Recent series of theoretical and experimental reports have driven attention to time-reversal symmetry-breaking spintronic and spin-splitting phenomena in materials with collinear-compensated magnetic order incompatible with conventional ferromagnetism or antiferromagnetism. Here we employ an approach based on nonrelativistic spin-symmetry groups that resolves the conflicting notions of unconventional ferromagnetism or antiferromagnetism by delimiting a third basic collinear magnetic phase. We derive that all materials hosting this collinear-compensated magnetic phase are characterized by crystal-rotation symmetries connecting opposite-spin sublattices separated in the real space and opposite-spin electronic states separated in the momentum space. We describe prominent extraordinary characteristics of the phase, including the alternating spin-splitting sign and broken time-reversal symmetry in the nonrelativistic band structure, the planar or bulk $d$-, $g$-, or $i$-wave symmetry of the spin-dependent Fermi surfaces, spin-degenerate nodal lines and surfaces, band anisotropy of individual spin channels, and spin-split general, as well as time-reversal invariant momenta. Guided by the spin-symmetry principles, we discover in ab initio calculations outlier materials with an extraordinary nonrelativistic spin splitting, whose eV-scale and momentum dependence are determined by the crystal potential of the nonmagnetic phase. This spin-splitting mechanism is distinct from conventional relativistic spin-orbit coupling and ferromagnetic exchange, as well as from the previously considered anisotropic exchange mechanism in compensated magnets. Our results, combined with our identification of material candidates for the phase ranging from insulators and metals to a parent crystal of cuprate superconductors, underpin research of novel quantum phenomena and spintronic functionalities in high-temperature magnets with light elements, vanishing net magnetization, and strong spin coherence. In the discussion, we argue that the conflicting notions of unconventional ferromagnetism or antiferromagnetism, on the one hand, and our symmetry-based delimitation of the third phase, on the other hand, favor a distinct term referring to the phase. The alternating spin polarizations in both the real-space crystal structure and the momentum-space band structure characteristic of this unconventional magnetic phase suggest a term altermagnetism. We point out that $d$-wave altermagnetism represents a realization of the long-sought-after counterpart in magnetism of the unconventional $d$-wave superconductivity.

中文翻译：

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超越常规的铁磁性和反铁磁性：具有非相对论自旋和晶体旋转对称性的相

最近的一系列理论和实验报告引起了人们对具有与传统铁磁性或反铁磁性不相容的共线补偿磁序的材料中的时间反转对称破坏自旋电子学和自旋分裂现象的关注。在这里，我们采用了一种基于非相对论自旋对称群的方法，该方法通过界定第三个基本共线磁相来解决非常规铁磁性或反铁磁性的冲突概念。我们得出，所有具有这种共线补偿磁相的材料都具有晶体旋转对称性，这些材料连接在真实空间中分离的反自旋子晶格和在动量空间中分离的反自旋电子态。我们描述了该阶段的显着非凡特征，$d$-,$G$-， 或者$\mathrm{一世}$- 自旋相关费米表面的波对称性、自旋简并节线和表面、单个自旋通道的带各向异性和自旋分裂一般，以及时间反转不变动量。在自旋对称原理的指导下，我们从头开始发现计算具有非凡非相对论自旋分裂的异常材料，其 eV 尺度和动量依赖性由非磁性相的晶体势决定。这种自旋分裂机制不同于传统的相对论自旋轨道耦合和铁磁交换，也不同于先前考虑的补偿磁体中的各向异性交换机制。我们的研究结果，结合我们对从绝缘体和金属到铜酸盐超导体母体晶体等相的材料候选者的识别，为具有轻元素、净磁化消失和强自旋相干。在讨论中，我们认为，一方面，非常规铁磁性或反铁磁性的相互矛盾的概念，另一方面，我们基于对称性对第三相的界定，倾向于使用一个不同的术语来指代相。这种非常规磁相的实空间晶体结构和动量空间带结构特征中的交替自旋极化表明术语交替磁性。我们指出$d$- 波交替磁性代表了非常规磁性中备受追捧的对应物的实现$d$波超导性。