ARTICLE IN PRESS https://doi.org/10.1038/s41535-026-00907-2 Received: 18 December 2025 Accepted: 28 May 2026 Cite this article as: Hou, J.-X., Li, C., Hu, L.-H. et al. Field-free Josephson diode and tunable ϕ0-junction in chiral kagome antiferromagnets. npj Quantum Mater. (2026). https:// doi.org/10.1038/s41535-026-00907-2 Jin-Xing Hou, Chuang Li, Lun-Hui Hu & Song-Bo Zhang We are providing an unedited version of this manuscript to give early access to its findings. Before final publication, the manuscript will undergo further editing. Please note there may be errors present which affect the content, and all legal disclaimers apply. If this paper is publishing under a Transparent Peer Review model then Peer Review reports will publish with the final article. © The Author(s) 2026. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. npj Quantum Materials Article in Press Field-free Josephson diode and tunable ϕ0-junction in chiral kagome antiferromagnets
ARTICLE IN PRESS ARTICLE IN PRESS Field-free Josephson diode and tunable ϕ0-junction in chiral kagome antiferromagnets Jin-Xing Hou,1, ∗Chuang Li,2, 1, ∗Lun-Hui Hu,2, † and Song-Bo Zhang1, 3, ‡ 1Hefei National Laboratory, Hefei, 230088, China 2Center for Correlated Matter and School of Physics, Zhejiang University, Hangzhou 310058, China 3School of Emerging Technology, University of Science and Technology of China, Hefei, 230026, China Abstract The recent realization of superconducting proximity effects in chiral antiferromagnets (cAFMs) opens a new route to nonreciprocal superconducting transport of fundamental interest and practical importance. Using microscopic modeling and symmetry analysis, we show that Josephson junctions formed by conventional s-wave superconductors (SCs) and cAFMs on the kagome lattice exhibit both Josephson diode effects and anomalous Josephson effects (a ϕ0-junction state with a finite supercurrent at zero phase bias) when spacial inversion I, time-reversal T , and the combined mirror-time-reversal T Mz symmetries are simultaneously broken. We propose two setups to realize these phenomena and achieve high diode efficiency. (i) An SC/cAFM/SC junction with spin-orbit coupling, which enables a field-free diode effect together with a tunable anomalous Josephson supercurrent. (ii) An SC/cAFM/cAFM′/SC junction, where two cAFM layers with different in-plane order orientations, under an out-of-plane Zeeman field, produce a pronounced diode effect and anomalous Josephson response. These results establish a direct link between T Mz symmetry breaking and nonreciprocal superconductivity, suggesting cAFMs as versatile platforms for symmetry-engineered Josephson diodes and tunable ϕ0-junctions. Introduction The dc Josephson effect is a macroscopic quantum phe- nomenon in which a supercurrent flows between two su- perconductors (SCs) without an applied voltage [1]. It stems from phase-coherent tunneling of Cooper pairs and underlies a broad range of superconducting phenomena and technologies. In conventional junctions that preserve time-reversal and inversion symmetries, the current-phase relation is odd, I(ϕ) = −I(−ϕ), which enforces equal forward and backward critical currents, I→ c = I← c . When these symmetries are broken, the current-phase relation can become asymmetric under ϕ →−ϕ and a Josephson diode effect may emerge, I→ c ̸= I← c [2–11]. Similar to the intrinsic superconducting diode effect realized with- out junctions [12–24], this nonreciprocal superconducting transport in Josephson junctions enables dissipationless rectification [25–27] and is driving intensive interest for quantum information processing [28–32] and supercon- ducting spintronic devices [33–43]. Beyond unequal critical currents, the same symmetry breaking can also induce an anomalous phase bias in the Josephson response. In particular, the current–phase rela- tion can become phase-shifted so that a finite supercurrent flows even at zero phase difference, Is(0) ̸= 0, giving rise to an anomalous Josephson effect. This behavior corresponds to a ϕ0-junction, in which the ground-state phase differ- ence ϕ0 (at the free-energy minimum, where Is(ϕ0) = 0) deviates from the conventional value 0 [44–47]. Such anomalous phase shifts provide sensitive probes of un- derlying symmetries and offer opportunities for supercon- ducting qubit [25, 48, 49], phase batteries [25, 50–54], etc. ∗These authors contributed equally. † [email protected] ‡ [email protected] Josephson diode and ϕ0-junction state have been explored in diverse platforms, including Rashba spin-orbit coupled systems [26, 55, 56], topological materials [8, 32, 57–59], van der Waals heterostructures [60–63], as well as the re- cently discovered collinear altermagnets [64–69]. Despite extensive theoretical [2–4, 30, 70, 71] and experimental progress [15, 17, 72, 73], the realization of field-free, siz- able, tunable Josephson diodes and anomalous Joseph- son currents in zero-net-magnetization platforms remains highly desirable from a device perspective. Recently, the superconducting proximity effect has been realized in chiral antiferromagnets (cAFMs) on the kagome lattice, most notably in Mn3Ge thin films [74, 75]. These materials host noncollinear order in the kagome plane, which breaks time-reversal symmetry T and mirror symmetry Mz about the kagome plane, while retaining zero net magnetization. As a consequence, they exhibit prominent nonrelativistic spin splitting in the electronic bands [76–85]. Owing to these characteristics, which bear similarities to those of altermagnets, cAFMs have sometimes been discussed as a distinct noncollinear vari- ant within extended symmetry-based classifications of altermagnetism [86]. These intrinsic properties give rise to many exotic quantum transport phenomena such as giant anomalous Hall effect [76–79], anomalous Nernst effect [80, 87, 88], and current-induced spin torques [89– 93]. Moreover, their peculiar spin texture and spin split- ting have been shown to promote pronounced spin-triplet pairing correlations that contributes substantially to the supercurrent [94–96]. While the bulk cAFMs are inver- sion symmetric, inversion symmetry can be broken at interfaces in practical cAFM-based hybrid junctions due to the kagome lattice geometry and structural asymme- try. Despite these favorable ingredients and experimental feasibility, their potential for nonreciprocal Josephson transport has yet to be systematically investigated. Motivated by this, we study Josephson junctions com-
ARTICLE IN PRESS ARTICLE IN PRESS posed of conventional s-wave SCs and cAFMs on kagome lattices, where the noncollinear chiral spin texture pro- vides an intrinsic route toward nonreciprocal supercon- ducting transport without requiring a net magnetization or external magnetic fields. We find that breaking the combined mirror-time-reversal symmetry T Mz is a key requirement for the Josephson diode effect beyond the breaking of time-reversal and spatial inversion alone. This symmetry principle further provides a general guideline for engineering nonreciprocal Josephson responses in kagome- based cAFM systems. This symmetry principle provides a general guideline for engineering nonreciprocal Josephson responses in kagome-based cAFM systems. To illustrate this, we propose two experimentally feasible device im- plementations, as sketched in Fig. 1. The first setup is an SC/cAFM/SC junction in which spin–orbit coupling (SOC) is introduced to explicitly break T Mz and inver- sion I at the junction interfaces. This enables a field-free Josephson diode and a tunable ϕ0-junction. The second setup is an SC/cAFM/cAFM′/SC junction operated with an out-of-plane Zeeman exchange field (or spin canting), where the relative orientation of the cAFM orders in the two cAFM layers controls sizable diode responses and anomalous phase shifts. Notably, these effects do not oc- cur in ferromagnetic counterparts, indicating the essential role of cAFM order. Our results suggest that combin- ing kagome geometry with chiral magnetic textures and controlled symmetry breaking provides a promising route toward realizing Josephson diodes and related supercon- ducting spintronic functionalities. Results Model Hamiltonian and symmetry analysis We start with the SC/cAFM/SC Josephson junction, in which a cAFM is sandwiched between two conventional s- wave superconducting leads [see Fig. 1(a)]. The junction is oriented along the y-direction and translational symmetry is assumed in the x-direction. Thus, the momentum kx remains a good quantum number and the system can be treated as a stack of “rows” of triangular magnetic unit cells along the y-direction. The total Hamiltonian of the junction is given by HSNS = H0 + H∆+ HcAFM + Hso, (1) where H0 is the prototypical tight-binding Hamiltonian on the kagome lattice H0 = X kx,l Ψ† ℓH(kx)Ψℓ+ Ψ† ℓV(kx)Ψℓ+1 + h.c. , (2) with the spinor basis Ψ† ℓ= (c† ℓ,1,↑, c† ℓ,2,↑, c† ℓ,3,↑, c† ℓ,1,↓, c† ℓ,2,↓, c† ℓ,3,↓). The operator c† ℓ,ν,σ creates an electron with spin σ ∈{↑, ↓} at sublattice ν ∈{1, 2, 3} of the ℓ-th row of unit cells along the y-direction. Here, for ease of notation, we suppress kx dependence in the basis cℓ,ν,σ ≡ckx,ℓ,ν,σ. SC cAFM cAFM SC SC SC cAFM (a) (b) 2 3 1 SOC Fig. 1. Schematic of two Josephson setups based on cAFM on the kagome lattice. (a) The SC/cAFM/SC junction with SOC: the site arrows show the local magnetic moments, and the bond arrows mark the coplanar unit vectors nµν associated with SOC in the cAFM. (b) The SC/cAFM/cAFM′/SC junction with two cAFM layers in middle: the two noncollinear orders differ by a relative angle θr and an out-of-plane Zeeman field Bz is applied there. h.c. indicates the Hermitian conjugate of the preceding term. H(kx) = s0h(kx) and V(kx) = s0V (kx) correspond to the local term for each row and the hoping matrix between neighboring rows, respectively. s0 is the unit matrix and s = (sx, sy, sz) is the Pauli matrix vector in spin space. The matrices h(kx) and V (kx) are h(kx) = −t µ/t −1 1 + e−ikx 1 1 + eikx µ/t −1 1 1 1 µ/t −1 , (3) V (kx) = −t 0 0 0 0 0 0 1 e−ikx 0 , (4) where t is the hopping amplitude between neighboring lattice sites, and µ is the chemical potential. We set the lattice constant a = 1, t to be the unit of energy, and the Dirac points of the kagome model at zero energy. The second term H∆in Eq. (1) describes the s-wave pairing potential in the superconducting leads H∆= X ℓ ∆ℓ[Ψ† ℓ,↑(Ψ† ℓ,↓)T −Ψ† ℓ,↓(Ψ† ℓ,↑)T ] + h.c., (5) where Ψ† ℓ,σ = (c† ℓ,1,σ, c† ℓ,2,σ, c† ℓ,3,σ) with σ ∈{↑, ↓}, and ∆ℓ= Θ(ℓ+ NL/2)∆0e−iϕ/2 + Θ(ℓ−NL/2)∆0eiϕ/2. ∆0 is the magnitude of the pairing potential, ϕ is the phase difference between the two SCs, Θ(x) is the Heaviside step function, and NL is the distance between two SCs (i.e., the length of cAFM region) in units of row spacing √ 3a. The third term in Eq. (1) is an on-site exchange describing the non-collinear magnetic order in the junction HcAFM = X ℓ,ν Ψ† ℓ,ν mℓ,ν · s Ψℓ,ν, (6) where mℓ,ν = Jℓ(cos θν, sin θν, 0) denotes the local mag- netic moment on the ν-sublattice (ν ∈{1, 2, 3}) in the
ARTICLE IN PRESS ARTICLE IN PRESS ℓ-th row of unit cells. The exchange strength is given by Jℓ = JΘ(ℓ + NL/2 + 1)Θ(−ℓ + NL/2 + 1)and in-plane an- gles θν specify the in-plane magnetic configuration. The sublattice moments mν form 120◦ in-plane pattern [see Fig. 1], which we parameterize them asθ2/3 = θ1 ± 2π/3 or θ2/3 = θ1 ∓ 2π/3. In the kagome antiferromagnet, SOC may exist intrin- sically, as evidenced by large anomalous Hall effects in the materials [78,97–100]. It can be described by [76] Hso = itso X ⟨··· ⟩,σ,σ′ λµνnµν · sσσ ′c† j,ℓ,µ,σcj′,ℓ′,ν,σ′, (7) where c† j,ℓ,ν,σ (cj,ℓ,ν,σ) creates (annihilates) an electrons on the sublatticeν ∈ {1, 2, 3} of the unit cell located at column j and row ℓ, and the spin indices areσ, σ′ ∈ {↑ ,↓}. The notation ⟨· · · ⟩ ≡ ⟨j, ℓ, µ; j′, ℓ′, ν⟩restricts the summation to nearest neighbors, tso denotes the SOC strength, and λµν = −λνµ with λ12 = λ23 = λ31 = 1. nµν = nνµ = (cos φµν, sin φµν, 0) are three coplanar unit vectors on the bond connecting sublatticesµ and ν, where φ12 = π/2, φ23 = 7π/6 and φ31 = −π/6. In the basis introduced in Eq. (2), the SOC terms is rewritten as Hso = Xh LX ℓ=−L fµν(kx)n µν ·s σσ ′ c† ℓ,µ,σcℓ,ν,σ′ + L−1X ℓ=−L gµν(kx)n µν ·s σσ ′ c† ℓ,µ,σcℓ+1,ν,σ′ + L−1X ℓ=−L g† µν(kx)n µν ·s σσ ′ c† ℓ+1,µ,σcℓ,ν,σ′ i ,(8) where P ≡ P kx,µ,ν,σ,σ′, L = ⌈NL/2⌉ denotes rounding a number up to the nearest integer ofNL/2, the matrix functions f (kx) and g(kx) are given by f (kx) = itso 0 −1 − e−ikx −1 1 + eikx 0 −1 1 1 0 , (9) g(kx) = itso 0 0 0 0 0 0 1 e−ikx 0 . (10) In cAFMs such as the Mn3Ge family, the noncollinear magnetic order is stabilized primarily by strong exchange interactions, while relativistic SOC is typically a sub- leading energy scale and expected to cause at most a moderate canting of the magnetic moments rather than a qualitative reconstruction of the cAFM texture [76]. Since our focus is the symmetry-breaking role of SOC in the electronic Hamiltonian and its impact on Joseph- son transport, we treat SOC as a parameter acting on the itinerant electrons and keep the cAFM background fixed. A fully self-consistent magnetic calculation would require additional microscopic input beyond the scope of this work and is not expected to qualitatively affect the conclusions in the exchange-dominated regime. In the presence of any symmetry that flips the sign of the superconducting phase difference in the system Hamil- tonian, i.e., H(ϕ) → H(−ϕ), while simultaneously revers- ing the supercurrent direction, Is → −Is, the current- phase relation must satisfy Is(ϕ) = −Is(−ϕ). Thus, to realize a Josephson diode effect, all such symmetries must be broken. In the kagome-cAFM-based Josephson junction, the relevant symmetries include inversion I, time-reversal T , and the combined mirror-time-reversal symmetry T Mz. We first examine inversion symmetry and it microscopic action on the kagome lattice. For concreteness and with- out loss of generality, we choose the inversion center at the sublattice ‘3’ of the central row ℓ = 0 in the cAFM. With this choice, inversion acts on the annihilation operators as cj,ℓ,1,σ → c−j,−ℓ+1,1,σ, (11) cj,ℓ,2,σ → c−j−1,−ℓ+1,2,σ, (12) cj,ℓ,3,σ → c−j,−ℓ,3,σ. (13) This transformation leaves the sublattice index unchanged but shifts the unit-cell coordinates according to the kagome geometry. In the junction with translation sym- metry along thex-direction but broken in they-direction, it is instructive to work in the hybrid real-momentum representation. To this end, we perform a partial Fourier transform and recast the inversion operation as ckx,ℓ,1,σ → c−kx,−ℓ+1,1,σ, (14) ckx,ℓ,2,σ → eikx c−kx,−ℓ+1,2,σ, (15) ckx,ℓ,3,σ → c−kx,−ℓ,3,σ, (16) where the eikx factor in Eq.(15) arises from the trans- lation of sublattice “2” along the column indexj (the x direction). The action of inversion on creation operators follows analogously, with i → −i. For a bulk kagome system with homogeneous parameters andϕ = 0, all the terms in the Hamiltonian respect inversion symmetry. In a finite Josephson junction, however, inversion symmetry will be broken at the interfaces because the kagome unit cells at the boundaries do not map onto themselves. This geometric asymmetry already affects the local terms, such as H∆ and HcAFM, it becomes even more pronounced for thenonlocalSOCterm. Explicitly, theinversionoperation yields IHsoI −1 = Hso + ∆Hinter, where the additional interface terms Hinter = X kx,σ,σ′ sσσ ′ · f21(kx)n 21 � c† kx,L+1,2,σ ×c kx,L+1,1,σ′ −c † kx,−L,2,σckx,−L,1,σ′ +g 31(kx)n 31 � c† kx,L,3,σckx,L+1,1,σ′ −c † kx,−L,3,σckx,−L+1,1,σ′ +g 32(kx)n 32 � c† kx,L,3,σckx,L+1,2,σ′ −c † kx,−L,3,σckx,−L+1,2,σ′ +h.c. (17) ARTICLE IN PRESS ARTICLE IN PRESS arise because inversion maps SOC bonds at one inter- face to bonds that do not exist at the opposite interface. Therefore, Hso becomes intrinsically inversion-asymmetric when restricted to the finite junction. The junction thus explicitly lacks inversion symmetry, providing one of the essential prerequisites for a finite Josephson diode effect. Time-reversal symmetry T is a local antiunitary sym- metry that flips both spin and momentum [101, 102]. In the (spin ⊗sublattice) basis Ψ, the time-reversal operator reads T = isyI3×3K, (18) where the three-by-three unit matrix I3×3 acts in sub- lattice space and K denotes the complex conjugation. For ϕ = 0, the terms H0, Hso and H∆are invariant un- der time reversal. However, the cAFM exchange term HcAFM filps sign, T HcAFMT −1 = −HcAFM. Thus, T is also explicitly broken in the cAFM region. This provides another necessary ingredient for realizing the Josephson diode effect. Finally, we analyze mirror reflection Mz about the kagome plane. This mirror symmetry flips in-plane spin components while preserving out-of-plane component, Mz : (sx, sy, sz) →(−sx, −sy, sz). Its matrix representa- tion can be written as [103, 104] Mz = iszI3×3. (19) As the magnetic moments of the cAFM lie in the plane, MzHcAFMM−1 z = −HcAFM. Thus, the coplanar cAFM order breaks T and Mz individually. However, because the texture is coplanar, the combined symmetry TMz is preserved, as indicated by (T Mz)HcAFM(T Mz)−1 = HcAFM. Thus, for the whole junction Hamiltonian H(ϕ) (including HcAFM), one has (T Mz)H(ϕ)(T Mz)−1 = H(−ϕ), which enforces a reciprocal current–phase relation. To generate a Josephson diode in kagome cAFMs, the ad- ditional breaking of T Mz is required. The SOC provides precisely such a symmetry breaking. time-reversal T flips the spin and T iT −1 = −i, leading to T HsoT −1 = Hso. In contrast, mirror reflection Mz flips the in-plane spin components and gives MzHsoM−1 z = −Hso. Therefore, the SOC intrinsically breaks the T Mz symmetry of the kagome system, i.e., (T Mz)Hso(T Mz)−1 = −Hso [76]. Field-free Josephson diode effect We now demonstrate the emergence of the Josephson diode effect in the kagome SC/cAFMs/SC junction. The Josephson supercurrent can be obtained from the free en- ergy [1] or from the Green function [105–107], see Methods for details. For numerical efficiency and to avoid finite- size effects associated with the superconducting leads, we alternatively compute it employing the surface Green function approach. Figure 2(a) displays representative current–phase rela- tions for a fixed cAFM strength (J = t) and several SOC strengths (tso = 0, 0.1t, and 0.2t). These current-phase (a) (c) (d) Fig. 2. Josephson diode effect in the SC/cAFM/SC junction with SOC. (a) Current-phase relations for cAFM strength J = t, junction length is NL = 40 and different SOC strengths tso = 0, 0.1t and 0.2t. (b) Diode efficiency η as a function of J and tso for NL = 40. (c) η as a function of J for tso = 0.1t and NL = 40, 60 and 80. (d) η as a function of tso for J = t, NL = 40, 60 and 80. Other parameters are µAFM = 0.2t, µS = 0.2t, ∆= 0.02t and temperature kBT = 0.02∆. relations can also be obtained from the derivative of the free energy shown in the inset of Fig. 3. For illustration, we take the junction length NL = 40 (in units of row spacing √ 3a), set the chemical potentials of the cAFM and the SC to µAFM = µS = 0.2t, the superconducting pairing potential to ∆= 0.02t, and the temperature to kBT = 0.02∆. In the absence of SOC (tso = 0), the current-phase relation is anti-symmetric under ϕ →−ϕ, indicating identical forward and backward critical cur- rents and thus no diode effect. Strikingly, once a finite SOC (tso ̸= 0) is introduced, this antisymmetry is lifted and the forward and backward critical currents, I→ c and I← c , become unequal. The resulting diode efficiency, de- fined as η ≡(I→ c −I← c )/(I→ c +I← c ), is therefore finite only when SOC is present. This demonstrates that the coexis- tence of SOC and cAFM order is capable of generating a Josephson diode effect in the kagome junction. A sizable diode effect arises only when the symmetries protecting reciprocal supercurrent are sufficiently lifted, thus a finite diode efficiency η is rooted in this symmetry breaking. In particular, the SOC breaks inversion I at the junction interface as well as the combined symmetry T Mz, while the cAFM order breaks time-reversal sym- metry T . To study the connection between the diode effect and the interplay between cAFM and SOC system- atically, we calculate the diode efficiency η as a function of the cAFM strength J and the SOC strength tso, the results of which are presented in Fig. 2(b). Strikingly, a pronounced efficiency exceeding 30% can be achieved. Furthermore, the efficiency is an odd function of both