ARTICLE IN PRESS ARTICLE IN PRESS two experimentally feasible Josephson setups that exhibit sizable diode efficiencies and anomalous Josephson effects. In the SC/cAFM/SC junction, the presence of SOC en- ables a field-free Josephson diode effect and a continuously tunable anomalous Josephson response. Both the diode efficiency and the ϕ0-junction state (associated with the anomalous Josephson current) can be controlled via the cAFM and SOC strengths, while remaining robust against variations in the junction length, demonstrating stabil- ity in extended systems. In the SC/cAFM/cAFM′/SC setup, a relative angle between the cAFM orders in the two cAFM regions, combined with an out-of-plane Zee- man field, gives rise to significant diode response and anomalous Josephson effect. These effects are otherwise absent under in-plane Zeeman fields or in the ferromag- netic counterparts, highlighting the essential role of T Mz symmetry breaking and the distinctive non-collinear na- ture of cAFM order. Taken together, our results establish a direct connection between T Mz symmetry breaking, cAFM order, and nonreciprocal superconducting trans- port, positioning kagome cAFM systems as promising platforms for symmetry-engineered Josephson diodes and tunable anomalous Josephson effects. Finally, we comment on experimental relevance of our proposal. Mn3Ge thin films hosts the desired chiral mag- netic order and have been successfully interfaced with conventional s-wave superconductors such as Nb and Al [74, 75]. The cAFM order can be tuned via electric- field or current-driven switching [109–116], and can also be controlled by applying a magnetic field [78, 117]. In our setting, the SOC is an intrinsic material property. Its effective strength in the low-energy description may be influenced by materials choice, for example, by using related cAFMs with heavier constituent elements within the Mn3X family. These features establish a feasible plat- form for realizing tunable Josephson diode effects and ϕ0-junctions. When completing this work, we became aware of a related preprint [96] investigating the superconducting proximity effect in a similar kagome cAFM based Joseph- son junction with spin canting. That work also identified the role of asymmetric interfaces arising from the kagome lattice geometry, while focusing primarily on the topo- logical phase diagram and spin-polarized superconductiv- ity. We further note a recent preprint [118] studying the Josephson diode effect in junctions where a barrier host- ing noncollinear spin textures is sandwiched between two spin-triplet superconductors. These concurrent studies further highlight the growing interest in noncollinear mag- netic textures as a promising platform for unconventional and nonreciprocal Josephson phenomena. Methods The Josephson supercurrent Is was determined by two complementary approaches: a thermodynamic formula- tion based on the system free energy and a microscopic treatment using the Green function formalism. In nu- merical implementations, minor technical differences may arise. In the free-energy approach, the superconducting leads are treated as finite in length, whereas in the Green- function method they are effectively semi-infinite, with the Green function constructed iteratively from the far ends toward the cAFM region. Quantitative agreement therefore requires the superconducting leads in the free- energy calculation to be sufficiently long (i.e., exceeding the superconducting coherence length). When this con- dition is met, both methods yield consistent results and are both suited for capturing phase-dependent transport in superconducting junctions. Free-energy formulation In equilibrium, the supercurrent originates from the depen- dence of the total free energy F(ϕ) on the superconducting phase difference ϕ between the two superconductors. The current is given by Is = 2e ℏ ∂F(ϕ) ∂ϕ , (22) as established in the standard theory of Josephson junc- tions [1]. At finite temperature T, the free energy is evaluated from the partition function Z = P i e−Ei/(kBT ) through F = −kBT ln Z, where Ei denotes the many- body energy of the i-th state. In the zero-temperature limit, F reduces to the ground-state energy, obtained as the sum of all occupied (negative) single-particle eigenval- ues of the system Hamiltonian. These eigenenergies are obtained via exact diagonalization of the Bogoliubov–de Gennes Hamiltonian for a given phase difference ϕ. Green-function formulation For systems with spatially dependent order parameters or magnetic textures, a Green-function treatment is numer- ically more efficient. Within the Matsubara formalism, the Josephson current can be written as [106, 107] Is = −ie 2ℏkBT Z dkx 2π X ωn Tr[ˇτ3 ˇV+ ˇG(ℓ, ℓ+ 1) −ˇτ3 ˇV−ˇG(ℓ+ 1, ℓ)], (23) where ωn = (2n + 1)πkBT are the fermionic Matsubara frequencies. Here, ˇG(ℓ, ℓ′) is the Nambu Green function connecting two neighboring rows ℓand ℓ′, ˇV± represent the inter-row hopping matrices ( ˇV+ = ˇV † −), and ˇτ3 is the third Pauli matrix in particle-hole space. The trace is taken over spin, particle–hole, and sublattice spaces. To simplify notation, the dependencies on kx and ωn are omitted, i.e., ˇG(ℓ, ℓ′) ≡ˇGωn(ℓ, ℓ′; kx) and ˇV± ≡ˇV±(kx). The Green functions satisfy the recursion relations ˇG(ℓ+ 1, ℓ) = ˇG(ℓ+ 1, ℓ+ 1) ˇV+ ˇG(ℓ, ℓ), (24) ˇG(ℓ, ℓ+ 1) = ˇG(ℓ, ℓ) ˇV−ˇG(ℓ+ 1, ℓ+ 1), (25) which allow the full Green functions of the system to be constructed iteratively [105, 119].
ARTICLE IN PRESS ARTICLE IN PRESS In the Nambu representation, the Green function at row ℓtakes the block-matrix form ˇG(ℓ, ℓ) = ˆGe,ℓ ˆFeh,ℓ ˆFhe,ℓ ˆGh,ℓ , (26) where ˆGe,ℓand ˆGh,ℓdenote the normal Green functions for electrons and holes, respectively, while ˆFeh,ℓand ˆFhe,ℓ describe the anomalous (electron–hole) propagators that encode the superconducting correlations. The hopping matrices between adjacent rows can be written in the particle–hole basis as ˇV+ = ˆV + e ˆ0 ˆ0 ˆV + h , ˇV−= ˆV − e ˆ0 ˆ0 ˆV − h , (27) where ˆV ± e and ˆV ± h correspond to electron and hole hop- ping between neighboring rows. Substituting Eqs. (26) and (27) into the general current expression [Eq. (23)] and the recursion relations [Eq. (24) and Eq. (25)], the Josephson current can be recast as Is = −ie 2πℏkBT Z dkx X ωn Tr ˆV + e ˆFeh,ℓˆV − h ˆFhe,ℓ+1 −ˆV + h ˆFhe,ℓˆV − e ˆFeh,ℓ+1 . (28) This form highlights that the Josephson current is gov- erned by the convolution of the anomalous Green func- tions ˆFeh and ˆFhe, which describe the pair correlations penetrating across the junction. In other words, the super- current results from the coherent transfer of Cooper pairs mediated by these anomalous propagators, with their spa- tial dependence reflecting the proximity and magnetic effects within the junction. Data availability The datasets generated and/or analyzed during the cur- rent study are not publicly available due to the theoreti- cal nature of the work and the absence of experimental datasets, but are available from the corresponding author on reasonable request. References [1] M. Tinkham, Introduction to superconductivity, Vol. 1 (Courier Corporation, 2004). [2] J. Hu, C. Wu, and X. Dai, Proposed design of a Joseph- son diode, Phys. Rev. Lett. 99, 067004 (2007). [3] M. Davydova, S. Prembabu, and L. Fu, Universal Joseph- son diode effect, Sci. Adv. 8, eabo0309 (2022). [4] Y. Zhang, Y. Gu, P. Li, J. Hu, and K. Jiang, General theory of Josephson diodes, Phys. Rev. X 12, 041013 (2022). [5] M. Nadeem, M. S. Fuhrer, and X. Wang, The supercon- ducting diode effect, Nat. Rev. Phys. 5, 558 (2023). [6] J. J. He, Y. Tanaka, and N. Nagaosa, A phenomenologi- cal theory of superconductor diodes, New J. Phys. 24, 053014 (2022). [7] K. Misaki and N. Nagaosa, Theory of the nonreciprocal Josephson effect, Phys. Rev. B 103, 245302 (2021). [8] Y. Tanaka, B. Lu, and N. Nagaosa, Theory of giant diode effect in d-wave superconductor junctions on the surface of a topological insulator, Phys. Rev. B 106, 214524 (2022). [9] K. Jiang and J. Hu, Superconducting diode effects, Nat. Phys. 18, 1145 (2022). [10] D. Wang, Q.-H. Wang, and C. Wu, Current- reversion symmetry breaking and the dc Josephson diode effect, Science Bulletin https://doi.org/10.1016/j.scib.2025.11.011 (2025). [11] D. Wang, Q.-H. Wang, and C. Wu, Josephson diode effect: a phenomenological perspective, arXiv e-prints , arXiv:2506.23200 (2025). [12] T. Le, Z. Pan, Z. Xu, J. Liu, J. Wang, Z. Lou, X. Yang, Z. Wang, Y. Yao, C. Wu, et al., Superconducting diode effect and interference patterns in kagome CsV3Sb5, Na- ture 630, 64 (2024). [13] D. Chakraborty and A. M. Black-Schaffer, Perfect su- perconducting diode effect in altermagnets, Phys. Rev. Lett. 135, 026001 (2025). [14] Y.-Y. Lyu, J. Jiang, Y.-L. Wang, Z.-L. Xiao, S. Dong, Q.- H. Chen, M. V. Milošević, H. Wang, R. Divan, J. E. Pear- son, et al., Superconducting diode effect via conformal- mapped nanoholes, Nat. Commun. 12, 2703 (2021). [15] Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F. Ritter, D. Z. Haxell, M. Davydova, S. Ilić, O. Glezakou- Elbert, A. Varambally, F. S. Bergeret, A. Kamra, L. Fu, P. A. Lee, and J. S. Moodera, Ubiquitous superconduct- ing diode effect in superconductor thin films, Phys. Rev. Lett. 131, 027001 (2023). [16] A. Daido, Y. Ikeda, and Y. Yanase, Intrinsic super- conducting diode effect, Phys. Rev. Lett. 128, 037001 (2022). [17] F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Obser- vation of superconducting diode effect, Nature 584, 373 (2020). [18] S. Banerjee and M. S. Scheurer, Altermagnetic supercon- ducting diode effect, Phys. Rev. B 110, 024503 (2024). [19] S. Ilić, P. Virtanen, D. Crawford, T. T. Heikkilä, and F. S. Bergeret, Superconducting diode effect in diffusive superconductors and Josephson junctions with Rashba spin-orbit coupling, Phys. Rev. B 110, L140501 (2024). [20] R. Wang and N. Hao, Universal diagnostic criterion for intrinsic superconducting diode effect, arXiv e-prints , arXiv:2507.04876 (2025). [21] S. Fracassi, S. Traverso, S. Heun, M. Sassetti, M. Carrega, and N. T. Ziani, Intrinsic and tunable superconducting diode effect in quantum spin hall systems, arXiv e-prints , arXiv:2512.02575 (2025). [22] S. Bhowmik, D. Samanta, A. K. Nandy, A. Saha, and S. K. Ghosh, Optimizing one dimensional superconduct- ing diodes: interplay of Rashba spin-orbit coupling and magnetic fields, Commun Phys 8, 260 (2025). [23] J. Ma, R. Zhan, and X. Lin, Superconducting diode ef- fects: Mechanisms, materials and applications, Advanced Physics Research , 2400180 (2025). [24] A. Pal, D. Mondal, T. Nag, and A. Saha, Topological superconductivity and superconducting diode effect me-
ARTICLE IN PRESS ARTICLE IN PRESS diated via unconventional magnet and ising spin-orbit coupling, arXiv preprint , arXiv:2512.01266 (2025). [25] M. Amundsen, J. Linder, J. W. A. Robinson, I. Žutić, and N. Banerjee, Colloquium: Spin-orbit effects in su- perconducting hybrid structures, Rev. Mod. Phys. 96, 021003 (2024). [26] D. Debnath and P. Dutta, Gate-tunable Josephson diode effect in Rashba spin-orbit coupled quantum dot junc- tions, Phys. Rev. B 109, 174511 (2024). [27] P. J. Moll and V. B. Geshkenbein, Evolution of super- conducting diodes, Nat. Phys. 19, 1379 (2023). [28] T. Golod and V. M. Krasnov, Demonstration of a su- perconducting diode-with-memory, operational at zero magnetic field with switchable nonreciprocity, Nat. Com- mun. 13, 3658 (2022). [29] M. Gupta, G. V. Graziano, M. Pendharkar, J. T. Dong, C. P. Dempsey, C. Palmstrøm, and V. S. Pribiag, Gate- tunable superconducting diode effect in a three-terminal Josephson device, Nat. Commun. 14, 3078 (2023). [30] R. S. Souto, M. Leijnse, and C. Schrade, Josephson diode effect in supercurrent interferometers, Phys. Rev. Lett. 129, 267702 (2022). [31] E. Nikodem, J. Schluck, M. Geier, M. Papaj, H. F. Legg, J. Feng, M. Bagchi, L. Fu, and Y. Ando, Tunable super- conducting diode effect in a topological nano-SQUID, Sci. Adv. 11, eadw4898 (2025). [32] Z. Liu, L. Huang, and J. Wang, Josephson diode effect in topological superconductors, Phys. Rev. B 110, 014519 (2024). [33] J. J. He, Y. Tanaka, and N. Nagaosa, The supercurrent diode effect and nonreciprocal paraconductivity due to the chiral structure of nanotubes, Nat. Commun. 14, 3330 (2023). [34] N. F. Yuan and L. Fu, Supercurrent diode effect and finite-momentum superconductors, Proc. Natl. Acad. Sci. 119, e2119548119 (2022). [35] N. Nagaosa and Y. Yanase, Nonreciprocal transport and optical phenomena in quantum materials, Annu. Rev. Condens. Matter Phys. 15, 63 (2024). [36] H. Narita, J. Ishizuka, R. Kawarazaki, D. Kan, Y. Sh- iota, T. Moriyama, Y. Shimakawa, A. V. Ognev, A. S. Samardak, Y. Yanase, et al., Field-free superconduct- ing diode effect in noncentrosymmetric superconduc- tor/ferromagnet multilayers, Nat. Nanotechnol. 17, 823 (2022). [37] F. Liu, Y. M. Itahashi, S. Aoki, Y. Dong, Z. Wang, N. Ogawa, T. Ideue, and Y. Iwasa, Superconducting diode effect under time-reversal symmetry, Sci. Adv. 10, eado1502 (2024). [38] A. Costa, O. Kanehira, H. Matsueda, and J. Fabian, Unconventional Josephson supercurrent diode effect in- duced by chiral spin-orbit coupling, Phys. Rev. B 111, L140506 (2025). [39] N. L. Schulz, D. Nikolić, and M. Eschrig, Theory of quantum-geometric charge and spin Josephson diode effects in strongly spin-polarized hybrid structures with noncoplanar spin textures, Phys. Rev. B 112, 104515 (2025). [40] N. L. Schulz, D. Nikolić, and M. Eschrig, Quantum- geometric spin and charge Josephson diode effects, Phys. Rev. B 112, 104514 (2025). [41] D. Nikolić, N. L. Schulz, A. I. Buzdin, and M. Es- chrig, Spin-resolved Josephson diode effect through strongly spin-polarized conical magnets, Phys. Rev. B 112, 224507 (2025). [42] S. Djurdjević and Z. Popović, Josephson diode effect in d-wave superconductor/ferromagnet/d-wave super- conductor junction with interfacial Rashba spin–orbit coupling, Prog. theor. exp. phys. 2025, 103I01 (2025). [43] S. Reinhardt, T. Ascherl, A. Costa, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, J. Fabian, D. Kochan, et al., Link between supercurrent diode and anomalous Josephson effect revealed by gate-controlled interferometry, Nat. Commun. 15, 4413 (2024). [44] G.-L. Guo, X.-H. Pan, and X. Liu, ϕ0 junction and Josephson diode effect in high-temperature superconduc- tors, Phys. Rev. B 112, 014509 (2025). [45] H.-P. Sun, S.-B. Zhang, C.-A. Li, and B. Trauzettel, Tunable second harmonic in altermagnetic Josephson junctions, Phys. Rev. B 111, 165406 (2025). [46] B. Lu, K. Maeda, H. Ito, K. Yada, and Y. Tanaka, φ Josephson junction induced by altermagnetism, Phys. Rev. Lett. 133, 226002 (2024). [47] A. Buzdin and A. E. Koshelev, Periodic alternating 0- and π-junction structures as realization of φ-Josephson junctions, Phys. Rev. B 67, 220504 (2003). [48] F. Tafuri, Fundamentals and frontiers of the Josephson effect, Vol. 286 (Springer Nature, 2019). [49] G.-L. Guo, H.-B. Leng, Y. Hu, and X. Liu, 0−π qubit with one Josephson junction, Phys. Rev. B 105, L180502 (2022). [50] T. Golod, A. Rydh, and V. M. Krasnov, Detection of the phase shift from a single abrikosov vortex, Phys. Rev. Lett. 104, 227003 (2010). [51] 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). [52] A. Assouline, C. Feuillet-Palma, N. Bergeal, T. Zhang, A. Mottaghizadeh, A. Zimmers, E. Lhuillier, M. Eddrie, P. Atkinson, M. Aprili, et al., Spin-orbit induced phase- shift in Bi2Se3 Josephson junctions, Nat Commun 10, 126 (2019). [53] E. Goldobin, H. Sickinger, M. Weides, N. Ruppelt, H. Kohlstedt, R. Kleiner, and D. Koelle, Memory cell based on a φ Josephson junction, Appl. Phys. Lett. 102, 24 (2013). [54] E. Strambini, A. Iorio, O. Durante, R. Citro, C. Sanz- Fernández, C. Guarcello, I. V. Tokatly, A. Braggio, M. Rocci, N. Ligato, et al., A Josephson phase battery, Nat. Nanotechnol. 15, 656 (2020). [55] C. Baumgartner, L. Fuchs, A. Costa, J. Picó-Cortés, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. F. Junior, et al., Effect of Rashba and dresselhaus spin–orbit coupling on supercurrent rectifica- tion and magnetochiral anisotropy of ballistic Josephson junctions, J. Phys.: Condens. Matter. 34, 154005 (2022). [56] S. Mondal, P.-H. Fu, and J. Cayao, Josephson diode effect with andreev and majorana bound states, Phys. Rev. B 112, 144506 (2025). [57] B. Pal, A. Chakraborty, P. K. Sivakumar, M. Davydova, A. K. Gopi, A. K. Pandeya, J. A. Krieger, Y. Zhang, M. Date, S. Ju, et al., Josephson diode effect from Cooper pair momentum in a topological semimetal, Nat. Phys. 18, 1228 (2022). [58] B. Lu, S. Ikegaya, P. Burset, Y. Tanaka, and N. Na- gaosa, Tunable Josephson diode effect on the surface of topological insulators, Phys. Rev. Lett. 131, 096001