High-order path-integral Monte Carlo methods for solving quantum dot problems

The conventional second-order path-integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of antisymmetric free-fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work we show that optimized fourth-order path-integral Monte Carlo methods, which use no more than five free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.

Figure 1

A two-propagator calculation using the exact, HO fermion propagator (2) in 2D. $E$ is the energy in units of $\hslash \omega$ and $\tau =\beta \omega$. Symbol sizes are larger than statistical error bars. For 70 to 100 fermions, the exact energies are 554, 676, 806, and 945. The results obtained here at the largest value of $\tau$ are 554.044(5), 676.045(9), 806.312(6), and 945.22(2), respectively. The residual errors are not statistical, but are due to software precision limiting the maximum value of $\tau$ that can be reached. See text for details.

Figure 2

The convergences of the Hamiltonian energy (H) vs the thermodynamics energy (TH) in a second-order PA PIMC calculation for $N=8$ polarized electrons at $\lambda =8$. The number is the number of PA propagators used in that calculation. The black triangle and red circles are the CW and Hamiltonian energies, respectively. The error bars are computed from 60 to 100 block averages of $5\times {10}^4$ configurations, and are mostly smaller than the plotting symbols. The black line is the PIMC result of Egger et al. [3].

Figure 3

Comparing the Hamiltonian energies from optimized fourth-order propagators vs the PA Hamiltonian energies from Fig. 2, for the same $N=8$ quantum dot. The optimized propagator results are labeled as “Best Bead” (BB), with 3–5 free-fermion propagators. B2 has no free-parameter to optimize the energy.

Figure 4

Convergence of the Hamiltonian energy for $N=20$ polarized electrons using the optimized, fourth-order three- and four-bead propagators. Error bars are computed from 200 to 300 block average of $5\times {10}^4$ configurations of all 20 electrons.