Full Uncertainty Quantification of Sign-Problem-Free Quantum Monte Carlo Methods and Nuclear Lattice Effective Field Theory Benchmarks

Technical Summary: Full Uncertainty Quantification of Sign-Problem-Free Quantum Monte Carlo Methods and Nuclear Lattice Effective Field Theory Benchmarks

Problem Statement
Modern ab initio nuclear theory has achieved significant success in reproducing observables for light nuclei and infinite nuclear matter by combining chiral effective field theory (EFT) with many-body solvers. However, extending these methods to medium-mass and heavy nuclei (up to $^{100}$Sn) faces a critical bottleneck: the lack of rigorous uncertainty quantification for systematic algorithmic errors. Without such control, discrepancies between theory and experiment are often attributed to the underlying nuclear forces, leading to the "readjustment" of low-energy constants (LECs) to compensate for solver deficiencies. This masks the true origin of theoretical errors and compromises the predictive power of ab initio approaches.

Specifically, recent criticisms (Ref. [1]) have challenged the validity of Nuclear Lattice Effective Field Theory (NLEFT) benchmarks, citing issues with variational bounds, correlation energies, and nuclear saturation mechanisms. These criticisms often stem from comparing results derived from different regulated lattice theories (e.g., mixing transfer-matrix and Hamiltonian formalisms) or inconsistent finite-volume treatments, rather than genuine failures of the many-body methods.

Methodology
The authors employ sign-problem-free Quantum Monte Carlo (QMC) methods, specifically Auxiliary-Field Quantum Monte Carlo (AFQMC), within the NLEFT framework. The study focuses on the recently developed LAT-OPT1 action, a minimal sign-problem-free lattice Hamiltonian that incorporates a nonperturbative spin-orbit interaction while preserving the antiunitary symmetry required for positive Monte Carlo weights in even-even nuclei.

The methodology involves a comprehensive error budget analysis and a diagnostic re-examination of benchmark comparisons:

1. Uncertainty Quantification: The authors systematically quantify statistical, imaginary-time, finite-volume, temporal-discretization, auxiliary-field-induced, and residual-sign uncertainties for nuclei ranging from $^4$He to $^{100}$Sn.
2. Nonperturbative Spin-Orbit Treatment: Unlike previous approaches where spin-orbit forces were treated as perturbative insertions, LAT-OPT1 embeds the spin-orbit density directly into the Euclidean-time projection Hamiltonian.
3. Regulator Consistency Checks: The paper rigorously compares different computational formalisms:
   • Transfer-Matrix vs. Hamiltonian: Analyzing the differences between finite temporal lattice spacing ($a_t$) transfer-matrix calculations and the $a_t \to 0$ Hamiltonian limit.
   • Boundary Conditions: Investigating the impact of Periodic Boundary Conditions (PBC) versus Average Twisted Boundary Conditions (ATBC) on finite-volume shell effects in nuclear matter.
   • Renormalization: Examining the continuum limit and the renormalization of short-range interactions.

Key Contributions and Results

• Quantitative Control of Systematic Errors:
  The study demonstrates that the cumulative many-body computational uncertainty in ground-state energies for doubly magic nuclei up to $^{100}$Sn is well below the percent level.

  • Imaginary-Time & Finite-Volume: Extrapolations to infinite projection time ($\tau \to \infty$) and infinite volume ($L \to \infty$) are stable. A box size of $L=11$ (with $a=1.32$ fm) is shown to be sufficiently converged for the studied nuclei.
  • Temporal Discretization: The use of a transfer-matrix formalism with finite $a_t$ induces an effective Hamiltonian ($H_{eff}$) that differs from the target Hamiltonian ($H$). The authors show that while $H_{eff}$ depends on $a_t$, the energy error scales as $O(a_t^2)$, allowing for precise control.
  • Auxiliary-Field Induced Terms: The square-completion required to maintain sign-problem-free sampling for the spin-orbit term generates induced two-body ($\rho_s^2$) and three-body terms. These are quantified and found to be small corrections (a few MeV), confirming the validity of the construction.
  • Residual Sign Problem: For odd-$A$ and odd-odd nuclei, a mild residual sign problem persists. However, the average phase remains close to unity (0.94–0.98), and statistical uncertainties remain comparable to even-even systems.

• Nonperturbative Role of Spin-Orbit Coupling:
  A critical finding is that the spin-orbit interaction must be included nonperturbatively in the Euclidean-time projection. Treating it only as a perturbative insertion on an SU(4)-symmetric wave function misses shell-structure contributions of several to ten MeV in medium-mass nuclei. Embedding it in the propagation dynamically generates the correct shell structure.

• Resolution of Benchmark Discrepancies:
  The paper addresses recent criticisms (Ref. [1]) by identifying conceptual and technical errors in the comparison of different lattice theories:

  • Inequivalent Comparisons: Comparing Hartree-Fock (HF) bounds from an $a_t \to 0$ Hamiltonian with AFQMC results from a finite-$a_t$ transfer matrix is invalid. The finite-$a_t$ transfer matrix defines a distinct regulated theory with induced interactions. When the same regulated theory is used consistently, the variational principle holds (HF energies bound projected energies).
  • Correlation Energies: Small correlation energies in neutron matter for the "Essential Elements" (EE) interaction do not imply trivial wave functions. Detailed analysis of irreducible densities reveals significant multimodal superfluidity (s-wave, p-wave, and quartet correlations) despite small energy differences.
  • Nuclear Saturation: The flattening of interaction energy at high densities is not a lattice-packing artifact. Analytical derivations in continuous space with Galilean-invariant regulators show that interaction-energy flattening is a natural consequence of momentum-space regulation. The saturation properties of NLEFT interactions track with the smearing-induced regulator scale, not the lattice site-occupancy scale.
  • Continuum Limit: The claim that lattice interactions lead to system collapse in the continuum limit is refuted. The scaling argument used in the criticism relies on an incorrect Born-level renormalization; proper renormalization to physical scattering observables yields a different scaling behavior ($C_c \propto a$) that prevents collapse.

Significance and Claims
The paper claims to establish a solid foundation for full error analysis in NLEFT simulations with full chiral forces. By rigorously quantifying uncertainties and demonstrating that sign-problem-free methods can achieve sub-percent precision for heavy nuclei, the work validates NLEFT as a reliable benchmark for nuclear many-body methods.

The authors assert that the apparent discrepancies in recent literature are resolved when comparisons are made between equivalent regulated theories with consistent parameters, volumes, and boundary conditions. The LAT-OPT1 action is presented not merely as a phenomenological tool but as an exactly solvable, nonperturbative benchmark model that captures essential shell structure dynamics. This framework enables the separation of genuine nuclear-force deficiencies from many-body solver errors, providing a pathway for predictive studies of shell evolution, spectroscopy, and dense matter in medium-mass and heavy nuclei.