ARTICLE IN PRESS ARTICLE IN PRESS parameters, η(−J) = −η(J) and η(−tso) = −η(tso), and it vanishes whenever J = 0 or tso = 0. These features show that the requisite symmetry breaking is provided cooperatively by the cAFM order and SOC, indicating that both ingredients are essential. They also suggest that the direction of the nonreciprocal transport can be controlled by reversing the sign of either J or tso. Figures 2(c) and 2(d) further examine how the diode effect depends on the cAFM strength J, SOC strength tso and junction length NL. Specifically, Fig. 2(c) shows η as a function of J for a fixed tso = 0.1t and increasing junction lengths, NL = 40, 60, and 80, while Fig. 2(d) shows η as a function of tso for a fixed J = t and the same set of junction lengths. Remarkably, although η varies with J and tso, its overall profile remains quali- tatively robust against changes in NL, exhibiting only modest finite-size effects. This robustness is consistent with long-range equal-spin triplet pairing induced across the junction [94, 95], which primarily carry the nonrecip- rocal supercurrent and sustains a sizable diode efficiency even for large NL. Moreover, the non-monotonic depen- dence of η on J and tso reflects a competition between two trends: increasing with J and tso strengthens the relevant symmetry breaking and enhances the current asymmetry, whereas overly large J or tso suppresses the proximity-induced superconducting correlations and re- duces higher-harmonic components in the current-phase relation. Consequently, η exhibits a maximal value. Altogether, these results demonstrate that the interplay among the kagome lattice geometry, cAFM order, and SOC naturally produces a sizable Josephson diode effect even in the absence of external magnetic fields. We note that a similar diode effect driven by interfacial asymmetry has been recently discussed in a Josephson junction based on WTe2 [108]. Tunable ϕ0-junction state and anomalous Joseph- son effect Beyond the diode effect, the kagome SC/cAFM/SC junction also supports tunable ϕ0-junction states and anomalous Josephson currents. We define ϕ∗ F as the phase that minimizes the junction free energy F(ϕ), F(ϕ∗ F ) = minϕ F(ϕ) ≡F0, which determines the ground- state phase difference of the junction. As Is(ϕ) = (2e/ℏ)∂F(ϕ)/∂ϕ, this equilibrium phase satisfies Is(ϕ∗ F ) = 0. In conventional Josephson junctions, ϕ∗ F is pinned to 0 or π. A ϕ0-junction corresponds to ϕ∗ F ̸= 0, π. In the present system, the interplay between the cAFM order and SOC shifts and tunes ϕ∗ F , realizing a controllable ϕ0-junction state. This anomalous phase shift is reflected in the evolution of the free-energy landscape F(ϕ) −F0. The inset of Fig. 3(a) shows that, without SOC (tso = 0), the free energy is symmetric in ϕ, while a finite tso breaks this symmetry and generally shifts the minimum to ϕ∗ F ̸= 0, π. Figure 3(a) further displays ϕ∗ F as a function of tso at fixed J = t for two representative superconducting chemical (a) (b) (c) (d) Fig. 3. ϕ0-junction state and anomalous Josephson current in the SC/cAFM/SC junction with SOC. (a) Phase position ϕ∗ F of the lowest free energy F0 ≡minϕ[F(ϕ)] as a function of SOC strength tso for J = t, µS = 0.2t (blue) and µS = 0.6t (yellow). Inset: free energy F measured relative to F0 as a function of the superconducting phase difference ϕ for J = t and tso = 0, 0.1t, 0.2t. (b) ϕ∗ F of F0 as a function of J for tso = 0.1t and µS = 0.2t. Phase diagram of (c) the phase position ϕ∗ F and (d) Anomalous Josephson current Ia as functions of J and tso for µS = 0.2t. Other parameters are µAFM = 0.2t, ∆= 0.02t, NL = 40, and kBT = 0.02∆. potentials, µS = 0.2t (blue) and µS = 0.6t (yellow). We see that in presence of SOC, ϕ∗ F generally deviates from 0 and π. For high interface transparency (small Fermi- level mismatch µAFM = µS = 0.2t), F(ϕ) may develop two degenerate minima. The presence of SOC lifts this degeneracy so that the global minimum can switch be- tween the competing minima, resulting in abrupt changes of ϕ∗ F in the weak-tso regime. In contrast, for interface reduced transparency (e.g., large mismatch, µAFM = 0.2t, µS = 0.6t), the energy degeneracy is suppressed and ϕ∗ F evolves smoothly with tso. Figure 3(b) further shows that, at fixed tso, changing the cAFM strength J can sweep ϕ∗ F , highlighting the broad controllability of the ground-state phase. The phase diagram ϕ∗ F (J, tso) in Fig. 3(c) further confirms the wide tunability. Notably, for J < |µAFM|, two Fermi surfaces are present at each valley. In this regime, the Josephson current receives multiple coherent contributions(from the two pockets and inter-pocket pro- cesses that can correspond to finite-momentum pairing). The resulting interference leads to oscillations in the phase position. In contrast, when J > |µAFM|, only a single Fermi surface remains per valley, inter-pocket channels are absent, and ϕ∗ F evolve smoothly with J and tso. The finite shift of ϕ∗ F in the junction is accompanied by an anomalous Josephson effect, namely a finite supercur- rent at zero phase difference, Is(0) ̸= 0. To quantify this response, we define the anomalous Josephson current as Ia ≡Is(ϕ = 0). A nonzero Ia implies a deviation from an odd current-phase relation and, in our system, requires
ARTICLE IN PRESS ARTICLE IN PRESS the simultaneous breaking of I, T , and T Mz, as for the diode effect. As shown in Fig. 2(a), in the absence of SOC, the junction preserves T Mz, which enforces Ia = 0, whereas introducing SOC breaks this symmetry and yields Ia ̸= 0. The phase diagram of Ia as a function of J and tso further shows that Ia = 0 whenever either J = 0 or tso = 0 [see Fig. 3(d)], consistent with the requirement of simultaneous symmetry breaking. Moreover, Ia can be tuned by these parameters, and its sign (i.e., the direction of the anomalous supercurrent) reverses upon changing the sign of either J or tso, demonstrating controllable inversion of the anomalous Josephson response. Diode effect in the SC/cAFM/cAFM′/SC junction induced by Zeeman fields To further demonstrate that breaking the combined T Mz symmetry, rather than SOC itself, is crucial for realizing the Josephson diode effect, we now propose and study an SC/cAFM/cAFM′/SC junction without SOC, as il- lustrated in Fig. 1(b). From the previous setup, we have seen that, in the absence of SOC, inversion symmetry breaking induced solely by the junction interface is in- sufficient to produce a diode effect. To overcome this limitation, in the present second setup, we consider two cAFM regions in the junction with a relative angle θr, which naturally breaks spatial inversion symmetry. Such a bi-cAFM configuration still preserves the T Mz sym- metry and therefore cannot by itself generate a diode effect, as we will show below. This motivates us to further introduce a Zeeman exchange field into the junction. The total Hamiltonian of the junction is then given by HSNNS = H0 + H∆+ ˜HcAFM + HB. (20) Here, the pairing term H∆is given by Eq. (5) but with a spatially depdent order paramter ∆ℓ= Θ(ℓ+ NL1)∆0e−iϕ/2+Θ(ℓ−NL2)∆0eiϕ/2. NL1 and NL2 denote the lengths of the two cAFM regions. Accordingly, the cAFM term ˜HcAFM is given by Eq. (6) but with mℓ,ν = ˜m1,νΘ(ℓ+NL1 +1)Θ(−ℓ+1)+ ˜m2,νΘ(ℓ)Θ(−ℓ+NL2 +1). ˜mα,ν = J(cos ˜θα,ν, sin ˜θα,ν, 0) is the magnetic moment at sublattice ν with strength J and direction ˜θα,ν = 2(3 −ν)π/3 + θ∗ α in the α-th (α = 1, 2) cAFM regions. The relative angle between the chiral anti-ferromagnetic orders in the two cAFMs is defined as θr ≡θ∗ 2 −θ∗ 1. Finally, the Zeeman term is written HB = X ℓ,ν Ψ† ℓ,ν Bℓ· s Ψℓ,ν, (21) where Bℓ= BΘ(ℓ+ NL1 + 1)Θ(−ℓ+ NL2 + 1) and B = (Bx, By, Bz) is the Zeeman exchange field. An in-plane Zeeman field, B = (Bx, By, 0), breaks T and Mz individually, but each operation flips the in-plane spin once. Thus, the in-plane Zeeman field preserves the combined symmetry T Mz. In contrast, an out-of-plane Zeeman field, B = (0, 0, Bz), breaks T while preserving Mz, and hence violates T Mz. (a) (b) (c) (d) Fig. 4. Josephson diode effect in the SC/cAFM/cAFM′/SC junction. (a) Current-phase relations for J = 0.4t, under an out-of-plane Zeeman field Bz = 0.1t and for relative cAFM angles θr = 0, 0.25π, 0.5π and 0.75π. The corresponding diode efficiencies are η = 0, 0.018, 0.067 and 0.173, respectively. (b) Same as panel (a) but for an in-plane Zeeman field applied along the x-direction (Bx = 0.1t, solid line) and the y-direction (By = 0.1t, dashed line). No diode effect is observed for either in-plane field. (c) Diode efficiency η as a function of the relative cAFM angle θr for J = 0.4t with Zeeman fields B = (0.1t, 0, 0), (0, 0.1t, 0), and (0, 0, 0.1t) in the x-, y-, and z- directions, respectively. (d) η as a function of J for θr = 0.5π, with Zeeman fields B = (0.1t, 0, 0), (0, 0.1t, 0), and (0, 0, 0.1t) applied along the x-, y-, and z-directions, respectively. Other parameters are µAFM = 0.2t, µS = 0.2t, ∆= 0.02t, NL1 = NL2 = 20, and kBT = 0.02∆. We discuss the Josephson current and how it can be tuned by the relative angle of the cAFM orders and the Zeeman field in the SC/cAFM/cAFM′/SC junc- tion. In the presence of an out-of-plane Zeeman field B = (0, 0, 0.1t), the current-phase relation is symmetric for θr = 0 or π, while becomes asymmetric when the cAFM orders acquire a relative angle θr (different from π, e.g., 0.25π, 0.5π, 0.75π) [see Fig. 4(a)]. Correspond- ingly, a pronounced diode effect develops (with the diode efficiencies η = 0.018, 0.067, 0.173, respectively). These results demonstrate that nonuniform cAFM order pro- vides a natural source of inversion-symmetry breaking, while a Zeeman exchange field in the z-direction further breaks the T Mz combined symmetry, thereby enabling nonreciprocal supercurrent transport. In contrast, Zee- man fields applied in the x- or y- directions preserve T Mz, leading to a symmetric current-phase relation and prohibiting the diode effect [see Fig. 4(b)]. The θr de- pendence of the diode efficiency η in Fig. 4(c) highlights this symmetry constraint: η is finite only for out-of-plane Zeeman fields and is odd in θr. Moreover, it shows that the diode efficiency can be continuously tuned by chang- ing the relative angle θr and can reach substantial values
ARTICLE IN PRESS ARTICLE IN PRESS (a) (b) (c) (d) Fig. 5. ϕ0-junction state and anomalous Josephson current in the SC/cAFM/cAFM′/SC junction. (a) Phase position ϕ∗ F of lowest free energy F0 as a function of the relative angle θr between the two cAFM orders for J = 0.4t with (Bz = 0.3t) and without (Bz = 0) Zeeman field along z-direction. (b) Anomalous Josephson supercurrent Ia as a function of θr for J = 0.4t with (Bz = 0.3t) and without (Bz = 0) Zeeman field along z-direction. Phase diagram of (c) ϕ∗ F and (d) Ia as functions of J and θr for Bz = 0.3. Other parameters are µAFM = 0.2t, µS = 2t, ∆= 0.02t, NL1 = NL2 = 20, and kBT = 0.02∆. (exceeding 20%). At θr = 0.5π, η exhibits pronounced oscillations with cAFM strength J for a fixed out-of-plane Zeeman field (Bz = 0.1t) and is even in J, but remains negligible for in-plane Zeeman fields [see Fig. 4(d)]. For a bilayer ferromagnetic counterpart, where the Néel vectors in the two ferromagnets have an arbitrary relative angle, no diode effect is found. This comparison indicates that a non-collinear chiral antiferromagnetic order is essential for realizing the diode effect. Taken together, these findings establish that the Josephson diode effect in this system originates from the cooperative breaking of I, T , and T Mz symmetries by nonuniform cAFM order and an out-of-plane Zeeman field. Finally, we investigate the ϕ0-junction and anomalous Josephson current in the SC/cAFM/cAFM′/SC structure. Figures 5(a,b) show the phase position ϕ∗ F (defined by the minimum of the free energy F0) and the anomalous Josephson current Ia as functions of the relative angle θr at fixed J = 0.4t, both without and with an out-of-plane Zeeman field Bz. In the absence of Zeeman fields (Bz = 0, red), ϕ∗ F is pinned at π, indicating a robust π junction and the T Mz symmetry enforces Ia = 0 for all θr. Applying a Zeeman field (Bz = 0.3t, blue) breaks T Mz, shifting the free energy minimum away from 0 (≡2π) and π and allowing ϕ∗ F to vary smoothly with θr over the full interval [0, 2π], thus realizing a tunable ϕ0-junction. At the same time, a finite anomalous Josephson current Ia emerges and evolves continuously with changing θr. Notably, Ia (a) (b) Fig. 6. Diode effect and ϕ0-junction state for Bz ∈[−2∆, 2∆]. (a) Diode efficiency η as a function of out-of-plane Zeeman field Bz for J = −0.36t. (b) Phase position ϕ∗ F as a function of Bz at J = −0.36t. Other parameters are NL1 = NL2 = 20, θr = 0.5π, µAFM = 0.2t, µS = 0.2t, ∆= 0.02t and kBT = 0.02∆. vanishes at θr = 0 and π, and reaches its maximum near θr = π/2 and 3π/2, demonstrating pronounced tunability controlled by the relative orientation of the two cAFM layers. To further elucidate this behavior, Figs. 5(c) and (d) present the phase diagrams of ϕ∗ F and Ia as functions of J and θr. Both quantities exhibit smooth and continuous evolution across the parameter space, showing that the ϕ0-junction and anomalous Josephson current can be substantially controlled by the cAFM strength and the relative angle. In the above discussion, we used relatively large value of Bz (e.g., Bz = 0.1t or 0.3t) to highlight the mechanism. Note, however, that the qualitative conclusions remain valid in the weak Bz regime, which may be experimentally more relevant. To illustrate this, we calculate the diode efficiency η and the phase position ϕ∗ F as functions of Bz within Bz ∈[−2∆, 2∆] for J = −0.36t. As shown in Fig. 6(a), the diode efficiency η increases from zero as the magnitude of Bz grows, reaches a maximum (η ∼22%) at intermediate fields, and then decreases for larger |Bz|. Meanwhile, the emergence of a finite ϕ∗ F shift of the free- energy minimum away from ϕ∗ F = 0, π, indicating the ϕ0-junction state. Figure 6(b) shows that ϕ∗ F evolves con- tinuously with increasing Bz. These results demonstrate that a sizable diode effect and anomalous phase shift can already be induced by weak Zeeman fields compatible with the pairing potential ∆of the superconducting leads. Discussion We have investigated the Josephson diode and anomalous Josephson effects in planar Josephson junctions incorpo- rating s-wave superconductors and cAFMs on the kagome lattice, where the noncollinear chiral magnetic texture provides an intrinsic platform for directional Cooper-pair transport without relying on net magnetization. Our study demonstrates that both nonreciprocal Josephson transport and the anomalous Josephson effect emerge only when inversion I, time-reversal T , and combined mirror-time-reversal T Mz symmetries are simultaneously broken. Guided by this symmetry analysis, we propose