Renormalization of spectra by phase competition in the half-filled Hubbard-Holstein model

We present electron and phonon spectral functions calculated from determinant quantum Monte Carlo simulations of the half-filled two-dimensional Hubbard-Holstein model on a square lattice. By tuning the relative electron-electron $(e-e)$ and electron-phonon $(e-ph)$ interaction strengths, we show the electron spectral function evolving between antiferromagnetic insulating, metallic, and charge-density-wave (CDW) insulating phases. The phonon spectra concurrently gain a strong momentum dependence and soften in energy upon approaching the CDW phase. In particular, we study how the $e-e$ and $e-ph$ interactions renormalize the spectra and find that the presence of both interactions suppresses the amount of renormalization at low energy, thus allowing the emergence of a metallic phase at intermediate coupling strengths. In addition, we find a modest enhancement of the $d$-wave pairing susceptibility in the metallic regime, although spin and charge correlations are still dominant at the temperatures considered in our study. These findings demonstrate the importance of considering the influence of multiple interactions in spectroscopically determining any one interaction strength in strongly correlated materials.

Figure 1

(a)–(f) Spectral functions $A(\mathbf{k},\omega )$ along high-symmetry cuts through the Brillouin zone for various $e-ph$ interaction strengths $\lambda$. The solid red line indicates the noninteracting tight-binding band structure, and the dashed red line indicates the band shifted by ($\pi , \pi$). (g) Density of states $N\left(\omega \right)$ and (h) $A(\mathbf{k}=(0,0),\omega )$ for increasing $\lambda$ (bottom to top). The black squares in (h) denote the maxima of the spectra. Note that the spectra in (g) and (h) are artificially offset for clarity. The remaining simulation parameters are $U=6t, \Omega =t, \beta =4/t$, and $N=8\times 8$.

Figure 2

Real part of the self-energy ${\Sigma }^{\prime }(\mathbf{k},\omega )$ along high-symmetry cuts through the Brillouin zone for (a) $\lambda =0$, (b) $\lambda =0.5$, (c) $\lambda =0.8$, and (d) $\lambda =1$. (e)–(h) Real part of the self-energy at $\mathbf{k}=(\pi /2,\pi /2)$ for $\lambda =0.5-0.8$. The remaining simulation parameters are $U=6t, \beta =4/t, \Omega =t$, and $N=8\times 8$.

Figure 3

${\lambda }_{\mathrm{eff}}$ as defined in Eq. (5) as a function of $e-ph$ coupling strength $\lambda$ with the $e-e$ interaction set to $U=6t$. Other simulation parameters are $\beta =4/t, \Omega =t, N=8\times 8$.

Figure 4

Temperature dependence of the spin and charge susceptibilities ${\chi }_s\left(\mathbf{q}\right)$ and ${\chi }_c\left(\mathbf{q}\right)$ for $U=6t$ evaluated in several ways. (a) ${\chi }_s(\pi ,\pi )$ (open symbols) and ${\chi }_c(\pi ,\pi )$ (solid symbols) as a function of $\lambda$. (b) ${\chi }_s\left(\mathbf{q}\right)$ for $\lambda =0$, and (c) ${\chi }_c\left(\mathbf{q}\right)$ for $\lambda =0.9$ for several temperatures. The points are the DQMC data, while the lines are Lorentzian fits. (d) Coherence length $\xi$ of ${\chi }_s\left(\mathbf{q}\right)$ and ${\chi }_c\left(\mathbf{q}\right)$ from (b) and (c) as a function of temperature. DOS $N\left(\omega \right)$ for (e) $\lambda =0$ and (f) $\lambda =0.9$. Other simulation parameters are $\Omega =t$ and $N=8\times 8$.

Figure 5

Spectral functions $A(\mathbf{k},\omega )$ for high-symmetry cuts through the Brillouin zone for (a) $\Omega =t$, (b) $\Omega =1.5t$, and (c) $\Omega =2t$. The remaining simulation parameters are $U=6t, \lambda =0.9, \beta =6/t$, and $N=8\times 8$.

Figure 6

(a)–(e) Phonon spectral function $B(\mathbf{q},\omega )$ along high-symmetry Brillouin-zone cuts for various $e-ph$ strengths $\lambda$. The maximum at each $\mathbf{q}$ is indicated by blue squares. (f) The integrated phonon spectral weight as a function of $\lambda$ for several momentum points. (g) the phonon density of states $N_{ph}\left(\omega \right)$. The remaining simulation parameters are $U=6t, \beta =4/t, \Omega =t$, and $N=8\times 8$.

Figure 7

Low-energy spectral weight $\beta G(\mathbf{r}=0,\tau =\beta /2)$ as a function of $\lambda$ at various temperatures for phonon frequencies $\Omega$ equal to (a) $t/2$, (b) $t$, and (c) $2t$. The remaining simulation parameters are $U=5t$ and $N=8\times 8$.

Figure 8

$d$-wave pairing susceptibility at several temperatures for $U$ equal to (a) $2t$, (b) $4t$, (c) $5t$, and (d) $6t$. $s$-wave pairing susceptibility for $U$ equal to (e) $2t$, (f) $4t$, (g) $5t$, and (h) $6t$. Other simulation parameters are $\Omega =t$ and $N=8\times 8$.

Figure 9

Spectra obtained from clusters with different sizes $N$ at various momentum space points and $e-ph$ coupling strengths. Note that $(\pi ,0)$ is accessible from all three clusters considered, while the $N=8\times 8$ results in (d)–(f) and the $N=10\times 10$ results in (d) and (f) are interpolated. The remaining simulation parameters are $U=6t, \Omega =t$, and $\beta =4/t$.