ARTICLE IN PRESS ARTICLE IN PRESS (2023). [59] C.-Z. Chen, J. J. He, M. N. Ali, G.-H. Lee, K. C. Fong, and K. T. Law, Asymmetric Josephson effect in inversion symmetry breaking topological materials, Phys. Rev. B 98, 075430 (2018). [60] H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, The field-free Josephson diode in a van der Waals heterostructure, Nature 604, 653 (2022). [61] G. Qiu, H.-Y. Yang, L. Hu, H. Zhang, C.-Y. Chen, Y. Lyu, C. Eckberg, P. Deng, S. Krylyuk, A. V. Davydov, et al., Emergent ferromagnetism with superconductivity in Fe (Te, Se) van der waals Josephson junctions, Nat Commun 14, 6691 (2023). [62] G. Hu, Y. Han, H. Guo, S. Lv, T. Gao, Y. Wang, Z. Zhao, K. Zhu, Q. Qi, G. Xian, et al., Polarity-reversible zero- field diode effect in van der waals ferromagnetic Joseph- son junction for logic operation, Adv. Mater. , e13434 (2025). [63] J.-X. Hu, Z.-T. Sun, Y.-M. Xie, and K. T. Law, Joseph- son diode effect induced by valley polarization in twisted bilayer graphene, Phys. Rev. Lett. 130, 266003 (2023). [64] D. Chakraborty and A. M. Black-Schaffer, Perfect su- perconducting diode effect in altermagnets, Phys. Rev. Lett. 135, 026001 (2025). [65] Q. Cheng, Y. Mao, and Q.-F. Sun, Field-free Josephson diode effect in altermagnet/normal metal/altermagnet junctions, Phys. Rev. B 110, 014518 (2024). [66] L. Sharma and M. Thakurathi, Tunable Josephson diode effect in singlet superconductor-altermagnet-triplet superconductor junctions, Phys. Rev. B 112, 104506 (2025). [67] A. Boruah, S. Acharjee, and P. K. Saikia, Field-free Josephson diode effect in an ising- superconductor/altermagnet/ising-superconductor Josephson junction, Phys. Rev. B 112, 054505 (2025). [68] D. Debnath, A. Saha, and P. Dutta, Spin polar- ization and diode effect in thermoelectric current through altermagnet-based superconductor heterostruc- tures, Phys. Rev. B 113, 104508 (2026). [69] B. K. Sahoo and A. Soori, Field-free transverse Joseph- son diode effect in altermagnets, arXiv preprint , arXiv:2509.14109 (2025). [70] W.-T. Lu, T.-F. Fang, Q. Cheng, and Q.-F. Sun, Elec- trically controlled field-free Josephson diode effect in two-dimensional antiferromagnets, Phys. Rev. B 112, 184505 (2025). [71] Q.-K. Shen and Y. Zhang, Josephson diodes induced by loop current states, Phys. Rev. B 111, 174515 (2025). [72] H. Sickinger, A. Lipman, M. Weides, R. G. Mints, H. Kohlstedt, D. Koelle, R. Kleiner, and E. Goldobin, Experimental evidence of a φ Josephson junction, Phys. Rev. Lett. 109, 107002 (2012). [73] H.-X. Lou, J.-J. Chen, X.-G. Ye, Z.-B. Tan, A.-Q. Wang, Q. Yin, X. Liao, J.-Z. Fang, X.-Y. Liu, Y.-L. He, Z.- T. Zhang, C. Li, Z.-M. Wei, X.-M. Ma, D. Yu, and Z.-M. Liao, Radio-frequency quantum rectification in kagome superconductor CsV3Sb5, arXiv preprint , arXiv: 2508.15266 (2025). [74] K.-R. Jeon, B. K. Hazra, K. Cho, A. Chakraborty, J.-C. Jeon, H. Han, H. L. Meyerheim, T. Kontos, and S. S. Parkin, Long-range supercurrents through a chiral non- collinear antiferromagnet in lateral Josephson junctions, Nat. Mater. 20, 1358 (2021). [75] K.-R. Jeon, B. K. Hazra, J.-K. Kim, J.-C. Jeon, H. Han, H. L. Meyerheim, T. Kontos, A. Cottet, and S. S. Parkin, Chiral antiferromagnetic Josephson junctions as spin- triplet supercurrent spin valves and dc squids, Nat. Nan- otechnol. 18, 747 (2023). [76] H. Chen, Q. Niu, and A. H. MacDonald, Anomalous hall effect arising from noncollinear antiferromagnetism, Phys. Rev. Lett. 112, 017205 (2014). [77] N. Kiyohara, T. Tomita, and S. Nakatsuji, Giant anoma- lous hall effect in the chiral antiferromagnet Mn3Ge, Phys. Rev. Appl. 5, 064009 (2016). [78] S. Nakatsuji, N. Kiyohara, and T. Higo, Large anomalous hall effect in a non-collinear antiferromagnet at room temperature, Nature 527, 212 (2015). [79] Y. Zhang, Y. Sun, H. Yang, J. Železn`y, S. P. Parkin, C. Felser, and B. Yan, Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn3X (X = Ge, Sn, Ga, Ir, Rh, and Pt), Phys. Rev. B 95, 075128 (2017). [80] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki, D. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakatsuji, Large anomalous nernst effect at room temperature in a chiral antiferromagnet, Nat. Phys. 13, 1085 (2017). [81] L.-H. Hu and S.-B. Zhang, Spin magnetization in un- conventional antiferromagnets with collinear and non- collinear spins, Sci. China Phys. Mech. Astron. 68, 247211 (2025). [82] S.-B. Zhang and L.-H. Hu, Finite-momentum mixed singlet-triplet pairing in chiral antiferromagnets induced by even-parity spin texture, Phys. Rev. B 112, L100501 (2025). [83] R.-Y. Sun, H.-K. Jin, H.-H. Tu, and Y. Zhou, Possible chiral spin liquid state in the s = 1/2 kagome Heisenberg model, npj Quantum Materials 9, 16 (2024). [84] Q. Wang, H. Lei, Y. Qi, and C. Felser, Intriguing kagome topological materials, npj Quantum Materials 10, 72 (2025). [85] Z. Xiong, T. Datta, and D.-X. Yao, Resonant inelastic x-ray scattering study of vector chiral ordered kagome antiferromagnet, npj Quantum Materials 5, 78 (2020). [86] S.-W. Cheong and F.-T. Huang, Altermagnetism classi- fication, npj Quantum Mater. 10, 38 (2025). [87] R. Cheng, S. Okamoto, and D. Xiao, Spin Nernst effect of magnons in collinear antiferromagnets, Phys. Rev. Lett. 117, 217202 (2016). [88] H. Narita, M. Ikhlas, M. Kimata, A. A. Nugroho, S. Nakatsuji, and Y. Otani, Anomalous Nernst effect in a microfabricated thermoelectric element made of chiral antiferromagnet Mn3Sn, Appl. Phys. Lett. 111, 202404 (2017). [89] Q. Shao, P. Li, L. Liu, H. Yang, S. Fukami, A. Razavi, H. Wu, K. Wang, F. Freimuth, Y. Mokrousov, et al., Roadmap of spin–orbit torques, IEEE transactions on magnetics 57, 1 (2021). [90] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Antiferromagnetic spintronics, Rev. Mod. Phys. 90, 015005 (2018). [91] M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P. K. Muduli, M. Ikhlas, Y. Omori, T. Tomita, A. H. MacDon- ald, S. Nakatsuji, et al., Magnetic and magnetic inverse spin hall effects in a non-collinear antiferromagnet, Na- ture 565, 627 (2019). [92] J. Holanda, H. Saglam, V. Karakas, Z. Zang, Y. Li, R. Divan, Y. Liu, O. Ozatay, V. Novosad, J. E. Pearson,
ARTICLE IN PRESS ARTICLE IN PRESS and A. Hoffmann, Magnetic damping modulation in IrMn3/Ni80Fe20 via the magnetic spin hall effect, Phys. Rev. Lett. 124, 087204 (2020). [93] Y. Liu, Y. Liu, M. Chen, S. Srivastava, P. He, K. L. Teo, T. Phung, S.-H. Yang, and H. Yang, Current-induced out-of-plane spin accumulation on the (001) surface of the IrMn3 antiferromagnet, Phys. Rev. Appl. 12, 064046 (2019). [94] S.-B. Zhang, L.-H. Hu, Q. Niu, and Z. Zhang, Spin- valley locking and pure spin-triplet superconductivity in noncollinear antiferromagnets proximitized to conven- tional superconductors, arXiv e-prints , arXiv:2507.11921 (2025). [95] J.-X. Hou, H.-P. Sun, B. Trauzettel, and S.-B. Zhang, Enhancement of Josephson supercurrent in a π-junction state by chiral antiferromagnetism, Phys. Rev. B 112, 075305 (2025). [96] A. Y. Chaou, G. Lemut, F. von Oppen, and P. W. Brouwer, Proximity superconductivity in chiral kagome antiferromagnets, arXiv preprint , arXiv:2508.08372 (2025). [97] J.-X. Yin, B. Lian, and M. Z. Hasan, Topological kagome magnets and superconductors, Nature 612, 647 (2022). [98] J.-X. Yin, S. S. Zhang, H. Li, K. Jiang, G. Chang, B. Zhang, B. Lian, C. Xiang, I. Belopolski, H. Zheng, et al., Giant and anisotropic many-body spin–orbit tun- ability in a strongly correlated kagome magnet, Nature 562, 91 (2018). [99] Z. Liu, M. Wei, W. Peng, D. Hou, Y. Gao, and Q. Niu, Multipolar anisotropy in anomalous hall effect from spin- group symmetry breaking, Phys. Rev. X 15, 031006 (2025). [100] P. Liu, J. Li, J. Han, X. Wan, and Q. Liu, Spin-group symmetry in magnetic materials with negligible spin- orbit coupling, Phys. Rev. X 12, 021016 (2022). [101] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with sym- metries, Rev. Mod. Phys. 88, 035005 (2016). [102] B. A. Bernevig, Topological insulators and topological su- perconductors, in Topological Insulators and Topological Superconductors (Princeton university press, 2013). [103] Z. Gao, M. Hua, H. Zhang, and X. Zhang, Classification of stable dirac and weyl semimetals with reflection and rotational symmetry, Phys. Rev. B 93, 205109 (2016). [104] J. Rodríguez-Carvajal and F. Bourée, Symmetry and magnetic structures, in EPJ Web of Conferences, Vol. 22 (EDP Sciences, 2012) p. 00010. [105] M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, and J. Rubio, Highly convergent schemes for the calculation of bulk and surface Green functions, J. Phys. F: Met. Phys. 15, 851 (1985). [106] Y. Asano, Numerical method for dc Josephson current be- tween d-wave superconductors, Phys. Rev. B 63, 052512 (2001). [107] S.-B. Zhang and B. Trauzettel, Detection of second- order topological superconductors by Josephson junc- tions, Phys. Rev. Res. 2, 012018 (2020). [108] G.-L. Guo, X.-H. Pan, H. Dong, and X. Liu, Edge depen- dent Josephson diode effect in WTe2-based Josephson junction, arXiv preprint , arXiv:2508.21357 (2025). [109] Z. Zhou, Y. Cao, Z. Pan, Y. Zhang, S. Liang, F. Pan, and C. Song, Field-free full switching of chiral antiferro- magnetic order, Nature 651, 341–347 (2026). [110] T. Higo, K. Kondou, T. Nomoto, M. Shiga, S. Sakamoto, X. Chen, D. Nishio-Hamane, R. Arita, Y. Otani, S. Miwa, et al., Perpendicular full switching of chiral antiferromag- netic order by current, Nature 607, 474 (2022). [111] H. Tsai, T. Higo, K. Kondou, T. Nomoto, A. Sakai, A. Kobayashi, T. Nakano, K. Yakushiji, R. Arita, S. Miwa, et al., Electrical manipulation of a topolog- ical antiferromagnetic state, Nature 580, 608 (2020). [112] B. Pal, B. K. Hazra, B. Göbel, J.-C. Jeon, A. K. Pandeya, A. Chakraborty, O. Busch, A. K. Srivastava, H. Deniz, J. M. Taylor, et al., Setting of the magnetic structure of chiral kagome antiferromagnets by a seeded spin-orbit torque, Sci. Adv. 8, eabo5930 (2022). [113] Z. Zheng, L. Jia, Z. Zhang, Q. Shen, G. Zhou, Z. Cui, L. Ren, Z. Chen, N. F. Jamaludin, T. Zhao, et al., All-electrical perpendicular switching of chiral antiferro- magnetic order, Nat. Mater. 24, 1407 (2025). [114] H. Xie, X. Chen, Q. Zhang, Z. Mu, X. Zhang, B. Yan, and Y. Wu, Magnetization switching in polycrystalline mn3sn thin film induced by self-generated spin-polarized current, Nat Commun 13, 5744 (2022). [115] Z. Zheng, T. Zeng, T. Zhao, S. Shi, L. Ren, T. Zhang, L. Jia, Y. Gu, R. Xiao, H. Zhou, et al., Effective electrical manipulation of a topological antiferromagnet by orbital torques, Nat Commun 15, 745 (2024). [116] Y. Takeuchi, Y. Yamane, J.-Y. Yoon, R. Itoh, B. Jinnai, S. Kanai, J. Ieda, S. Fukami, and H. Ohno, Chiral-spin rotation of non-collinear antiferromagnet by spin-orbit torque, Nat. Mater. 20, 1364 (2021). [117] Z. Xu, Y. Zhou, X. Zhang, Y. Qiao, Z. Xu, D. Shao, and Z. Zhu, Spin-orbit torque-driven chiral domain wall motion in Mn3Sn, Appl. Phys. Lett. 127, 7 (2025). [118] G. R. Frazier and Y. Li, Anisotropic Josephson coupling of d-vectors arising from interplay with frustrated spin textures, arXiv preprint , arXiv:2510.25756 (2025). [119] M. P. L. Sancho, J. M. L. Sancho, and J. Rubio, Quick iterative scheme for the calculation of transfer matrices: Application to Mo (100), J. Phys. F: Met. Phys. 14, 1205 (1984). Acknowledgments We thank James Jun He, Jian Li, and Alex Westström for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 12488101) and Quantum Science and Technology- National Science and Technology Major Project (Grant No. 2021ZD0302801). C.L. and L.H.H. were supported by National Key R&D Program of China (Grant No. 2025YFA1411501), the National Natural Science Foun- dation of China (Grant Nos. 12561160109, 12574148), the Fundamental Research Funds for the Central Uni- versities (Grant No. 226-2024-00068). C.L. was also supported by central fiscal special-purpose fund (Grant No.2021ZD0302500). Author contributions Jin-Xing Hou and Song-Bo Zhang conceived, designed, and promoted the project. Song-Bo Zhang and Lun-Hui