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Symmetries of three-dimensional periodic scalar fields are described by 230 space groups (SGs). Symmetries of three-dimensional periodic (pseudo)vector fields, however, are described by the spin-space groups (SSGs), which were initially used to describe the symmetries of magnetic orders. In SSGs, the real-space and spin degrees of freedom are unlocked in the sense that an operation could have different spatial and spin rotations. SSGs give a complete symmetry description of magnetic structures and have natural applications in the band theory of itinerary electrons in magnetically ordered systems with weak spin-orbit coupling. Altermagnetism, a concept raised recently that belongs to the symmetry-compensated collinear magnetic orders but has nonrelativistic spin plitting, is well described by SSGs. Because of the vast number and complicated group structures, SSGs have not yet been systematically enumerated. In this work, we exhaust SSGs based on the invariant subgroups of SGs, with spin operations constructed from three-dimensional (3D) real representations of the quotient groups for the invariant subgroups. For collinear and coplanar magnetic orders, the spin operations can be reduced into lower-dimensional real representations. As the number of SSGs is infinite, we consider only SSGs that describe magnetic unit cells up to 12 times crystal unit cells. We obtain 157 289 noncoplanar, 24 788 coplanar-noncollinear, and 1421 collinear SSGs. The enumerated SSGs are stored in an online database with a user-friendly interface. We develop an algorithm to identify SSGs for realistic materials and find SSGs for 1626 magnetic materials. We also discuss several potential applications of SSGs, including the representation theory, topological states protected by SSGs, structures of spin textures, and refinement of magnetic neutron diffraction patterns using SSGs. Our results serve as a solid starting point for further studies of symmetry and topology in magnetically ordered materials.

Background and purposeDiabetes mellitus (DM) is a well-established cardiovascular risk factor for atherosclerotic disease; however, its effect on the risk of rupture of intracranial aneurysms remains controversial. Herein, we aimed to perform a case–control study to investigate the relationship between DM and aneurysmal subarachnoid hemorrhage (aSAH).MethodsWe retrospectively reviewed the data of patients with ruptured or unruptured aneurysms who were treated between 2013 and 2023. Univariate and multivariate analyses were performed. Propensity score matching (PSM) analysis was conducted to evaluate the relationship between DM and risk of aSAH.ResultsA total of 4,787 patients with 5,768 intracranial aneurysms were included. Among them, 2,957 (61.8%) were females, 1765 (36.9%) had ruptured aneurysms, and 531 (11.1%) presented with DM. Female sex, current drinking, and hypercholesterolemia were associated with a higher risk of aSAH, whereas old age, former smoking, and DM were associated with a lower risk of aSAH in multivariate analysis (p