Constrained-path auxiliary-field quantum Monte Carlo for coupled electrons and phonons

We present an extension of constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) for the treatment of correlated electronic systems coupled to phonons. The algorithm follows the standard CP-AFQMC approach for description of the electronic degrees of freedom while phonons are described in first quantization and propagated via a diffusion Monte Carlo approach. Our method is tested on the one- and two-dimensional Holstein and Hubbard-Holstein models. With a simple semiclassical trial wave function, our approach is remarkably accurate for $\omega /\left(2dt\lambda \right)<1$ for all parameters in the Holstein model considered in this study where $d$ is the dimensionality, $\omega$ is the phonon frequency, $t$ is the electronic hopping strength, and $\lambda$ is the dimensionless electron-phonon coupling strength. In addition, we empirically show that the autocorrelation timescales as $1/\omega$ for $\omega /t\lesssim 1$, which is an improvement over the $1/{\omega }^2$ scaling of the conventional determinant quantum Monte Carlo algorithm. In the Hubbard-Holstein model, the accuracy of our algorithm is found to be consistent with that of standard CP-AFQMC for the Hubbard model when the Hubbard $U$ term dominates the physics of the model, and is nearly exact when the ground state is dominated by the electron-phonon coupling scale $\lambda$. The ap- proach developed in this work should be valuable for understanding the complex physics arising from the interplay between electrons and phonons in both model lattice problems and ab initio systems.

Figure 1

Born-Oppenheimer potential energy surfaces in units of $t$ for the two-electron and two-site Holstein model: (a) $\omega =t$, $\lambda =0.1$, and $g=0.447t$ and (b) $\omega =t$, $\lambda =1$, and $g=1.414t$. The minimum in (a) is $E=-2.40t$ at ($X_1=0.63$, $X_2=0.63$), while the two minima in (b) are $E=-8.25t$ at ($X_1=3.90$, $X_2=0.10$) and ($X_1=0.10$, $X_2=3.90$), respectively.

Figure 2

Error in the total energy in units of $t$ for the two-site two-electron Holstein model as a function of $\omega$ for various $\lambda$ values: (a) CSMP2, (b) LFPT2, and (c) AFQMC results. For $\lambda =0.1$, LFPT2 energy errors lie outside the plotted range. In (c), for $\lambda \ge 0.5$ AFQMC/TP(11) results are shown, while for $\lambda =0.1$ we present AFQMC/S results. Note the different vertical scales in (c).

Figure 3

Total energy per site in units of $t$ for the 20-site 20-electron 1D Holstein model as a function of $\omega$ for various values of $\lambda$: (a) $\lambda =0.1$ results, (b) $\lambda =0.3$ results, (c) $\lambda =0.8$ results, and (d) $\lambda =2.0$ results. Note that all error bars for AFQMC are too small to be seen on the plotted scales, DMRG reference values are unavailable for $\omega <0.8$ when $\lambda =0.8$ and all values of $\omega$ when $\lambda =2.0$, and AFQMC/TP(11) is not presented for $\lambda =0.1$ because the results for this trial wave function are nearly identical to those of AFQMC/S. The inset shows energy differences from DMRG for AFQMC/S (blue) and AFQMC/TP(11) (red) ($E_{\text{AFQMC}}-E_{\text{DMRG}}$).

Figure 4

Total energy per site in units of $t$ for the $4\times 4$ 2D Holstein model at half-filling as a function of $\omega$ for various $\lambda$ values: (a) $\lambda =0.1$ results, (b) $\lambda =0.3$ results, (c) $\lambda =0.5$ results, and (d) $\lambda =2.0$ results. Note that all error bars of CP-AFQMC/S are too small to be seen on the plotted scales, The inset shows energy differences between CP-AFQMC/S and DMRG ($E_{\text{AFQMC}}-E_{\text{DMRG}}$), for $\omega \ge 0.8$ when $\lambda <2.0$ where DRMG results are available.

Figure 5

Log-log plot of autocorrelation time ${\tau }_{\text{ac}}/\mathrm{\Delta }\tau$ and phonon frequency $\omega /t$ for $\lambda =0.1$ and $1.0$ for the $4\times 4$ 2D Holstein model. The black dotted lines are linear fits for each curve. For $\lambda =0.1$, the slope is $-0.9876$ with $R^2=0.9993$ and for $\lambda =1.0$, the slope is $-0.8617$ with $R^2=0.9942$.

Figure 6

The total energy per site in units of $t$ for the 20-site 1D Hubbard-Holstein model at half-filling with $U=4t$ as a function of $\lambda$ for various $\omega$ values: (a) $\omega =0.1$, (b) $\omega =0.4$, (c) $\omega =1.0$, and (d) $\omega =2.0$. Note that all error bars of CP-AFQMC are too small to be seen on the plotted scales. Note that the DMRG results are unavailable for $\omega =0.1$ and $0.4$ as well as for $\lambda >2$.

Figure 7

Error in total energy per site compared to DMRG in units of $0.001t$ for the 20-site 1D Hubbard-Holstein model at half-filling with $U/t=4$ as a function of $\omega$ for various $\lambda$ values: (a) $\lambda =0.1$ results and (b) $\lambda =1.0$ results. The black dotted line in (a) indicates the electronic CP-AFQMC energy error (i.e., $\lambda =0.0$). In (b), DMRG energies are only available for $\omega =1.0,1.5,2.0$.

Figure 8

Total energy per site in units of $t$ for the $4\times 4$ 2D Hubbard-Holstein model with $U/t=4$ as a function of $\omega$: (a) $\omega =0.1$ results at half-filling, (b) $\omega =2.0$ results at half-filling, (c) $\omega =0.1$ results at $\frac{1}{8}$ hole doping, and (d) $\omega =2.0$ results at $\frac{1}{8}$ hole doping. Note that all error bars of CP-AFQMC are too small to be seen on the plotted scales.