Single-particle excitations and phonon softening in the one-dimensional spinless Holstein model

We investigate the influence of the electron-phonon coupling in the one-dimensional spinless Holstein model at half-filling using both a recently developed projector-based renormalization method (PRM) and an refined exact diagonalization technique in combination with the kernel polynomial method. At finite phonon frequencies the system shows a metal-insulator transition accompanied by the appearance of a Peierls distorted state at a finite critical electron-phonon coupling. We analyze the opening of a gap in terms of the (inverse) photoemission spectral functions which are evaluated in both approaches. Moreover, the PRM approach reveals the softening of a phonon at the Brillouin-zone boundary which can be understood as precursor effect of the gap formation.

Figure 1

Low-energy excitations of the single-particle spectral function $A_K^{+}\left(\omega \right)$ and the corresponding integrated spectral density $[S_K^{+}\left(\omega \right)={\int }_{-\infty }^{\omega }{A}_K^{+}\left({\omega }^{\prime }\right)d{\omega }^{\prime }]$ at $K=\pm \pi ∕2$ for ${\epsilon }_p∕t=0.6$ and ${\omega }_0∕t=0.1$. Dashed (solid) lines show the spectrum including all phonon modes (without the $Q=0$ phonon mode).

Figure 2

(Color online) Wave-number-resolved spectral densities for photoemission[$A_K^{-}\left(\omega \right)$; dashed (red) lines] and inverse photoemission [$A_K^{+}\left(\omega \right)$; solid (black) lines] in the metallic state $({\epsilon }_p∕t⪡1)$. The corresponding integrated densities $S_K^{\pm }\left(\omega \right)$ are given by bold lines. All ED data were obtained for an eight-site system with periodic boundary conditions. Note that the sum rules are fulfilled.

Figure 3

(Color online) PE [dashed (red) lines] and IPE [solid (black) lines] spectra near the metal to Peierls insulator transition point.

Figure 4

(Color online) PE and IPE spectra in the Peierls phase.

Figure 5

(Color online) Phonon distribution in the ground state of the spinless Holstein model for three characteristic coupling strengths.

Figure 6

Results from the PRM approach for $g∕t=0.10$ $({\epsilon }_p∕t=0.1)$: (a) Electronic spectral functions $A_k^{+}\left(\omega \right)$ (black) and $A_k^{-}\left(\omega \right)$ (red) where the wave number $k$ is fixed to $k_F=\pi ∕2$. The coherent excitation peak (dotted line) is at $\omega =0$. (b) Renormalized phonon dispersion ${\tilde{\omega }}_q$. The original phonon frequency is ${\omega }_0∕t=0.1$.

Figure 7

Same quantities as in Fig. 6 from the PRM approach for a $g$-value of $g∕t=0.30$ $({\epsilon }_p∕t=0.9)$ somewhat below the critical value $g_c∕t\approx 0.31$ $({\epsilon }_p∕t=0.96)$.

Figure 8

Same quantities as in Fig. 6 from the PRM approach for $g∕t=0.34$ $({\epsilon }_p∕t=1.16)$ which is somewhat larger than $g_c∕t\approx 0.31$ $({\epsilon }_p∕t=0.96)$. Note that the coherent pole has vanished and a charge gap has opened (a).

Figure 9

Coherent pole strength ${\tilde{\alpha }}_{k_F}^2$ from the PRM approach as function of the electron-phonon coupling $g$ for a system with 1000 lattice sites.