Tackling the Sign Problem in the Doped Hubbard Model with Normalizing Flows

We train a pair of autoregressive models to construct zero-mean control variates to mitigate the sign problem in quantum Monte Carlo simulations. The two autoregressive networks are confined to the positive- and negative-sign sectors with strictly disjoint support, and each is exactly normalized over its sector. Their difference is therefore structurally zero-mean, providing an unbiased auxiliary observable whose correlation with the sign estimator controls the variance reduction. We implement the method within the stochastic series expansion framework, which we extend to frustrated lattices by developing an incremental loop-topology update. Sign-ergodic sampling is achieved through a twist channel, which is the unique sign-changing mechanism on non-bipartite lattices. We implement the control variates as autoregressive transformers with an end-of-sequence parity mask that enforces exact sign-sector resolution, while the incremental loop-count change and cumulative frustration parity are incorporated as topological features. On the triangular-lattice Heisenberg antiferromagnet, we benchmark the method in the small-$N$ limit. The control variate reduces the standard error of the average sign by up to an order of magnitude and that of the energy estimator by a factor of three to five, remaining effective even when the average sign drops below $10^{-3}$. This work lays out the framework and provides a proof-of-principle demonstration that autoregressive control variates can effectively mitigate the sign problem. Scaling to larger systems with physics-informed architectures is the subject of future work.

We introduce a novel technique for scalable sampling of spin-system states with continuous symmetries using diffusion models. By applying our approach to the XY model, a fundamental continuous-spin model in condensed matter physics, we show that our technique addresses the shortfalls of the Markov chain Monte Carlo (MCMC) in generalization to varying system sizes. More specifically, we show that training a temperature-conditioned diffusion model on smaller-size XY model lattices enables the generation of accurate samples in larger lattice sizes. By tracking physically important observables of the model, such as spin correlations, our experiments demonstrate that diffusion sampling followed by a few MCMC steps reduces the thermalization time by an order of magnitude relative to the standard MCMC with random initialization. Our study provides valuable insight as to how generative models can be used to study continuous-state condensed matter systems at scale.

Learned dynamical proposals can generate configurations without providing a tractable endpoint density. Nonequilibrium path probabilities offer a way to correct such proposals, but correction alone does not determine their ability to connect separated regions. We introduce Neural Non-Equilibrium Hamiltonian Monte Carlo (NHMC), which combines conditional momentum distributions with reversible, volume preserving dynamics. The forward--reverse path ratio gives the work used for training, importance weighting, normalizer estimation, and Metropolis correction. We then construct a configuration space round-trip kernel whose reverse and forward paths share an intermediate configuration. Conditional on that configuration, its path record update is independence Metropolis--Hastings. We compare its stationary inter-region flow with the flow obtained under exact conditional matching and bound their difference by the conditional path mismatch. This separates errors in the conditional proposal from dependence between the endpoint region and the intermediate configuration. Many well calculations test normalizers and mode probabilities; a controlled lattice $φ^4$ calculation compares ideal and accepted inter-sector flows and measures sector relaxation. In two dimensional $U(1)$, $\mathrm{SU}(2)$, and $\mathrm{SU}(3)$ gauge theories, corrected path weight estimates reproduce finite volume Wilson loops; $U(1)$ topological susceptibilities are also compared with analytic predictions across several lattice volumes.