Time-Dependent Quantum Monte Carlo for fermions: from Bayesian conditioning to spinor dynamics
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 is what keeps the cost polynomial. Here that conditional interaction is derived rather than postulated. The method has so far been applied mostly to opposite-spin electrons, where the exchange is dormant rather than absent, and the present formulation addresses the regime in which it is active. Treating the walker as a localization of its electron to a finite resolution, Bayes theorem and a single empirical substitution yield the Nadaraya-Watson form which the method has used heuristically. The nonlocality length thereby acquires a meaning as the width of the conditioning rather than as a fitted coupling, and the pair and mean-field limits follow from one construction. For fermions, exchange cannot be carried by the walkers and stays in the wave sector, which is what leaves the positive walker sampling free of the sign problem. Because the conditioning removes the gauge freedom which eliminates the orthonormality multipliers in Hartree-Fock, orthonormality is enforced here by a term derived from the constraint. The formulation is generalized 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.