Pseudo-BCS wave function from density matrix decomposition: Application in auxiliary-field quantum Monte Carlo

We present a method to construct pseudo-BCS wave functions from the one-body density matrix. The resulting many-body wave function, which can be produced for any fermion systems, including those with purely repulsive interactions, has the form of a number-projected BCS form, or antisymmetrized geminal power (AGP). Such wave functions provide a better ansatz for correlated fermion systems than a single Slater determinant, and often better than a linear combination of Slater determinants (for example from a truncated active space calculation). We describe a procedure to build such a wave function conveniently from a given reduced density matrix of the system, rather than from a mean-field solution (which gives a Slater determinant for repulsive interactions). The pseudo-BCS wave function thus obtained reproduces the density matrix or minimizes the difference between the input and resulting density matrices. One application of the pseudo-BCS wave function is in auxiliary-field quantum Monte Carlo (AFQMC) calculations as the trial wave function to control the sign/phase problem. AFQMC is often among the most accurate general methods for correlated fermion systems. We show that the pseudo-BCS form further reduces the constraint bias and leads to improved accuracy compared to the usual Slater determinant trial wave functions, using the two-dimensional Hubbard model as an example. Furthermore, the pseudo-BCS trial wave function allows a new systematically improvable self-consistent approach, with pseudo-BCS trial wave function iteratively generated by AFQMC via the one-body density matrix.

Figure 1

Illustration of the density matrix decomposition to construct the pseudo-BCS wave function. The system is a $4\times 4$ Hubbard model, with $N_{\uparrow }=N_{\downarrow }=7$ (i.e., $1/8$ doping), $U=4t$, with periodic boundary condition and overall momentum $(0,\pi )$ or $(\pi ,0)$. “SD” indicates natural orbital, i.e., the result of a single Slater determinant constructed from the eigenvectors of the density matrix. “pBCS” indicates the pseudo-BCS wave function constructed from the procedure in Sec. 2b.

Figure 2

The density matrix $G_{i,j}$ computed from AFQMC, shown as a function of the distance $\vec{r}$ between sites $i$ and $j$ under PBC, compared with the exact result from ED. The inset shows the error $\delta G_{i,j}$ with respect to the ED result. The system and setup are the same as in Fig. 1. Statistical error bars are shown but are smaller than symbol size.

Figure 3

Convergence of the self-consistent AFQMC calculation using a trial wave function of the pseudo-BCS form versus a single Slater determinant from natural orbitals. The system is a $4\times 8$ Hubbard cylinder at half-filling, with pinning field applied on both edges, with $U=4t$ and $t^{\prime }=0.3t$. (a) The ground-state energy and (b) the mean squared error ${\delta }_{S^z}^2$ in the spin density with respect to DMRG. The SC procedures were repeated multiple times with different random number seeds to estimate the uncertainties in the convergence process, with the standard deviations shown by the shading on the curve.

Figure 4

Resolving different magnetic orders. The computed spin densities with self-consistent AFQMC, using either pseudo-BCS (SC AFQMC/pBCS) or single determinant (SC AFQMC/SD) form of trial wave functions, are shown in four systems with different magnetic correlations, together with the one-shot AFQMC/FE and reference DMRG results for comparison. All systems have $U=4t$ and are at half-filling, with cylindrical boundary condition and edge pinning fields applied; the system size and next-nearest-neighbor hopping value $t^{\prime }$ are indicated in each panel. All four panels use the same symbols as indicated in the top left panel. The inset in the lower right panel shows a magnified view of the middle region. The spin density is plotted versus site label along a cut in the $y$ direction of the cylinder, with $i_x=2$. For all systems, the results are statistically indistinguishable (after accounting for the alternating sign) with respect to $i_x$.