Linear Ensemble-Constrained Quantum Monte Carlo: From Dilute Fermi Gas to Flat-Band Ferromagnetism

We present a finite-temperature canonical-ensemble determinant quantum Monte Carlo algorithm that enforces an exact fermion number and enables numerically stable simulations of correlated lattice fermions. We propose a stabilized QR update that reduces the computational complexity from standard cubic scaling O(βN^{3}) to linear scaling O(βN  N_{e}^{2}) with respect to the system size N, where N_{e} is the particle number. This yields a dramatic speedup in dilute regimes (N_{e}≪N), opening unbiased access to large-scale simulations of strongly correlated low-density phases. We validate the method on the dilute Fermi gas with onsite Hubbard interactions, observing the suppression of the fermion sign problem in the dilute limit. Furthermore, we apply this approach to a one-dimensional flat-band system, where the canonical ensemble allows for precise control over filling. We reveal a ferromagnetic instability at low temperatures in the half-filling regime. Our linear-scaling approach provides a powerful tool for investigating emergent phenomena in dilute quantum matter.