Dimensional and doping stability of Peierls charge density waves

Abstract

The Peierls instability, the spontaneous dimerization of a one-dimensional metallic chain at half filling, is a paradigmatic mechanism for charge-density-wave (CDW) formation. Here we test its robustness under finite doping and interchain hybridization in finite-thickness arrays of identical chains. We find that the stacking geometry plays a decisive role in stabilizing CDW order away from half filling. In particular, parallel-coupled chains exhibit a bistable regime where the normal and dimerized states coexist as local minima of the total energy, while skew-coupled chains display reentrant CDW order upon doping. Our results demonstrate that even minimal models of coupled atomic chains host rich phase diagrams controlled by doping, lattice rigidity, and interchain coupling geometry.

Figures (5)

• Figure 1: (a) Total energy curves for six different values of $\kappa$ between $\kappa=0.5t^{-1}$ and $\kappa=t^{-1}$. Dots represent the position of the global minima of each curve, computed using Eq. (\ref{['Eq:Delta0']}). (b) Phase diagram where the normal (blue) and the Peierls CDW order (yellow) regions are separated by a red dashed curve given analytically in Eq. (\ref{['Eq:DeltaIsMinimum']}). The inset shows the two structural configuration we considered in this work.
• Figure 2: (a) Total energy of two parallel-coupled atomic chains for different values of $\gamma$. As the coupling increases, the total energy transitions from two to three minima before $\Delta=0$ establishes as the global minimum. (b) Phase diagram of two parallel-coupled chains. In between the phase separation, there is a green region denoting bistability. (c) Phase diagram as a function of doping for $\gamma=0.025t$. The level of doping at $\sim0.4\%$ that enhances the CDW order corresponds to the Fermi level at which a gap is opened in the band structure at the edge of the Brillouin zone $k=-\pi/a$, as shown in the inset.
• Figure 3: (a) Phase diagram of parallel-coupled chains as a function interchain coupling $\gamma$ and rigidity $\kappa$, with number of chains ranging from $N=3$ (leftmost) to $N=6$ (right most). Light blue represents a weak CDW order due to gapping of zero-energy bands that must exist for odd $N$, electron-hole symmetric spectra. All even-number chains share the same CDW/bistable regime phase boundary (see appendix \ref{['App:2N']}). (b) Phase diagram as a function of doping and rigidity, with $\gamma=0.025t$. All subbands open a gap at the edge of the Brillouin zone $k=-\pi/2a$. The doping level at these energy points maximizes the energy reduction of the system and enhances the CDW order.
• Figure 4: (a) Total energy $E_{\mathrm{tot}}(\Delta)$ of two skew-coupled chains for $\kappa=0.7t^{-1}$ and $\gamma=0.02t$, shown for several representative electron dopings. The evolution of the global minimum, from $\Delta\neq0$ to $\Delta=0$ and back to $\Delta\neq0$, demonstrates reentrant CDW order. (b) Phase diagram as a function of electron doping $n$ and rigidity $\kappa$. Inset: band structure near the Brillouin-zone edge, illustrating that CDW order is enhanced when the chemical potential lies close to a spectral gap (in particular at $n=0$ and $n\simeq0.45\%$).
• Figure 5: Phase diagrams of skew-coupled stacks with $N=3,\ldots,6$ chains for $\gamma=0.02t$, shown as a function of electron doping $n$ and rigidity $\kappa$. The colors denote the ground state obtained from the global minimum of $E_{\rm tot}(\Delta)$: normal state ($\Delta=0$), zig-zag CDW, and nematic CDW. Insets: corresponding band structures near the Brillouin-zone edge $k=-\pi/2a$. At low doping, the zig-zag configuration is favored, while at intermediate doping the nematic configuration becomes energetically preferred over an extended parameter range, consistent with the relative size and location of the spectral gaps.