Time-Dependent Quantum Monte Carlo for fermions: from Bayesian conditioning to spinor dynamics

Abstract

The time-dependent quantum Monte Carlo method represents a many-electron state by an ensemble of replicas, in which each electron is described by a cloud of walkers which samples its density in physical space, one walker and one guide wave per replica, and it replaces the Hartree potential by a conditional interaction built from the walker positions. The sampling is in physical space rather than in configuration space, which keeps the cost polynomial. Here we derive that conditional interaction. Treating the walker as a localization of its electron to a finite resolution, we show that Bayes' theorem and a single empirical substitution yield the Nadaraya-Watson form which the method has used heuristically. The nonlocality length then becomes the width of the conditioning, and the pair and mean-field limits follow from one construction. Because the conditioning removes the gauge freedom which eliminates the orthonormality multipliers in Hartree-Fock, we enforce orthonormality by a term derived from the constraint. We generalize the formulation to spinors, where the Pauli suppression at coincidence becomes graded by the local spin alignment, and it reduces through the collinear and mean-field limits to the known two-particle spin equations. For two electrons in a double quantum dot in an inhomogeneous magnetic field, compared with the exact solution, the spinor equations reproduce the non-collinear spin geometry which collinear calculations cannot represent and recover the larger part of the entanglement missing from a single determinant. In the crossover between the exchange and the Zeeman regimes, where the singlet-triplet coherence sets the spin geometry, the replicas must carry that coherence themselves, as in the spin-coupled pair of valence-bond theory.