FFT-Accelerated Auxiliary Variable MCMC for Fermionic Lattice Models: A Determinant-Free Approach with $O(N\log N)$ Complexity

TL;DR

This work tackles the determinant bottleneck in fermionic lattice simulations by introducing a determinant-free MCMC that jointly samples worldlines and Hubbard-Stratonovich auxiliary fields. By deriving a momentum-space transfer kernel with a convolution structure, the authors exploit FFTs to achieve near-linear per-sweep complexity $O(N \log N)$ and implement FFBS and exact Gibbs updates for efficient, parallelizable sampling. Empirical validation on 2D Hubbard models up to $32\times32$ demonstrates results consistent with established theory and conventional $O(N^3)$ methods, with substantial speedups. The approach broadens the scope of scalable probabilistic inference for quantum many-body physics and inspires future work on continuous-time formulations and complex lattice geometries.

Abstract

We introduce a Markov Chain Monte Carlo (MCMC) algorithm that dramatically accelerates the simulation of quantum many-body systems, a grand challenge in computational science. State-of-the-art methods for these problems are severely limited by $O(N^3)$ computational complexity. Our method avoids this bottleneck, achieving near-linear $O(N \log N)$ scaling per sweep. Our approach samples a joint probability measure over two coupled variable sets: (1) particle trajectories of the fundamental fermions, and (2) auxiliary variables that decouple fermion interactions. The key innovation is a novel transition kernel for particle trajectories formulated in the Fourier domain, revealing the transition probability as a convolution that enables massive acceleration via the Fast Fourier Transform (FFT). The auxiliary variables admit closed-form, factorized conditional distributions, enabling efficient exact Gibbs sampling update. We validate our algorithm on benchmark quantum physics problems, accurately reproducing known theoretical results and matching traditional $O(N^3)$ algorithms on $32\times 32$ lattice simulations at a fraction of the wall-clock time, empirically demonstrating $N \log N$ scaling. By reformulating a long-standing physics simulation problem in machine learning language, our work provides a powerful tool for large-scale probabilistic inference and opens avenues for physics-inspired generative models.

Figures (2)

• Figure 1: Weak-coupling regime of the 2D Fermi-Hubbard model at $U/t = 1$, $\beta t = 32$ on a $32\times 32$ square lattice. Left: Momentum distribution $n_\mathbf{k}$ showing a sharp Fermi surface discontinuity at the non-interacting Fermi surface contour $\varepsilon(\mathbf{k}) = \cos(k_x) + \cos(k_y) = 0$ (white line), characteristic of Fermi liquid behavior. Middle: Real-space correlation function $C(\mathbf r)$ exhibiting long-range antiferromagnetic correlations with staggered pattern. Right: Spin structure factor $S(\mathbf{k})$ (with lattice size $8^2$, $16^2$, $32^2$) along $(0,0)\!\to\!(\pi,0)\!\to\!(\pi,\pi)\!\to\!(0,0)$ shows a sharp peak at $\mathbf{Q} = (\pi, \pi)$. The peak at $\mathbf Q$ increases with lattice size.
• Figure 2: Strong-coupling Mott insulating regime of the 2D Fermi-Hubbard model at $U/t = 20$, $\beta t = 32$ on a $32\times 32$ square lattice. Left: Charge distribution in the momentum space $n_\mathbf{k}$ in the Mott insulating regime. The absence of a sharp step across the noninteracting Fermi surface demonstrates the Mott character of the charge sector. Middle: Real-space spin correlation $C(\mathbf r)$ displaying a pronounced staggered pattern with exponential decay consistent with the effective Heisenberg description with $J_{\mathrm{eff}}=4t^2/U$. Right: Spin structure factor $S(\mathbf{k})$ (with lattice size $8^2$, $16^2$, $32^2$) along $(0,0)\!\to\!(\pi,0)\!\to\!(\pi,\pi)\!\to\!(0,0)$ shows a sharp peak at $\mathbf{Q} = (\pi, \pi)$. The peak at $\mathbf Q$ increases with lattice size.