High Energy Physics - Lattice

The path optimization method is applied to the Stephanov model and the chiral random matrix model, both of which share several properties with QCD, to mitigate the sign problem caused by the fermion determinant. The Stephanov model serves as a prototypical model of finite-density QCD, while the chiral random matrix model represents an ideal system featuring the Silver Blaze phenomenon. We show that the path optimization successfully improves the average phase factor in the Stephanov model at high chemical potential, reproducing the analytical results with reduced statistical errors. However, it fails to improve the average phase factor in the Stephanov model at low chemical potential, as well as in the chiral random matrix model. This tendency in the phase factor behavior seems to be closely related to the global sign problem.

Sampling from an unnormalized Boltzmann density requires proposals that move probability mass globally while retaining enough path-probability information for statistical correction. We introduce Neural Non-Equilibrium Hamiltonian Monte Carlo (NHMC), a train-then-correct learned Hamiltonian sampler. Starting from a tractable base distribution, NHMC learns stochastic Hamiltonian-style paths toward the target. Once training is complete, the learned proposal parameters are fixed; the proposal then generates complete paths and endpoint configurations, which are statistically corrected using the recorded non-equilibrium work. This dimensionless generalized work is determined by the probability ratio between the forward proposal path and a reverse reference path. During training, minimizing its mean reduces a path-space KL divergence and controls an upper bound on endpoint mismatch. During evaluation, the same quantity defines weights for self-normalized importance sampling on paths (path-SNIS), estimates normalizing constants or free-energy differences, and gives the acceptance ratio for path-space independent Metropolis-Hastings (path-IMH). The same forward-reverse laws also define a shared-bridge round-trip Metropolis kernel that acts directly on configurations and preserves the Boltzmann target. On double-well and finite-volume lattice $\phi^4$ targets, the NHMC construction gives corrected estimates when path overlap is sufficient; when overlap is poor, weight degeneracy, low acceptance, and long autocorrelation expose proposal failure. We additionally report a molecular internal-coordinate feasibility study using an MD prior and learned-force path proposal.

Euclidean Monte Carlo methods are effective when the path-integral weight is real and nonnegative, but finite-density fermion systems often produce sign-changing or complex scalar weights after the fermionic sector is traced out. Hamiltonian formulations avoid this complex-weight sampling problem but face rapid Hilbert-space growth. This paper studies a conditional Euclidean-Hamiltonian (CEH) reduction that combines these two descriptions. The calculation is organized around a Monte Carlo-tractable reference problem and a residual active sector. Instead of tracing the active sector into a determinant or scalar weight, CEH keeps it operator-valued and uses the reference calculation to determine projected correlation or transfer matrices. These matrices define a finite effective Hamiltonian, with the remaining finite-density dependence introduced after projection. At finite rank, the result is an effective model whose accuracy must be tested through basis enlargement, metric conditioning, stochastic matrix-element errors, and number-sector diagnostics. The construction is examined in three finite benchmarks. A two-channel oscillator tests conditional basis compression; a positive-measure stochastic calculation tests correlation-matrix harvesting and GEVP extraction, including a comparison with a PDMS-style estimator; and a four-site Hubbard ring combines a finite-budget determinant-sign stress test with finite-density continuation in non-target active spaces. The benchmarks support the proposed trade from scalar sign reweighting to a monitored active-space approximation in structured finite models, but they do not provide an end-to-end stochastic CEH treatment of the Hubbard sign problem or establish favorable scaling. Usefulness requires both low-rank approximability and efficient extraction of the required projected matrix data.

In critical lattice models, distance ($r$) dependent correlation functions contain power laws $r^{-2\Delta}$ governed by scaling dimensions $\Delta$ of an underlying continuum field theory. In Monte Carlo simulations, the leading dimensions can be extracted by data fitting, which is difficult when two or more powers contribute significantly. Here a method utilizing covariance between multiple lattice operators is developed where the r dependent eigenvalues of the covariance matrix reflect scaling dimensions of individual field operators. This disentangling is demonstrated explicitly for conformal field theories. The scheme is first tested on the critical point of the 2D Ising model, where the two primary scaling dimensions and their respective two lowest descendant dimensions are extracted. The 3D Ising model is studied next, revealing the two relevant primaries and their lowest descendants to high precision. The 2D tricritical Ising point is studied with the Blume-Capel model. Here the scaling dimensions of all three symmetric primary operators are successfully isolated along with the leading descendants. The eigenvectors are also studied and give useful information on the boundary between the ordered and disordered phases in the neighborhood of the tricritical point. Finally, the crossover from regular to tricritical Ising scaling is investigated on several points on the phase boundary of the Blume-Capel model away from its tricritical point. The scaling of the eigenvalues corresponding to tricritical descendant operators are found to be remarkably stable even far from the tricritical point. The covariance method represents a simple extension of standard analysis of correlation functions and can significantly enhance the utility of Monte Carlo simulations and other computational methods in studies of criticality, in particular conformal critical points.