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

Dominic Schuh; Lena Funcke; Janik Kreit; Thomas Luu; Simran Singh

cond-mat.str-elMar 18, 2026

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

Authors: Dominic Schuh, Lena Funcke, Janik Kreit, Thomas Luu, Simran Singh

Organizations: Transdisciplinary Research Area (TRA) Matter, University of Bonn, Bonn, Germany · Helmholtz Institute for Radiation and Nuclear Physics (HISKP), University of Bonn, Bonn, Germany · Institute for Advanced Simulation 4 (IAS-4), Forschungszentrum Jülich, Jülich, Germany

Abstract

The Hubbard model at finite chemical potential is a cornerstone for understanding doped correlated systems, but simulations are severely limited by the sign problem. In the auxiliary-field formulation, the spin basis mitigates the sign problem, yet severe ergodicity issues have limited its use. We extend recent advances with normalizing flows at half-filling to finite chemical potential by introducing an annealing scheme enabling ergodic sampling. Compared to state-of-the-art hybrid Monte Carlo in the charge basis, our approach accurately reproduces exact diagonalization results while reducing statistical uncertainties by an order of magnitude, opening a new path for simulations of doped correlated systems.

Figures & tables

Figure 1: Schematic illustration of the annealing procedure controlled by the parameter

\lambda

. At

\lambda=0

, the fermion determinant is absent and the target distribution is Gaussian. As

\lambda

is increased, fermionic effects are gradually introduced, deforming the distribution until the full interacting Hubbard action with its multimodal structure is recovered at

\lambda=1

Figure 2: Average sign

\Sigma

as a function of the chemical potential

\mu

for an eight-site hexagonal lattice with

U=2

\beta=8

, and

N_{t}=16

. Shown are results from the annealing-based NF approach in the spin basis (NF SP, yellow) and optimized HMC simulations in the charge basis [ 15 ] (O+HMC CH, grey).

Figure 3: One-body correlation functions for an eight-site hexagonal lattice with

U=2

\beta=8

\mu=1.75

, and

N_{t}=40

. Shown are continous exact diagonalization results (ED, grey dashed), state-of-the-art optimized HMC in the charge basis (O+HMC CH, grey), and our annealing-based NF ansatz in the spin basis (NF SP, yellow). Both HMC and NF results are generated with 1M samples each.

Figure 4: One-body correlation functions for an 18-site hexagonal lattice with

U=2

\beta=8

\mu=2.5

, and

N_{t}=40

. Shown are state-of-the-art optimized HMC in the charge basis (O+HMC CH, grey) and our annealing-based NF ansatz in the spin basis (NF SP, yellow). Both HMC and NF results are generated with 1M samples each.

Panel of Figure 5 — Figure 5: Distribution of the two-site Hubbard model in the spin basis ( $U=3$ , $\beta=5$ , and $N_{t}$ =20) for $\mu=0$ (left) and $\mu=2$ (right). Shown are the fields summed over the temporal direction, i.e., $\phi_{x}=\sum_{t=0}^{N_{t}-1}\phi_{xt}$ .

Figure 6: One-body correlation functions for the two-site Hubbard model with

U=3

\beta=5

\mu=2

, and

N_{t}=20

. Shown are exact diagonalization results at fixed discretization (ED, grey) and those obtained by our annealing-based NF ansatz in the spin basis (NF SP, yellow).

Figure 7: One-body correlation functions for the hexagonal eight-site Hubbard model with

U=2

\beta=8

\mu=1.75

, and

N_{t}=40

. Shown are continuum exact diagonalization results (ED, grey), HMC in the spin basis with two initializations (HMC SP Run 1/2, blue/grey), and our annealing-based NF ansatz with two initializations (NF SP Run 1/2, yellow/orange). Small deviations from the exact results at early times are due to discretization errors.

Figure 8: One-body correlation functions for the hexagonal eight-site Hubbard model with

U=3

\beta=6

\mu=2.75

, and

N_{t}=40

. Shown are continuum exact diagonalization results (ED, grey) and 1M samples generated by our annealing-based NF ansatz (NF SP, yellow).

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May 26, 2026J/K move · Enter open · S save

Neural Autoregressive Control Variates for the Quantum Monte Carlo Sign Problem

Bei Qiao, Lei Wang

Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China · University of Chinese Academy of Sciences, Beijing 100049, China

Quantum Many-Body Physics Variance Reduction Autoregressive Modeling

Open paper AskPDF arXiv

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.

Abstract

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