Computation of dynamical correlation functions for many-fermion systems with auxiliary-field quantum Monte Carlo

We address the calculation of dynamical correlation functions for many fermion systems at zero temperature, using the auxiliary-field quantum Monte Carlo method. The two-dimensional Hubbard hamiltonian is used as a model system. Although most of the calculations performed here are for cases where the sign problem is absent, the discussions are kept general for applications to physical problems when the sign problem does arise. We study the use of twisted boundary conditions to improve the extrapolation of the results to the thermodynamic limit. A strategy is proposed to drastically reduce finite size effects relying on a minimization among the twist angles. This approach is demonstrated by computing the charge gap at half filling. We obtain accurate results showing the scaling of the gap with the interaction strength $U$ in two dimensions, connecting to the scaling of the unrestricted Hartree-Fock method at small $U$ and Bethe ansatz exact result in one dimension at large $U$. An alternative algorithm is then proposed to compute dynamical Green functions and correlation functions which explicitly varies the number of particles during the random walks in the manifold of Slater determinants. In dilute systems, such as ultracold Fermi gases, this algorithm enables calculations with much more favorable complexity, with computational cost proportional to basis size or the number of lattice sites.

Figure 1

The calculation of imaginary-time correlation functions in an open-ended branching random walk: a sketch of the implementation.

Figure 2

Dynamical Green functions in real and momentum space, and the dependence of statistical errors on imaginary time. $G(R,\tau )$ was computed at $R=0$, and $G(Q,\tau )$ at a $Q$ close to the Fermi surface. The system was a $6\times 6$ lattice at half filling with $U/t=0.5$. In the main figure, statistical errors are much smaller than symbol size. The straight lines are exponential fits to the large imaginary time region (note semilog scale). The inset shows the dependence of the relative error bar on the imaginary time.

Figure 3

Charge gap measured from the dynamical Green function (filled squares) and from addition/removal (open circles), as a function of the inverse linear size of the system, $1/L$. The two panels are for two different interaction strengths: $U=0.5$ (upper) and $U=4$ (lower).

Figure 4

Charge gap measured from UHF calculations at $U=0.5$, as a function of $1/L$. The straight line indicates the UHF gap value at the thermodynamic limit, 0.0044272.

Figure 5

Finite-size effects in computing the gap, the use of twist boundary conditions, and special twist values. The top panel shows the noninteracting gap and the exact many-body gap as a function of the twist parameters. The bottom panel shows the corrected gaps and identifies the minimum. The system is a $14\times 14$ lattice at $U/t=0.5$.

Figure 6

Reduction of the finite-size effects and convergence to the thermodynamic limit in computing the charge gap. The charge gap at $U/t=0.5$ measured from the dynamical Green function is shown as a function of $1/L$, from a twist-averaging (TA) procedure together with the one-body correction (empty circles) and taking the minimum among the corrected gaps (filled squares). The dotted line is a quadratic fit to the twist-averaged data. The straight line is the estimation of the thermodynamic limit, obtained using the minimum gap estimator performing a linear fit. The inset shows the same data, together with the results from TA prior to the one-body correction also shown (filled circles) with those from periodic boundary conditions (PBC) (filled triangles), which contain large finite-size effects.

Figure 7

Charge gap at $U/t=1$ vs the inverse (linear) system size. The gases are measured from the dynamical Green function with twist-averaging and the one-body correction (empty circles) and with the minimum ${\mathrm{\Delta }}_{\min }$ (filled squares). The dotted line is a quadratic fit to the twist-averaged data. The straight line is the estimation of the thermodynamic limit, obtained using the minimum gap estimator.

Figure 8

Charge gap at $U/t=4$. Symbols and setup are the same as in Fig. 7.

Figure 9

Gap at half filling as a function of the interaction strength. Symbols are obtained from AFQMC calculations. Statistical error bars are shown but are smaller than symbol size. The (green) dashed line corresponds to a fit of the QMC data with a mean-field form allowing renormalized parameters. The (blue) dotted line is the actual mean-field result from unrestricted Hartree-Fock. The (orange) line at large $U$ is the Bethe ansatz prediction for one dimension. The inset shows a zoom of the main graph at small $U$.

Figure 10

Color plot of the spectral function $A(Q,\omega )$ as a function of momentum $Q$ (horizontal axis) along the principal directions in the Brillouin zone and frequency $\omega$ (vertical axis). The spectral function has been obtained by performing an analytic continuation of the calculated imaginary time Green functions in momentum space. The system was a $16\times 16$ lattice at $U/t=4$. The dotted line is the noninteracting dispersion relation.

Figure 11

Particle Green function $\tilde{G}(Q,\tau )$ for a large lattice in the repulsive Hubbard model: comparison between the alternative methodology and that described in Sec. 2b. A $12\times 12$ lattice with 72 spin-$\uparrow$ and 72 spin-$\downarrow$ particles is studied at $U/t=1$.