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Spin-orbit torque (SOT) plays an efficient and versatile role in electrical manipulation in spintronic devices at the nanoscale, which shows great promise for ultrafast and energy-efficient magnetic random-access memory (MRAM). To get high-performance SOT devices, their charge-to-spin conversion ratio must be sufficiently high and low current consumption is the desired one. Two-dimensional van der Waals (2D-vdW) materials possess strong tunability and spin-orbit coupling compared to conventional metals, which can efficiently achieve both things. This review covers a generalized introduction to SOT and its origin, their measurement techniques, SOTs observed in various 2D material-based heterostructures made of topological insulators (TIs), transition metal dichalcogenides (TMDs), and van der Waals (vdW) materials as they have excellent electronic properties down to their monolayer limit and ease of integration. Further, it covers the recent progress of SOT devices in each category, highlighting their potential for achieving high-performance and energy-efficient spintronic devices and their potential applications.

Definitions of orbital angular momentum based on Wigner distributions are used to discuss the connection between the Ji definition of the quark orbital angular momentum and that of Jaffe and Manohar. The difference between these two definitions can be interpreted as the change in the quark orbital angular momentum as it leaves the target in a DIS experiment. The mechanism responsible for that change is similar to the mechanism that causes transverse single-spin asymmetries in semi-inclusive deep-inelastic scattering.

We study how the rapidity evolution of gluon transverse momentum dependent distribution changes from nonlinear evolution at small x ≪ 1 to linear double-logarithmic evolution at moderate x ~ 1.

Generalized transverse-momentum dependent parton distributions (GTMDs) encode the most general parton structure of hadrons. In this contribution, which is largely based on a recent publication,¹ we focus on two twist-2 GTMDs which are denoted by F_1,4 and G_1,1 in parts of the literature. As already shown previously, both GTMDs have a close relation to orbital angular momentum of partons inside a hadron. However, recently even the mere existence of F_1,4 and G_1,1 has been doubted. We explain why this claim does not hold. We support our model-independent considerations by calculating the GTMDs in two spectator models and in perturbative QCD. For the model results we also explicitly check the relation to the orbital angular momentum of partons inside hadrons.

The study of isolated heavy quarkonia, such as $J/\psi$ and $\mathrm{{\rm Y}}$, produced in association with a photon in proton-proton collisions at the LHC, is probably the optimal way to get right away a first experimental determination of two gluon transverse-momentum-dependent distribution (TMDs) in an unpolarized proton, $f_1^g$ and $h_1^{\perp g}$, the latter giving the distribution of linearly polarized gluons. To substantiante this, we calculate the transverse-momentum-dependent effects that arise in the process under study and discuss the feasibility of their measurements.

Transverse-momentum dependent parton distribution functions (TMDs) provide a description of nucleon structure in terms of the parton transverse momentum and its transverse spin. At leading twist there are eight TMDs, each offering a unique feature of quarks in a polarized or an unpolarized nucleon. The Sivers distribution is one of the most interesting TMD due to its non-universality. It has been extracted using the data from semi-inclusive deep-inelastic scattering (SIDIS), but there is no data yet from spin-dependent Drell-Yan (DY) process. Such measurement will provide a crucial test of TMD formalism which predicts an equal magnitude and opposite sign for the Sivers function extracted from SIDIS and DY process. We will discuss key future measurements of TMDs using both SIDIS and DY process with a focus on Hall A SoLID SIDIS program at Jefferson Lab.

We present an algorithm to express Wilson lines that are defined on piecewise linear paths in function of their individual segments, reducing the number of diagrams needed to be calculated. The important step lies in the observation that different linear path topologies can be related to each other using their color structure. This framework allows one to easily switch results between different Wilson line topologies, which is helpful when testing different structures against each other.

Definitions of orbital angular momentum based on Wigner distributions are used to discuss the connection between the Ji definition of the quark orbital angular momentum and that of Jaffe and Manohar. The difference between these two definitions can be interpreted as the change in the quark orbital angular momentum as it leaves the target in a DIS experiment. The mechanism responsible for that change is similar to the mechanism that causes transverse single-spin asymmetries in semi-inclusive deep-inelastic scattering.

We consider the definition of a transverse-momentum-dependent parton distribution function (TMDPDF) and its evolution properties given a factorization theorem of Drell-Yan lepton pair production with a non-vanishing transverse momentum. We discuss rapidity divergences that exist in both limits of QCD: the soft and collinear. We argue that only when a specific combination of the soft and collinear matrix elements is formed, rapidity divergences cancel. We also argue that the soft matrix element, calculated in perturbation theory, contains only rapidity divergences and there is no genuine long-distance infra-red divergences.

Wilson lines are key objects in many QCD calculations. They are parallel transporters of the gauge field that can be used to render non-local operator products gauge invariant, which is especially useful for calculations concerning validation of factorization schemes and in calculations for constructing or modelling parton density functions. We develop an algorithm to express Wilson lines that are defined on piecewise linear paths in function of their Wilson segments, reducing the number of diagrams needed to be calculated. We show how different linear path topologies can be related using their color structure. This framework allows one to easily switch results between different Wilson line structures, which is helpful when testing different structures against each other, e.g. when checking universality properties of non-perturbative objects.

The investigation of the partonic degrees of freedom beyond collinear approximation (3D description) has been gained increasing interest in the last decade. At the HERMES experiment, azimuthal asymmetries in hard exclusive reactions and in semi-inclusive deep-inelastic scattering of electrons and positrons off a (polarized) hydrogen and deuterium target have been measured. Such asymmetries provide new insights on crucial aspects of the parton dynamics. By measuring various hadron types in the initial and final states, flavor sensitivity is achieved. Non zero signals are reported for azimuthal asymmetries with respect the transverse target polarization in real-photon exclusive-electroproduction, which are related (still in a model dependent way) to the elusive quark orbital motion. Evidence is reported of the poorly known transversity function and of naive-T-odd transverse-momentum-dependent parton distribution functions related to spin-orbit effects. Evidence of spin-orbit effects in quark fragmentation is also observed, which are opposite in sign for favored and disfavored processes.

The investigation of the partonic degrees of freedom beyond collinear approximation (3D description) has been gained increasing interest in the last decade. The Thomas Jefferson National Laboratory, after the CEBAF upgrade to 12 GeV, will become the most complete facility for the investigation of the hadron structure in the valence region by scattering of polarized electron off various polarized nucleon targets. A compendium of the planned experiments is here presented.

Parton distribution functions, which represent the flavor and spin structure of the nucleon, provide invaluable information in illuminating quantum chromodynamics in the confinement region. Among various processes that measure such parton distribution functions, semi-inclusive deep inelastic scattering is regarded as one of the golden channels to access transverse momentum dependent parton distribution functions, which provide a 3D view of the nucleon structure in momentum space. The Jefferson Lab experiment E06-010 focuses on measuring the target single and double spin asymmetries in the reaction with a transversely polarized ³He target in Hall A with a 5.89 GeV electron beam. A leading pion and the scattered electron are detected in coincidence by the left High-Resolution Spectrometer at 16° and the BigBite spectrometer at 30° beam right, respectively. The kinematic coverage concentrates in the valence quark region, x ~ 0.1–0.4, at Q² ~ 1–3 GeV². The Collins and Sivers asymmetries of ³He and neutron are extracted. In this review, an overview of the experiment and the final results are presented. Furthermore, an upcoming 12-GeV program with a large acceptance solenoidal device and the future possibilities at an electron–ion collider are discussed.

Fragmentation functions can be cleanly obtained from e⁺e⁻ annihilation. In the recent years various measurements related to unpolarized, polarized and transverse-momentum dependent fragmentation functions have become available from the Belle, BaBar and BESIII experiments. These fragmentation functions are absolutely essential in extracting the spin and flavor structure of the nucleon and will play an important role in fulfilling the scientific goals of the electron-ion collider.