Phase diagram of the one-dimensional Hubbard-Holstein model at half and quarter filling

The Hubbard-Holstein model is one of the simplest to incorporate both electron-electron and electron-phonon interactions. In one dimension at half filling, the Holstein electron-phonon coupling promotes on-site pairs of electrons and a Peierls charge-density wave, while the Hubbard on-site Coulomb repulsion $U$ promotes antiferromagnetic correlations and a Mott insulating state. Recent numerical studies have found a possible third intermediate phase between Peierls and Mott states. From direct calculations of charge and spin susceptibilities, we show that (i) as the electron-phonon coupling is increased, first a spin gap opens, followed by the Peierls transition. Between these two transitions, the metallic intermediate phase has a spin gap, no charge gap, and properties similar to the negative-$U$ Hubbard model. (ii) The transitions between Mott/intermediate and intermediate/Peierls states are of the Kosterlitz-Thouless form. (iii) For larger $U$, the two transitions merge at a tricritical point into a single first-order Mott/Peierls transition. In addition, we show that an intermediate phase also occurs in the quarter-filled model.

Figure 1

(Color online) Slope of the spin structure factor at wave vector $q_1=2\pi ∕N$ versus e-ph coupling $g$ for a 16 site half-filled system with $U=4$ and $\omega =1$. $\pi S_{\sigma }\left(q_1\right)∕q_1$ crossing 1 indicates the opening of a spin gap. Different symbols show the convergence with increasing phonon cutoff $N_p$.

Figure 2

(Color online) Integrated autocorrelation time for $N=16$, $U=2$, $\omega =1$, $\beta =32$, and $N_p=30$ as a function of e-ph coupling. Filled circles (squares) are the autocorrelation time for the charge (spin) structure factor $S\left(q_1\right)$ at $q_1=2\pi ∕N$. Open symbols are for the same observables, but calculated using quantum parallel tempering. Arrows indicate the location of the two transitions (see Sec. 3). We find that parallel tempering significantly reduces the autocorrelation time near the transitions.

Figure 3

(Color online) Finite-size scaling of the $q=\pi$ charge susceptibility for $U=0$ and $\omega =1$ at half filling. Data are for system sizes up to $N=128$ sites. In (a), we plot ${\chi }_{\rho }\left(\pi \right)∕N$ versus $N$. At critical coupling, ${\chi }_{\rho }\left(\pi \right)∕N$ approaches a constant for large $N$. Note that the $g=0$ curve corresponds to free fermions (no phonons). In (b), ${\chi }_{\rho }\left(\pi \right)∕N$ is plotted versus the effective e-ph coupling $2g∕\omega$, for system sizes $N=8$ (open circles), 16, 32, 64, and 128. The inset shows finite-size scaling of the transition point obtained by plotting the value of $2g^2∕\omega$ where ${\chi }_{\rho }\left(\pi \right)∕N$ for system size $N$ exceeds the susceptibility for system size $N∕2$. Line in the inset is a linear fit. We estimate the critical coupling as $2g_{c2}^2∕\omega \approx 1.00$ $(g_{c2}\approx 0.71)$.

Figure 4

(Color online) Long-wavelength spin and charge structure factor slopes for $U=0$ and $\omega =1$ at half filling. Open (filled) symbols are for charge (spin). Data are for system sizes of $N=16$, 32, 64, and 128 sites. For any $g>0$, $\pi S_{\sigma }\left(q_1\right)∕q_1$ is less than 1 indicating the presence of a spin gap. The inset shows the finite-size scaling of the point where $S_{\rho }\left(q_1\right)∕q_1$ crosses 1, with the line a fit to a quadratic. We estimate the critical coupling as $2g_{c2}^2∕\omega \approx 0.85$. The appearance of these data for $g<g_{c2}$ is similar to those for the negative-$U$ Hubbard model (Fig. 7 for $U<0$). The interpretation of these data is discussed in Sec. 3D.

Figure 5

(Color online) (a) Charge (open symbols) and spin (filled symbols) susceptibilities for the half-filled HHM with $U=2$, $\omega =1$. The first transition $\left(g_{c1}\right)$ occurs where ${\chi }_{\rho }={\chi }_{\sigma }$, corresponding to $U_{\mathrm{eff}}=0$. The second transition $\left(g_{c2}\right)$ is the Peierls transition, where ${\chi }_{\rho }\left(\pi \right)∕N$ diverges as in Fig. 3. Note that the spin susceptibility is also divided by $N$ to make the crossing at $U_{\mathrm{eff}}=0$ clear. (b) Long-wavelength spin structure factor for $U=2$, $\omega =1$. The point where $\pi S_{\sigma }\left(q_1\right)∕q_1$ crosses unity indicates the opening of the spin gap, identical to the point where ${\chi }_{\rho }={\chi }_{\sigma }$ in (a).

Figure 6

(Color online) First-order Mott/Peierls transition for large $U$. We show the CDW order parameter ${\left[m\left(\rho \right)\right]}^2$ (see text) vs $-U_{\mathit{eff}}=2g^2∕\omega -U$ for $\omega =1$ and $U=8$. The Mott/Peierls transition occurs for $U_{\mathrm{eff}}\sim -0.2$.

Figure 7

(Color online) LL exponents for the 1D Hubbard model [Eq. (1) with $g=0$] estimated from the long-wavelength charge and spin correlations. $K_{\rho }$ $\left(K_{\sigma }\right)$ is given by open (filled) symbols. In the infinite $N$ limit, $K_{\rho }=1$ for any $U<0$ and $K_{\rho }=0$ for $U>0$; $K_{\sigma }=1$ for any $U>0$ and $K_{\sigma }=0$ for any $U<0$. Observing these limiting values (shown by full and dashed horizontal lines) is difficult due to finite value of $q_1=2\pi ∕N$ and also the logarithmic scaling with $N$ for the exponent whose value is unity.

Figure 8

Phase diagram of the half-filled HHM for $\omega =0.5$, $\omega =1$, and $\omega =5$. The dashed line is given by $U=2g^2∕\omega$. All phase boundaries are determined using susceptibility and $K_{\sigma }$ data, with uncertainty approximately the size of the symbols. Lines are guides to the eyes. The three phases shown are Mott, (I)ntermediate, and Peierls. The Mott/I and I/Peierls boundaries merge into a single first-order Mott/Peierls boundary indicated by a heavy line for $U\gtrsim 4$ for $\omega =0.5$ and $U\gtrsim 5$ for $\omega =1$.

Figure 9

(Color online) (a) Charge (open symbols) and spin (filled symbols) susceptibilities for the quarter-filled HHM with $U=2$, $\omega =0.5$. (b) Long-wavelength spin structure factor for the same parameters. We find similar behavior to half filling, Fig. 5, with first a transition to a spin-gapped state, and second, the transition to the Peierls CDW state.

Figure 10

(Color online) Charge-charge correlations $⟨n_i{n}_j⟩$ versus distance $\mid i-j\mid$ for a 32 site quarter-filled system with $U=2$ and $\omega =0.5$. The three values of $2g^2∕\omega =1.69$, 2.25, and 2.89 correspond to LL, intermediate, and Peierls states, respectively. We find that in all three regions the charge correlations at quarter filling are of the form⋯2000⋯.