In determinant quantum Monte Carlo (DQMC) and related fermion simulations, the fermion sign problem forces estimates to be obtained by reweighting with the absolute value of the configuration weight, and the resulting bias of an observable is controlled by an exact amplification factor (1−⟨S⟩2)/(2⟨S⟩)(1-\langle\mathcal{S}\rangle^{2})/(2\langle\mathcal{S}\rangle) built from the average sign, together with the separation Δμ\Delta\mu between the sign-resolved (positive- and negative-sector) observable distributions. The source work established this mechanism quantitatively for the single-band square-lattice Hubbard model, but explicitly left open how the amplification factor and the sign-resolved distribution structure depend on lattice geometry, on the microscopic model, and on the QMC variant used, and which of the observed trends are universal rather than model-specific. This problem asks for that dependence to be mapped and classified: which features of the bias mechanism — the average-sign amplification, the sign-resolved histogram dissimilarity, and its temperature evolution — transfer across geometries, models, and algorithms, and which are contingent on the square-lattice Hubbard/DQMC setting in which they were measured.

Background

Fermion quantum Monte Carlo (QMC) methods suffer from the sign problem: for most models of correlated electrons (away from special parameter symmetries), the Monte Carlo configuration weights take both signs, so importance sampling must be performed with the absolute value of the weight and the physical expectation value is recovered by reweighting with the sign S=±1\mathcal{S}=\pm 1 of each configuration. Determinant quantum Monte Carlo (DQMC) — the variant studied in the source work, in which the electron-electron interaction is decoupled through a Hubbard-Stratonovich transformation (an exact rewriting that introduces auxiliary classical fields so that the fermion contribution to each configuration weight becomes a product of matrix determinants) — and world-line formulations (which instead sample discrete imaginary-time configurations of spins and fermion hops) all face versions of this problem. As the average sign ⟨S⟩\langle\mathcal{S}\rangle decays (exponentially in system size and inverse temperature), the statistical error of the reweighted estimator explodes, and in practice simulations are often run at parameters where ⟨S⟩\langle\mathcal{S}\rangle is small but estimable, so that a systematic bias remains when the sign is neglected or only partially accounted for. A complementary line of work characterizes the severity of the problem through the average sign itself: Mondaini, Tarat, and Scalettar (Phys. Rev. B 107, 245144 (2023); preprint arXiv:2207.09026) showed that ⟨S⟩\langle\mathcal{S}\rangle obeys universal scaling near quantum critical points (the continuous phase transitions reached at zero temperature by tuning a non-thermal parameter such as interaction strength) across three different models, and Tarat, Xiao, Mondaini, and Scalettar (Phys. Rev. B 105, 045107 (2022); preprint arXiv:2108.00553) deconvolved the mechanisms by which ⟨S⟩\langle\mathcal{S}\rangle decays; Xu, Shen, He, and Qin (arXiv:2501.11022) correlated sign-problem severity with entanglement in the doped two-dimensional Hubbard model. These studies treat the average sign (a single number per parameter point) but not the full distribution of observables within each sign sector. The source work (Larson, Mondaini, and Scalettar, Phys. Rev. B 113, 195124 (2026); preprint arXiv:2512.04056) went further by resolving the distribution P±(O)P_{\pm}(O) of an observable separately over configurations with positive and negative signs in DQMC for the single-band Hubbard model — the minimal lattice model of correlated electrons, with nearest-neighbor hopping tt and on-site repulsion UU — on the square lattice at intermediate coupling. It proved an exact relation: the difference between the correctly reweighted expectation value and the naively sampled (absolute-weight) one equals Δμ (1−⟨S⟩2)/(2⟨S⟩)\Delta\mu\,(1-\langle\mathcal{S}\rangle^{2})/(2\langle\mathcal{S}\rangle), where Δμ\Delta\mu is the difference of the sign-sector means μ+−μ−\mu_{+}-\mu_{-} of the observable and (1−⟨S⟩2)/(2⟨S⟩)(1-\langle\mathcal{S}\rangle^{2})/(2\langle\mathcal{S}\rangle) is the amplification factor, which diverges as ⟨S⟩→0\langle\mathcal{S}\rangle\to 0. Bias therefore arises only when both factors are present: sign decay alone is harmless if the two sign sectors sample observables identically (Δμ=0\Delta\mu=0), and the paper gave a histogram-equivalence criterion making this precise. It further quantified the sign-sector dissimilarity by the Wasserstein distance W1W_{1} (the earth-mover distance between the two normalized histograms, reported as W1/σtotW_{1}/\sigma_{\mathrm{tot}}, where σtot\sigma_{\mathrm{tot}} is the width of the full, sign-averaged observable distribution) and observed characteristic trends with temperature — an enhancement of histogram dissimilarity at high temperature and a low-temperature plateau of W1/σtotW_{1}/\sigma_{\mathrm{tot}} for T/t≲0.2T/t\lesssim 0.2 at U/t=6U/t=6 — with supporting evidence that the bias persists to the thermodynamic and continuum limits (the latter reached as the Trotter step, the imaginary-time discretization Δτ\Delta\tau used to factorize the partition function, goes to zero). The paper's summary states the natural open question: how the amplification factor and the sign-resolved structure evolve with lattice geometry, the model under investigation, and the type of QMC method used, since the amplification factor depends on all of them.

Problem Statement

Determine how the sign-problem bias mechanism of fermion quantum Monte Carlo generalizes beyond the single-band square-lattice Hubbard model studied with one DQMC implementation. Coverage is bounded as follows: a resolution requires at least one further representative along each of the three axes — one lattice geometry beyond the square lattice, one microscopic model beyond the single-band Hubbard model, and one QMC variant beyond the specific DQMC implementation used in the source work — each required representative differing from the source setting along exactly its own axis, with the other two axes held at their source values, so that the geometry-, model-, and algorithm-specific attributions asked for in (2) follow from controlled comparisons; each such representative is chosen and justified by the solver, and additional representatives on any axis are permitted but not required. Specifically: (1) For other lattice geometries, other microscopic models, and other QMC variants (different Hubbard-Stratonovich transformations, determinant-controlling schemes, or world-line formulations), measure the average sign ⟨S⟩\langle\mathcal{S}\rangle and the amplification factor (1−⟨S⟩2)/(2⟨S⟩)(1-\langle\mathcal{S}\rangle^{2})/(2\langle\mathcal{S}\rangle), and the full sign-resolved structure of observable distributions — the sector means μ±\mu_{\pm}, their separation Δμ=μ+−μ−\Delta\mu=\mu_{+}-\mu_{-}, and the dissimilarity of the normalized histograms P±(O)P_{\pm}(O) (e.g. via the Wasserstein distance W1/σtotW_{1}/\sigma_{\mathrm{tot}}) — using the exact bias identity ⟨O⟩W−⟨O⟩∣W∣=Δμ (1−⟨S⟩2)/(2⟨S⟩)\langle O\rangle_{W}-\langle O\rangle_{|W|}=\Delta\mu\,(1-\langle\mathcal{S}\rangle^{2})/(2\langle\mathcal{S}\rangle) to interpret the measured reweighted–absolute-weight difference as the product of the sector separation Δμ\Delta\mu and the amplification factor in each setting. (2) Classify which observed trends are universal and which are model-, geometry-, or algorithm-specific — in particular whether the high-temperature enhancement of histogram dissimilarity and the low-temperature plateau of W1/σtotW_{1}/\sigma_{\mathrm{tot}} found in the square-lattice Hubbard model recur across the other settings — and identify the microscopic or algorithmic features responsible when they do not. The common objective is a transferable understanding of when reweighting-based QMC estimates remain trustworthy: which observables, models, and algorithm choices keep Δμ\Delta\mu small even as ⟨S⟩\langle\mathcal{S}\rangle decays, so that sign-problem severity (a property of ⟨S⟩\langle\mathcal{S}\rangle alone) and actual estimator bias (a joint property of ⟨S⟩\langle\mathcal{S}\rangle and the sign-resolved distribution structure) can be distinguished across the field's simulation settings.

References

  1. Ryan Larson, Rubem Mondaini, Richard T. Scalettar, 'Sign-Resolved Statistics and the Origin of Bias in Quantum Monte Carlo', Phys. Rev. B 113, 195124 (2026), DOI 10.1103/x24t-bd2q; preprint arXiv:2512.04056, https://arxiv.org/abs/2512.04056
  2. Rubem Mondaini, Sabyasachi Tarat, Richard T. Scalettar, 'Universality and Critical Exponents of the Fermion Sign Problem', Phys. Rev. B 107, 245144 (2023), DOI 10.1103/PhysRevB.107.245144; preprint arXiv:2207.09026, https://arxiv.org/abs/2207.09026
  3. Sabyasachi Tarat, Bo Xiao, Rubem Mondaini, Richard T. Scalettar, 'Deconvolving the components of the sign problem', Phys. Rev. B 105, 045107 (2022), DOI 10.1103/PhysRevB.105.045107; preprint arXiv:2108.00553, https://arxiv.org/abs/2108.00553
  4. Ping Xu, Yang Shen, Yuan-Yao He, Mingpu Qin, 'On the correlation between entanglement and the negative sign problem', arXiv:2501.11022, https://arxiv.org/abs/2501.11022
  5. Wen Chen, Rubem Mondaini, 'Sign-problem-resilient singular-value probe in determinant quantum Monte Carlo', arXiv:2608.19028, https://arxiv.org/abs/2608.19028
  6. Ryan Larson, Rubem Mondaini, Richard T. Scalettar, 'Dataset for Sign-Resolved Statistics and the Origin of Bias in Quantum Monte Carlo', Zenodo (2025), DOI 10.5281/zenodo.17782714, https://zenodo.org/records/17782714