charge-density-wave regime at strong quasiperiodicity. We further use this controlled framework to investigate low-energy neutral excitations and the optical conductivity, identifying clear dynamical signatures distinguishing the different phases. Our results establish second-order band projection as a reliable tool for correlated quasiperiodic narrow-band systems and suggest a promising route for studying interacting quasiperiodic and moiré materials beyond one dimension.

Figures (9)

• Figure 1: (a) Single-particle IPR of the model given by Eq. \ref{['eq:1']} in the non-interacting limit as a function of the quasiperiodic potential strength, $V_{2},$ and energy, $E$, for $L=842$, $\tau=475/842$ and $\phi=0$. (b) Hopping modulation of the model, which creates a moiré pattern of length $L_{M}\simeq7.8$. (c) Phase diagram of the model in the plane of interaction strength $U$ and intensity of quasiperiodic hoppings $V_{2}$, adapted from Ref.$\,$gonccalves2024incommensurability.
• Figure 2: Feynman diagrams for the effective Hamiltonian. FB, OC and E lines are depicted in black, blue and red, respectively. Time flows horizontally to the right.
• Figure 3: LL-CDW phase transition determined by the finite-size scaling analysis of: (a) the fidelity susceptibility ($\chi_{F}$), defined by Eq. \ref{['eq:7']} with $\delta U=1.25\times10^{-2}$; (b) the charge gap ($\Delta_{C}$), defined by Eq. \ref{['eq:8']}; (c) the IPR of the average of the density fluctuations in momentum space ($IPR_{K}(\langle\delta\boldsymbol{n}\rangle)$), defined by Eq. \ref{['eq:9']}. Results are averaged over 10 random configurations of $\phi$ and $\theta$.
• Figure 4: (a) Fidelity susceptibility ($\chi_{F}$) for $V_{2}=2$. (b) Schematic phase diagram indicating the phase space points that are used for computing the density of states and the optical conductivity in panel (c). (c) Density of states computed with a Gaussian width of $\sigma=10^{-4}$ and $N=94$ for the following phase-space points: $V_{2}=0.9$, $U=0.075$ (LL); $V_{2}=0.9$, $U=0.25$ (CDW); $V_{2}=2$, $U=0.1$ (qf-CDW); $V_{2}=2$, $U=0.275$ (CDW). Inset shows a close-up at low energies.
• Figure 5: Real component of the regular part of the optical conductivity, $\mathfrak{Re}[\sigma_{reg}(\omega)]$, Eq. \ref{['eq:11']}, computed for $N=94$, with $\delta\omega=2\times10^{-4}$, Lorentzian width of $\eta=$$2\times10^{-4}$ and random shift and twist, for: (a) extended states ($V_{2}=0.9,U=0$), LL ($V_{2}=0.9,U=0.075$) and CDW ($V_{2}=0.9,U=0.25$); (b) critical states ($V_{2}=2,U=0$), qf-CDW ($V_{2}=2$, $U=0.1$) and CDW ($V_{2}=2$, $U=0.35$).