Spindrift: Learning quantum degeneracy from thermal purity in restricted path integral Monte Carlo

Abstract

Restricted path integral Monte Carlo (RPIMC) sidesteps the fluctuating Fermion sign problem by confining paths within nodal regions of a trial density matrix, thereby recovering polynomial scaling. However, this nodal surface must be provided from elsewhere; unless it is exact, it introduces a fixed-node energy error. Here we introduce \textsc{Spindrift}, a Variational Density Matrix approach that learns the many-body Fermionic density matrix from a regularised Bloch residual, evaluated on samples drawn by a restricted Worm algorithm. Motivated by the `purity' of quantum statistical mechanics at high temperature (where kinetic energy dominates), we train the density matrix along an imaginary-time (descending temperature) curriculum from an exact infinite-temperature heat-kernel starting point, learning the condensation of quantum correlations as temperature drops, through successive corrections to the previous reference. We parametrise our model with a permutation-equivariant continuous normalising flow to generate quasi-particle backflow trajectories, modulated by a symmetric Jastrow factor. This architecture guarantees exact Fermionic antisymmetry and spatial symmetry throughout training. Simulating $N=3$ interacting Fermions in a two-dimensional harmonic trap, we demonstrate stable curriculum training. The learnt velocity field smoothly deforms the nodal structure away from the free-particle reference. Open-Worm G-sector trapping provides a natural diagnostic for nodal accuracy. \textsc{Spindrift} systematically lowers the restricted energy relative to the free-particle reference across all temperatures and successfully reproduces the benchmark energy at $β=1$, establishing a stable, physics-informed framework for finite-temperature quantum Monte Carlo where the nodal structure is learnt self-consistently.