AFQMC

Auxiliary-field Quantum Monte Carlo (AFQMC) is a stochastic method for solving the non-relativistic quantum many-body problem, with particular use in electronic structure calculations for atoms, molecules, and solids. It projects the ground state by imaginary-time propagation using the operator exp(-tau H), where H is the many-body Hamiltonian.

The central idea is to decouple the two-body interaction via a Hubbard-Stratonovich transformation, introducing auxiliary fields that convert the evolution into a statistical average over one-body propagators. In AFQMC, the many-electron wavefunction is represented by a population of Slater-determinant walkers, and their amplitudes are updated by sampling the auxiliary fields at each imaginary-time step. Observables are estimated as averages over walkers with appropriate weights, using importance sampling and population control to manage statistical fluctuations.

A major challenge is the fermion sign (or phase) problem. AFQMC mitigates this with approximations such as constrained-path or phaseless constraints, which restrict walker evolution using a trial wavefunction. These constraints render the method efficient for a broad range of systems, at the cost of a controllable bias that depends on the choice of trial state. More exact, but computationally demanding, releases of the constraint can be used for benchmarking.

Variants include finite-temperature AFQMC and the phaseless AFQMC algorithm. The method relies on choices of trial wavefunction, basis set, and time-step discretization, which influence accuracy and efficiency.

Applications span electronic structure calculations for molecules and solids, including challenging transition-metal complexes and strongly correlated materials. AFQMC offers scalability and accuracy comparable to high-level quantum chemistry methods in many cases, while remaining computationally intensive and reliant on carefully chosen approximations. It has become a widely used tool in physics and chemistry research.