Using forward matrix elements of local leading-twist operators, we present a determination of the isovector third Mellin moments < x^2 > of nucleon unpolarized, polarized, and transversity parton distribution functions. Two lattice QCD ensembles at the physical pion mass are used, which were generated using a tree-level Symanzik-improved gauge action and 2+1 flavor tree-level improved Wilson Clover fermions coupling via 2-level HEX-smearing. Leveraging a wide set of operators, two extraction methods for the matrix elements, and the automatic inclusion of model uncertainties via bootstrapped model averages, we extract values of the third Mellin moments. This is the first direct calculation of these observables performed at the physical pion mass.

Generative models, particularly normalizing flows, have shown exceptional performance in learning probability distributions across various domains of physics, including statistical mechanics, collider physics, and lattice field theory. In the context of lattice field theory, normalizing flows have been successfully applied to accurately learn the Boltzmann distribution, enabling a range of tasks such as direct estimation of thermodynamic observables and sampling independent and identically distributed (i.i.d.) configurations. In this work, we present a proof-of-concept demonstration that normalizing flows can be used to learn the Boltzmann distribution for the Hubbard model. This model is widely employed to study the electronic structure of graphene and other carbon nanomaterials. State-of-the-art numerical simulations of the Hubbard model, such as those based on Hybrid Monte Carlo (HMC) methods, often suffer from ergodicity issues, potentially leading to biased estimates of physical observables. Our numerical experiments demonstrate that leveraging i.i.d. sampling from the normalizing flow effectively addresses these issues.

The sign problem that arises in Hybrid Monte Carlo calculations can be mitigated by deforming the integration manifold. While simple transformations are highly efficient for simulation, their efficacy systematically decreases with decreasing temperature and increasing interaction. Machine learning models have demonstrated the ability to push further, but require additional computational effort and upfront training. While neural networks possess the capacity to learn physical symmetries through proper training, there are anticipated advantages associated with encoding them into the network's structure. These include enhanced accuracy, accelerated training, and improved stability. The objective of the present study is twofold. First, we investigate the benefits of group convolutional models in comparison to fully connected networks, with a specific focus on the effects on the sign problem and on computational aspects. Second, we examine their capabilities for transfer learning, demonstrating the ability to further reduce training cost. We perform our investigations on the Hubbard model on select low-dimensional systems.

We present a Hamiltonian Monte Carlo study of doped perylene C_20H_12 described with the Hubbard model. Doped perylene can be used for organic light-emitting diodes (OLEDs) or as acceptor material in organic solar cells. Therefore, central to this study is a scan over charge chemical potential. A variational basis of operators allows for the extraction of the single-particle spectrum through a mostly automatic fitting procedure. Finite chemical potential simulations suffer from a sign problem which we ameliorate through contour deformation. The on-site interaction is kept at U/κ=2 . Discretization effects are handled through a continuum limit extrapolation. Our first-principles calculation shows significant deviation from non-interacting results especially at large chemical potentials.

We present one- and two-body measurements for the Hubbard model on the honeycomb (graphene) lattice from ab-initio quantum monte carlo simulations. Of particular interest is excitons, which are particle/hole excitations in low-dimensional systems. They are analogous to the pion in QCD, but without confinement, the question of whether they are bound and stable is of great interest in the condensed matter arena. By measuring one- and two-body correlators across various spin and isospin channels we can compute two-body energies relative to their thresholds, ultimately allowing us to check for stable states.

Abstract We present a Hamiltonian Monte Carlo study of doped perylene $$\textrm{C}_{20}\textrm{H}_{12}$$ C 20 H 12 described with the Hubbard model. Doped perylene can be used for organic light-emitting diodes (OLEDs) or as acceptor material in organic solar cells. Therefore, central to this study is a scan over charge chemical potential. A variational basis of operators allows for the extraction of the single-particle spectrum through a mostly automatic fitting procedure. Finite chemical potential simulations suffer from a sign problem which we ameliorate through contour deformation. The on-site interaction is kept at $$\nicefrac {U}{\kappa }=2$$ U κ = 2 . Discretization effects are handled through a continuum limit extrapolation. Our first-principles calculation shows significant deviation from non-interacting results especially at large chemical potentials. Graphic abstract

The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hybrid Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases.

The second Mellin moments ( x ) of the nucleon 's unpolarized, polarized, and transversity parton distribution functions are computed. Two lattice QCD ensembles at the physical pion mass are used: these were generated using a tree -level Symanzik-improved gauge action and 2 & thorn; 1 flavor tree -level improved Wilson Clover fermions coupling via 2 -level HEX -smearing. The moments are extracted from forward matrix elements of local leading twist operators. We determine renomalization factors in RI-(S)MOM and match to MS at scale 2 GeV. Our findings show that operators that exhibit vanishing kinematics at zero momentum can have significantly reduced excited -state contamination. The resulting polarized moment is used to quantify the longitudinal contribution to the quark spin -orbit correlation. All our results agree within two sigma with previous lattice results.

Quark orbital angular momentum in the proton is evaluated via a Lattice QCD calculation of the second Mellin moment of the twist-3 generalized parton distribution $\widetilde{E}_{2T} $ in the forward limit. The connection between this approach to quark orbital angular momentum and approaches previously utilized in Lattice QCD calculations, via generalized transverse momentum-dependent parton distributions and via Ji's sum rule, is reviewed. This connection can be given in terms of Lorentz invariance and equation of motion relations. The calculation of the second Mellin moment of $\widetilde{E}_{2T} $ proceeds via a finite-momentum proton matrix element of a quark bilocal operator with a straight-line gauge connection and separation in both the longitudinal and transverse directions. The dependence on the former component serves to extract the second Mellin moment, whereas the dependence on the latter component provides a transverse momentum cutoff for the matrix element. Furthermore, a derivative of the matrix element with respect to momentum transfer in the forward limit is required, which is obtained using a direct derivative method. The calculation utilizes a clover fermion ensemble at pion mass 317 MeV. The resulting quark orbital angular momentum is consistent with previous evaluations through alternative approaches, albeit with greater statistical uncertainty using a comparable number of samples.

We apply constant imaginary offsets to the path integral for a reduction of the sign problem in the Hubbard model. These simple transformations enhance the quality of results from HMC calcu- lations without compromising the speed of the algorithm. This method enables us to efficiently calculate systems that are otherwise inaccessible due to a severe sign problem. To support this claim, we present observables of the C20 and C60 fullerenes. Furthermore, we demonstrate that at a certain offset, the sign problem is completely lifted in the limit of large chemical potential.

The numerical sign problem poses a seemingly insurmountable barrier to the simulation of many fascinating systems. We apply neural networks to deform the region of integration, mitigating the sign problem of systems with strongly correlated electrons. In this talk we present our latest architectural developments as applied to contour deformation. We also demonstrate its applicability to real systems, namely perylene.

We present a Lattice QCD calculation of the second Mellin moments of the nucleon axial, vector and tensor parton distribution functions (PDFs). The calculation is performed at the physical pion mass with two different lattice spacings, and includes both zero and non-zero nucleon momenta. In our preliminary analysis, we identify operators that greatly reduce excited-state contamination.

Many fascinating systems suffer from a severe (complex action) sign problem preventing us from calculating them with Markov Chain Monte Carlo simulations.One promising method to alleviate the sign problem is the transformation of the integration domain towards Lefschetz Thimbles.Unfortunately, this suffers from poor scaling originating in numerically integrating of flow equations and evaluation of an induced Jacobian.In this proceedings we present a new preliminary Neural Network architecture based on complex-valued affine coupling layers.This network performs such a transformation efficiently, ultimately allowing simulation of systems with a severe sign problem.We test this method within the Hubbard Model at finite chemical potential, modelling strongly correlated electrons on a spatial lattice of ions.

Monte Carlo simulations away from half-filling suffer from a sign problem that can be reduced by deforming the contour of integration. Such a transformation, which induces a Jacobian determinant in the Boltzmann weight, can be implemented using neural networks. This additional determinant cost for a generic neural network scales cubically with the volume, preventing large-scale simulations. We implement a new architecture, based on complex-valued affine coupling layers, which reduces this to linear scaling. We demonstrate the efficacy of this method by successfully applying it to systems of different size, the largest of which is intractable by other Monte Carlo methods due to its severe sign problem.