Determinant quantum Monte Carlo study of the two-dimensional single-band Hubbard-Holstein model

We have performed numerical studies of the Hubbard-Holstein model in two dimensions using determinant quantum Monte Carlo (DQMC). Here, we present details of the method, emphasizing the treatment of the lattice degrees of freedom, and then study the filling and behavior of the fermion sign as a function of model parameters. We find a region of parameter space with large Holstein coupling where the fermion sign recovers despite large values of the Hubbard interaction. This indicates that studies of correlated polarons at finite carrier concentrations are likely accessible to DQMC simulations. We then restrict ourselves to the half-filled model and examine the evolution of the antiferromagnetic structure factor, other metrics for antiferromagnetic and charge-density-wave order, and energetics of the electronic and lattice degrees of freedom as a function of electron-phonon coupling. From this we find further evidence for a competition between charge-density-wave and antiferromagnetic orders at half-filling.

Figure 1

The finite-temperature ($\beta =4/t$) phase diagrams for the two-dimensional Hubbard-Holstein model at half-filling. The vertical axis is the strength of the Hubbard interaction, while the horizontal axis is the strength of the Holstein interaction measured in dimensionless units (see text). The color scale in the upper panel gives the average value of the double occupancy per site. In the lower panel, it gives the spectral weight at the Fermi surface. For reference, the point of maximum spectral weight is shown in the upper panel and the line where the double occupancy is one quarter is shown in the lower panel. The red line indicates the line where $U_{\mathrm{eff}}=0$ in the antiadiabatic limit.

Figure 2

The average value of the fermion sign as a function of $e$-$ph$ coupling $\lambda$ for various half-filled clusters. The parameters for these calculations are $\beta =8/t$, $\Delta \tau =t/10$, $U=4t$, and $\Omega =t$.

Figure 3

The average value of the fermion sign as a function of filling for $\lambda =0.25$ ($\circ$), $0.5$ ($▿$), $0.7$ ($\square$), and $0.9$ ($\diamond$). Results are shown for two sets of phonon frequencies $\Omega =t$ (panel a) and $\Omega =4t$ (panel b). All data sets are for the strongly correlated limit with $U=8t$. These results were obtained on a $N=8\times 8$ cluster with $\Delta \tau =0.1/t$. The inverse temperatures for panels (a) and (b) are $\beta =4/t$ and $3/t$, respectively. The solid lines are guides to the eye.

Figure 4

(a) The average sign for $\langle n\rangle =1$ as a function of $\lambda$ for $\Omega =t/2$ ($\circ$), $t$ ($▿$), $2t$ ($\square$), and $4t$ ($\Delta$). The Hubbard interaction strength is held fixed at $U=6/t$, and the inverse temperature is $\beta =4/t$. (b) The average value of the fermion sign at half-filling as a function of the phonon frequency $\Omega$ and fixed $e$-$ph$ coupling $\lambda =0.25$. Results are shown for $U=4t$, $\beta =4/t$ ($\circ$), $U=4t$, $\beta =6/t$ ($▿$), and $U=6t$, $\beta =4/t$ ($\diamond$). All results in panels (a) and (b) were obtained on an $N=8\times 8$ cluster with $\Delta \tau =0.1/t$.

Figure 5

The average value of the filling $\langle n\rangle$ as a function of chemical potential $\mu -W\lambda$ for the same parameter sets shown in Fig. 3. The $-W\lambda$ correction accounts for the global shift of the lattice equilibrium position (see main text).

Figure 6

The compressibility $\kappa$ as a function of chemical potential for the same parameter set shown in Fig. 5 (Ref. 83).

Figure 7

The (a) spin ${\chi }_s(\pi ,\pi )$ and (b) charge ${\chi }_c(\pi ,\pi )$ susceptibilities for several values of $U$ on an $N=8\times 8$ cluster, reproduced from Ref. 73. The inset of (b) shows ${\chi }_s$ (dashed lines) and ${\chi }_c$ (solid lines) at $U=4t$ for several lattice sizes. The error bars in the inset have been suppressed for clarity. The remaining parameters are $\beta =4/t$, $\Delta \tau =0.1/t$, and $\Omega =t$.

Figure 8

The structure factor $S(\pi ,\pi )$ as a function of inverse temperature $\beta$ for the half-filled Hubbard-Holstein model. Results are shown for clusters of linear dimension (a) $N=4$, (b) $N=6$, and (c) $N=8$ and for several values of the electron-phonon coupling strength $\lambda$, as indicated. The remaining parameters are $U=4t$, $\Delta \tau =0.1t$, and $\Omega =t$.

Figure 9

The real-space structure of the spin-spin correlation function $c(l_x,l_y)$ along the path indicated in the inset. For $\lambda =0$ (black $▿$), the antiferromagnetic correlations are evident. For increasing values of $\lambda$, the antiferromagnetism is suppressed. By $\lambda =0.5$ (blue $▵$), where $\lambda W\sim U$, all traces of antiferromagnetic correlations are gone.

Figure 10

The real-space structure of the density-density correlation function along the path indicated in the inset. For $\lambda =0$ (black $▿$), the density of the system is uniform within error bars. This uniform density persists for increasing values of $\lambda \le 0.5$ (blue $▵$). However, for $\lambda =0.7>U/W$ (red $\square$), a $(\pi ,\pi )$ charge-density-wave correlation begins to develop.

Figure 11

The negative of the average electron kinetic energy as a function of the $e$-$ph$ interaction strength $\lambda$ and $U=4t$ (green $\circ$), $6t$ (blue $▵$), $8t$ (red $\square$), and $10t$ (black $\diamond$). The arrows indicate the value of coupling when $W\lambda =U$ for each data set.

Figure 12

(a) The average potential energy of the electrons due to the Hubbard interaction $P_{el}$ as a function of $\lambda$ and $U=4t$ (green $\circ$), $6t$ (blue $▵$), $8t$ (red $\square$), and $10t$ (black $\diamond$). (b) The corresponding average value of the double occupancy. The dashed line indicates the value expected for the noninteracting metallic system. The arrows indicate the value of coupling when $W\lambda =U$ for each data set.

Figure 13

The average (a) kinetic and (b) potential energy of the lattice for the Hubbard-Holstein model as a function of the $e$-$ph$ interaction strength $\lambda$ and $U=4t$ (green $\circ$), $6t$ (blue $▵$), $8t$ (red $\square$), and $10t$ (black $\diamond$).

Figure 14

The average $e$-$ph$ interaction energy as a function of the $e$-$ph$ interaction strength $\lambda$ and $U=4t$ (green $\circ$), $6t$ (blue $▵$), $8t$ (red $\square$), and $10t$ (black $\diamond$).

Figure 15

The average value of the lattice displacement $\langle X\rangle$ for the half-filled model as a function of $e$-$ph$ coupling $\lambda$. Results are shown for $\Omega =t$ (red $\circ$) and $\Omega =4t$ (blue $\square$). The remaining parameters are as indicated. The solid lines are of the form $\langle X\rangle =\sqrt{W\lambda }/\Omega$.