For a long time, it has been believed that electrons' orbital angular momentum is quenched in metals unless it is induced by the spin-orbit coupling in magnetic materials. Contrary to the common belief, recent studies have revealed that electron eigenstates may have finite orbital angular momentum even without the spin-orbit coupling if the inversion symmetry is broken [1]. In centrosymmetric systems, on the other hand, although electron eigenstates do not have orbital angular momentum if the spin-orbit coupling is sufficiently weak, a flow of electrons with finite orbital angular momentum is generated in a transverse direction when an electric field is applied (orbital Hall effect) [2,3]. The first part of this talk discuss this orbital angular momentum in electron eigenstates and steady states. The second part presents recent magneto-optical detection of the orbital Hall effect. The orbital Hall effect generates an orbital angular momentum accumulation at the side surfaces of a material. The resulting orbital angular momentum accumulation is measured by utilizing the magneto-optical Kerr effect [4]. The third part discusses effective Hamiltonians for the electron orbital degree of freedom since they can facilitate the conceptual understanding of orbital dynamics. Just as effective Hamiltonians for the electron spin degree of freedom (such as spin Rashba Hamiltonian and spin Dresselhaus Hamiltonian) are useful for conceptual understanding of spin dynamics, effective Hamiltonians for the electron orbital degree of freedom can facilitate the understanding of orbital dynamics. In noncentrosymmetric systems, the simplest effective orbital Hamiltonians are linear in both orbital angular momentum and crystal momentum [1]. Due to their resemblance to the spin Rashba or spin Dresselhaus Hamiltonians, such orbital Hamiltonians may be called orbital Rashba or Dresselhaus Hamiltonians, and describe similar orbital angular momentum dynamics as the spin dynamics described by the spin Rashba Hamiltonian. One particular example is the orbital Edelstein effect, which is the orbital counterpart of the spin Edelstein effect. In centrosymmetric systems, on the other hand, the simplest effective orbital Hamiltonians are quadratic in both orbital angular momentum and crystal momentum [5]. The orbital Hamiltonian, which was used by Bernevig, Taylor, and Zhang (BTZ) in 2005 [2] to derive the orbital Hall effect in hole-doped silicon, is of this type. Such BTZ-type orbital Hamiltonians do not have spin counterparts, implying that the resulting orbital dynamics can differ from spin dynamics. The connection between the BTZ-type orbital Hamiltonians and the orbital texture will be clarified. If time allows, relaxation of orbital angular momentum will be discussed [6].

[1] S. R. Part et al., Phys. Rev. Lett. 107 (2011) 156803

[2] B. A. Bernevig et al., Phys. Rev. Lett. 95 (2005) 066601

[3] D. Go et al., Phys. Rev. Lett. 121 (2018) 086602

[4] Y.-G. Choi et al., Nature 619 (2023) 52

[5] S. Han et al., Phys. Rev. Lett. 128 (2022) 176601

[6] J. Sohn et al., arXiv:2404.11780 (accepted for publication in Phys. Rev. Lett.)