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

TL;DR

This work introduces Linear Ensemble Constrained Quantum Monte Carlo (LEC-QMC), a finite-temperature canonical-ensemble determinant QMC that exactly conserves particle number and uses a stabilized QR-update to achieve $O(\beta N N_e^2)$ scaling, enabling unbiased simulations of dilute, strongly correlated lattice fermions. The method employs an on-the-fly Fock-state update and particle-hole swaps to realize canonical sampling with numerical stability, and it demonstrates dramatic speedups and reduced sign problems in dilute regimes. The authors validate LEC-QMC on a dilute Hubbard system and apply it to a 1D flat-band model, where they prove ferromagnetic ground-state behavior via the Mielke-Tasaki mechanism and provide exact sublattice-spin fluctuation results. They also outline a constrained-path extension that decouples the ensemble constraint from the sign problem, broadening applicability to denser regimes and complex phases, with potential impact on moiré systems and ultracold atoms.

Abstract

We present a finite-temperature canonical-ensemble determinant quantum Monte Carlo algorithm that enforces an exact fermion number and enables stable simulations of correlated lattice electrons. 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 ($N_e \ll N$), opening unbiased access to large-scale simulations of strongly correlated low-density phases. We validate the method on the dilute electron gas with onsite Hubbard interactions, observing the suppression of the fermion sign problem in the dilute limit. Furthermore, we apply this approach to an 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.

Figures (6)

• Figure 1: Log-log plot of average CPU time per sweep vs. system size $N$. Parameters: $\Delta \tau = 0.1$, $U/t = 2.0$, $T/t = 1.0$, $N_e = 100$. The dashed lines show power-law fits, confirming the expected $O(N^3)$ scaling for DQMC (slope $\approx 3$) and the superior $O(N)$ scaling for LEC-QMC (slope $\approx 1$).
• Figure 2: Graph of the average sign $\langle \text{sign} \rangle$ versus system linear size $L$. When approaching the dilute limite $N = L^2 \gg N_e$, the system asymptotically has no sign problem.
• Figure 3: Illustration of the effective flat-band model. The onsite energy of the effective Hamiltonian ($t_1^2 + t_2^2$ for sublattice $A_1$ and $t_3^2 + t_4^2$ for sublattice $A_2$) is omitted for clarity. The gray shaded region indicates the spatial extent of a localized Wannier function, which overlaps with its neighboring Wannier function on two sites.
• Figure 4: Temperature evolution of (a) transverse and (b) longitudinal spin structure factors. The canonical ensemble constraint ($S^z_{\text{tot}} = 0$) confines the magnetic moment to the XY-plane, enforcing the sum rule $\langle S^{zz}(q=0)\rangle_{11} = \langle S^{zz}(q=0)\rangle_{22} = - \langle S^{zz}(q=0)\rangle_{12} = -\langle S^{zz}(q=0)\rangle_{21}$.
• Figure S1: The two fundamental update processes in the LEC-QMC method. (a) Adding a new particle (the orange column) to the system. We perform propagation on the new column. At every numerical stabilization step, we perform the modified Gram-Schmidt orthogonalization to get the QR decomposition of the new Fock state. (b) Removing a particle (the red column) from the system. We first apply a permutation to move the column to the last position, then apply Givens rotations at every numerical stabilization step to restore the triangular structure of the matrix. The QR decomposition of the new Fock state is obtained by truncating the last column and row of the matrices.