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Sliding phasons in Moiré Ladders

Paula Mellado, Francisco Muñoz, Javiera Cabezas-Escares

TL;DR

The work shows that a moiré potential arising from a relative leg shift in a four-band ladder flattens low-energy bands, enabling an interaction-driven incommensurate CDW with a rung-odd charge pattern. The CDW supports neutral, acoustic phasons whose velocity depends on the moiré parameter $\delta$ and the inter-leg tunneling, consistent with a sliding-CDW picture. Using mean-field theory and RPA, the authors identify the instability wavevector $Q^*$ set by moiré geometry and demonstrate a Goldstone phason mode emerging from the phase of the order parameter, later corroborated by DFT/DFPT studies on HfTe$_3$ that show corresponding miniband formation and chain-dependent distortions. Together, these results illuminate how structural incongruities and moiré patterns can promote excitonic, charge-ordered phases with tunable collective modes in layered materials, with potential applications in low-dissipation electronics and phonon-engineered devices.

Abstract

An incommensurate charge density wave is a periodic modulation of charge that breaks translational symmetry at a momentum that does not coincide with the primitive lattice vectors. Its Goldstone excitation, the phason, comprises collective gapless phase fluctuations. Aiming to unveil the mechanism behind the onset of incommensurate charge order in layered materials, we study a half-filled, four-band tight-binding model on a ladder with a relative shift $δ=p/q$ between the legs, induced by the dimerization of one of them. The shift results in a moiré supercell comprising q composite cells and a modulated inter-leg tunneling. The moiré potential compresses the leg bands into flat minibands near the Fermi level, resulting in additional low-energy peaks in the density of states. Including Coulomb interactions, we find an incommensurate charge-density-wave phase in which the charge modulation is out of phase between the legs. The collective excitations of this state are long-lived neutral, acoustic phasons whose speed is controlled by the moiré parameter $δ$ and the inter-leg tunneling amplitude. This model sheds light on the role of interlayer incongruities in the formation of excitonic charge-ordered phases in van der Waals and heterostructured materials.

Sliding phasons in Moiré Ladders

TL;DR

The work shows that a moiré potential arising from a relative leg shift in a four-band ladder flattens low-energy bands, enabling an interaction-driven incommensurate CDW with a rung-odd charge pattern. The CDW supports neutral, acoustic phasons whose velocity depends on the moiré parameter and the inter-leg tunneling, consistent with a sliding-CDW picture. Using mean-field theory and RPA, the authors identify the instability wavevector set by moiré geometry and demonstrate a Goldstone phason mode emerging from the phase of the order parameter, later corroborated by DFT/DFPT studies on HfTe that show corresponding miniband formation and chain-dependent distortions. Together, these results illuminate how structural incongruities and moiré patterns can promote excitonic, charge-ordered phases with tunable collective modes in layered materials, with potential applications in low-dissipation electronics and phonon-engineered devices.

Abstract

An incommensurate charge density wave is a periodic modulation of charge that breaks translational symmetry at a momentum that does not coincide with the primitive lattice vectors. Its Goldstone excitation, the phason, comprises collective gapless phase fluctuations. Aiming to unveil the mechanism behind the onset of incommensurate charge order in layered materials, we study a half-filled, four-band tight-binding model on a ladder with a relative shift between the legs, induced by the dimerization of one of them. The shift results in a moiré supercell comprising q composite cells and a modulated inter-leg tunneling. The moiré potential compresses the leg bands into flat minibands near the Fermi level, resulting in additional low-energy peaks in the density of states. Including Coulomb interactions, we find an incommensurate charge-density-wave phase in which the charge modulation is out of phase between the legs. The collective excitations of this state are long-lived neutral, acoustic phasons whose speed is controlled by the moiré parameter and the inter-leg tunneling amplitude. This model sheds light on the role of interlayer incongruities in the formation of excitonic charge-ordered phases in van der Waals and heterostructured materials.

Paper Structure

This paper contains 17 sections, 59 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Minibands in the RBZ
  4. Density of states and Lindhard Susceptibility
  5. Charge Density Waves
  6. Short Range Coulomb interactions in Mean Field Theory
  7. RPA susceptibility and Charge Density Waves
  8. Sliding Charge Density Waves
  9. Phason dynamics
  10. Moiré Phason
  11. DFT calculations and comparison with experiments
  12. Conclusions
  13. Rung harmonic $t_1$ from downfolding
  14. Details and Methods of DFT and DFPT calculations
  15. DFT
  16. ...and 2 more sections

Figures (9)

  • Figure 1: (a) Illustration of the moiré ladder system. (b) Band spectra of the kinetic Hamiltonian $H(k)$ in the EBZ, Eq.\ref{['eq:H4eff']}, at (from top to bottom), $\delta=1,\frac{19}{20},\frac{9}{10},\frac{17}{20}$. (c) Band spectra of the minibands $H_{\rm RBZ}$ in the RBZ, Eq.\ref{['eq:HRBZ']} at $\delta=\frac{19}{20}$.
  • Figure 2: DOS at T=0 and (a) $\delta=1$, (b) $\delta=1$ and no inter-leg hopping ($t_\perp =0$), (c) $\delta=\frac{19}{20}$ in the EBZ, (d) $\delta=\frac{19}{20}$ in the RBZ, (e) $\delta=\frac{17}{20}$ in the EBZ, (f) $\delta=\frac{17}{20}$ in RBZ.
  • Figure 3: Lindhard susceptibility $\chi_0(\kappa)$ at temperature (a) $T=\frac{t}{5}$ and (b) $T=5t$. (c) Site charge density associated with bands 1 and 2. (d) Largest eigenvalue $\lambda_{\max}^{(0)}(\kappa,T)$ of the bare density–density susceptibility matrix $\chi^{(0)}(\kappa,T)$ (blue), leg-1 projected (orange), leg-2 projected (green), at $T=\frac{t_1}{5}$. Q* is shown by the vertical dotted red line. In all figures $\delta=\frac{19}{20}$ and $\eta=10^{-5}$.
  • Figure 4: (a) Critical interaction $u_c$ versus temperature T at $\delta=\frac{19}{20}$. The inset shows a zoom in of the data at low temperatures. (b) Real space charge modulation $\delta n(x)$ at the sites of the unit cell (odd mode) at $Q*$.
  • Figure 5: Spectral function at $\delta=\frac{19}{20}$ and $T=1-\delta$ computed from (a) the single particle retarded Green function, Eq.\ref{['eq:spectral_scalar']} (b) the two-particle RPA susceptibility, Eq.\ref{['eq:spectralphason']}.
  • ...and 4 more figures