Phase diagram for the one-dimensional Hubbard-Holstein model: A density-matrix renormalization group study

Phase diagram of the Hubbard-Holstein model in the coexistence of electron-electron and electron-phonon interactions has been theoretically obtained with the density-matrix renormalization group method for one-dimensional systems, where an improved warm-up (the recursive sweep) procedure has enabled us to calculate various correlation functions. We have examined the cases of (i) the systems half-filled by electrons for the full parameter space spanned by the electron-electron and electron-phonon coupling constants and the phonon frequency, (ii) non-half-filled system, and (iii) trestle lattice. For (i), we have detected a region where both the charge and on-site pairing correlations decay with power laws in real space, which suggests a metallic behavior. While pairing correlations are not dominant in (i), we have found that they become dominant as the system is doped in (ii) or as the electronic band structure is modified (with a broken electron-hole symmetry) in (iii) in certain parameter regions.

Figure 1

The Hubbard-Holstein model. Each atom consists of an electron site (circle) and a local phonon with frequency ${\omega }_0$ (spring). The arrows denote up and down spin electrons, which are coupled to the phonons with a strength $\lambda$. An electron can hop between neighboring sites with the hopping amplitude $t$ and feel an on-site electron-electron repulsion U when encountered with another electron of opposite spin on the same site.

Figure 2

(Color online) Parameter space for the half-filled Hubbard-Holstein model spanned by the on-site electron-electron repulsion $U$, the phonon frequency ${\omega }_0$, and the electron-phonon coupling $\lambda$, where the unit of energy is the electronic transfer $t$. The model reduces to the Hubbard model when $\lambda =0$, and to the Holstein model when $U=0$. The ${\omega }_0∕t\to 0$ limit is the adiabatic limit, while the ${\omega }_0∕t\to \infty$ limit is the antiadiabatic limit.

Figure 3

(Color online) Log-log plot of correlation functions against the real-space distance for $(\lambda ,{\omega }_0)=(3.6,5.0)$ and $U=\left(\mathrm{a}\right)$ 8.0, (b) 3.6, and (c) 1.6 for the half-filled Hubbard-Holstein model.

Figure 4

(Color online) Exponents of correlation functions for the Hubbard-Holstein model with $(\lambda ,{\omega }_0)=(3.6,5.0)$.

Figure 5

(Color online) Phonon number $⟨n_{\text{phonon}}\left(i\right)⟩$ plotted against $i$ for $U=1.8,3.6,4.2$ and $(\lambda ,{\omega }_0)=(3.6,5.0)$ for a 64-site, half-filled Hubbard-Holstein chain.

Figure 6

(Color online) [(a)–(e)] Exponents of correlation functions determined by calculating exponents for a 20-site Hubbard-Holstein model for $\lambda ∕t=3.6$ with various values of ${\omega }_0∕t=0.05–5$. (f) The phonon number per site, normalized by Eq. (9).

Figure 7

(Color online) Correlation functions for $(U,\lambda ,\omega )=(2.1,3.6,2.0)$ on a 64-site, half-filled Hubbard-Holstein chain with $N_b=6$, where CDW correlation is dominant and almost constant.

Figure 8

(Color online) Charge gap, spin gap, and CDW amplitude plotted against $U$ for the half-filled Hubbard-Holstein model with $(\lambda ,{\omega }_0)=(3.6,2.0)$. Data obtained for $L=20,30,40,64$ have been linearly extrapolated to $1∕L\to 0$.

Figure 9

(Color online) (a) The summary phase diagram on the whole parameter space for the half-filled Hubbard-Holstein model as derived from (b) the results obtained for the $U\text{−}\lambda$ cross sections and (c) $U\text{−}\log {\omega }_0$ cross sections. The boundaries have been determined from the data points marked with dots.

Figure 10

(Color online) Correlation functions for a 64-site, hole-doped [$n=60∕64=0.94$ in (a) and $n=56∕64=0.88$ in (b)] Hubbard-Holstein chain with $(U,\lambda ,{\omega }_0)=(2.0,3.6,5.0)$. The exponents for the sSC correlation (dotted straight line) are estimated to be ${\eta }_{\mathrm{sSC}}=1.16\pm 0.02$ in (a) and ${\eta }_{\mathrm{sSC}}=1.09\pm 0.02$ in (b), while the CDW correlation to be ${\eta }_{\mathrm{CDW}}=1.30\pm 0.10$ in (a) and ${\eta }_{\mathrm{CDW}}=1.54\pm 0.13$ in (b).

Figure 11

(Color online) A trestle lattice (bottom) is equivalent to a chain that has the next-nearest-neighbor hopping $t^{\prime }$ (top).

Figure 12

(Color online) Correlation functions plotted against the real-space distance $r$ for a 64-site, half-filled Hubbard-Holstein model on the trestle lattice with $(U,\lambda ,{\omega }_0,t^{\prime })=(2.5,3.6,5.0,-0.4)$.

Figure 13

(Color online) Exponents of correlation functions plotted against $U$ for a 64-site, half-filled Hubbard-Holstein model on the trestle lattice for $(\lambda ,{\omega }_0)=(3.6,5.0)$ with $t^{\prime }∕t=-0.3$ (top panel) or $-0.6$ (bottom).

Figure 14

(Color online) Correlation exponents plotted against $-t^{\prime }$ for a 40-site, half-filled Hubbard-Holstein model on the trestle lattice with $(U,\lambda ,{\omega }_0)=(2.0,3.6,5.0)$. At least $m=630$ states have been retained in the last (tenth) sweep of the finite algorithm DMRG. The vertical dashed line represents the boundary at which the number of Fermi points changes from 2 to 4 as indicated by the band dispersion (top insets).