Fokker-Planck equation governing the distribution of walkers in AFQMC

TL;DR

This work derives a Fokker-Planck equation that governs the evolution of the walker distribution in AFQMC on the Grassmann manifold, transforming the stochastic Langevin dynamics into a deterministic PDE with diffusion, drift, and birth/death terms. By solving the FP equation with a Grassmann-parameterization based on Givens rotations, the authors show that the wavefunction actually sampled by AFQMC can be inexact even when the guiding state is exact, and that the constraint introduces a bias tied to boundary behavior in determinant space. The study provides a concrete, deterministic method to reproduce AFQMC results for small Hubbard-model systems and reveals the structural origins of bias, offering pathways to systematic improvements such as alternative boundary conditions, improved Hubbard-Stratonovich mappings, or machine-learning representations of the manifold. Overall, the FP formulation opens new avenues for understanding and enhancing AFQMC accuracy by treating walker populations through a PDE on the Grassmann manifold rather than solely through stochastic sampling.

Abstract

Auxiliary-field quantum Monte Carlo (AFQMC) is typically formulated as an open-ended random walk in an overcomplete space of Slater determinants, implemented through a Langevin equation. However, the explicit form of the underlying Fokker-Planck equation governing the walker population distribution has remained unknown. In this paper, we derive the Fokker-Planck equation for AFQMC and propose a novel numerical scheme to solve it. The solution of the Fokker-Planck equation reveals the wavefunction actually sampled by the AFQMC algorithm. Interestingly, we find that even when the exact ground state is used as a guiding wavefunction in constrained path AFQMC, contrary to the common assumption, the wavefunction sampled by AFQMC is not exact. Beyond clarifying several fundamental aspects of AFQMC, the availability of a Fokker-Planck equation formulation opens new avenues for systematically improving its accuracy, which we outline in this paper.

Figures (8)

• Figure 1: A 2-d representation of the random walk that is taken in the space of walkers. The arrows tracing out each path represents imaginary time or the number of steps in AFQMC. The thick red square represents the nodal surface of the ground state, that is, the set of walkers that are exactly orthogonal to the ground state. Three walkers, all starting from the same initial state are shown by green trajectories following stochastic paths. When some of the trajectories come in contact with the node the future propagation is shown with dotted lines.
• Figure 2: For a two-dimensional Fokker-Planck equation, we discretize the domain such that no grid points lie exactly on the boundary. The boundary conditions are represented schematically by the colored dots. Along the $\theta_{12}$ axis the wavefunction satisfies anti-periodic boundary condition $\phi(\pi/2+\delta, \theta_{13}) = -\phi(-\pi/2+\delta, \theta_{13})$ as indicated by the horizontal arrows on the left/right boundary points. In contrast, along the $\theta_{13}$ axis the wavefunction obeys a more unusual boundary condition: $\phi(\theta_{12}, \pi/2+\delta) = -\phi(-\theta_{12}, -\pi/2+\delta)$ which is illustrated by the vertical arrows.
• Figure 3: The figure shows the Hubbard model on the left with a general one-electron matrix. On the right we show the result of constraint path AFQMC calculations as a function of increasing $U$ with the ground state of the Hamiltonian as the guiding wavefunction. We also show the number of node encounters per imaginary time step which is strongly correlated with the bias in energy.
• Figure 4: The graph shows the convergence of Fokker-Planck results with increasing discretization using solids curves. The dotted orange lines is the exact ground state energy and the green lines with brown error bars correspond to the results from cp-AFQMC simulation. The table on the right shows that the converged energy from Fokker-Planck equation agrees with the cp-AFQMC simulations up to an error bar. The exact state ground state ($|\psi_0\rangle$) and the cp-AFQMC states ($|\psi_{AF}\rangle$) are shown below the table in site basis.
• Figure 5: The left panel shows the surface plot of $\psi(\vec{\theta})$ and the right plot shows its values on the two boundaries. Note that the function remains non-zero on the two boundaries. The value of the function at $\theta\approx\frac{\pi}{2}$ is negative and a mirror image of its value at $\theta\approx-\frac{\pi}{2}$ reflecting the unusual boundary condition of the Grassmann manifold.