Intermediate Phase of the One Dimensional Half-Filled Hubbard-Holstein Model

We present a numerical study of the Hubbard-Holstein model in one dimension at half filling, including finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-site interaction $U$ that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or Mott phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We confirm the intermediate phase, and show that the Luttinger liquid correlation exponent $K_{\rho }>1$ in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of Hubbard $U$, phonon coupling constant, and phonon frequency.

Figure 1

$U=0$, $\omega =1$ results for long-wavelength charge (open symbols) and spin (filled symbols) structure factors versus $g$ for periodic systems of $N=16$ (diamonds), $N=32$ (circles), and $N=64$ (squares) sites. Statistical errors are smaller than the symbols. The inset shows finite-size scaling of the critical coupling $g_c$ (indicated by arrow) where $K_{\rho }=1$.

Figure 2

Long-wavelength charge (open symbols) and spin (filled symbols) structure factors versus $g$. (a) $U=2$, $\omega =0.5$; arrow indicates finite-size scaled $g_c$. (b) $U=2$, $\omega =5$; arrows indicate finite-size scaled transitions $g_{c1}$ and $g_{c2}$.

Figure 3

Charge (a) and spin (b) stiffnesses versus $g$ for $U=2$, $\omega =5$. Open (closed) symbols are for $N=32$ (16) site lattices. Arrows mark the transition points $g_{c1}$ and $g_{c2}$ determined from Fig. 2b.

Figure 4

Evolution of the intermediate phase versus $\omega$. (a) $\omega =0.5$; (b) $\omega =1$; (c) $\omega =5$. Open circles indicate the boundary $g_{c1}$ between Mott and SC phases, filled diamonds the boundary $g_{c2}$ between SC and Peierls phases, and open squares the boundary between Mott and Peierls phases. Dashed line indicates $U=2g^2/\omega$.