Transverse Anomalous Josephson Effect

• Transverse AJE is a phenomenon in Josephson junctions where a supercurrent flows perpendicular to the applied phase bias due to symmetry breaking from altermagnetic order and Rashba SOC.
• It is realized in a four-terminal cross-shaped device that leverages momentum-selective pairing to generate nonreciprocal, diode-like supercurrent behavior without external magnetic fields.
• Theoretical models predict tunable transverse currents with high diode coefficients, offering promising applications in field-free superconducting logic and spintronic devices.

The transverse anomalous Josephson effect (AJE) refers to the generation of a supercurrent perpendicular to the applied phase bias in a Josephson junction, where the current exhibits an intrinsic phase offset (anomalous phase) and diode-like nonreciprocal behavior. This phenomenon requires the simultaneous breaking of specific symmetries in the junction's central region. In altermagnets with Rashba spin–orbit coupling (SOC), the effect can be realized in multiterminal Josephson devices entirely without external magnetic fields, owing to their distinctive symmetry and electronic structure properties (Sahoo et al., 17 Sep 2025).

1. Microscopic Mechanism in Altermagnetic Systems

Altermagnets combine features of antiferromagnets and ferromagnets: their bands display spin-polarized Fermi surfaces but, unlike ferromagnets, their net magnetization vanishes because contributions from different sublattices or momentum sectors cancel macroscopically. The defining property is a momentum-dependent spin splitting—electronic states at $\mathbf{k}$ and $-\mathbf{k}$ have opposite spin polarizations—leading to time-reversal symmetry breaking at the microscale despite a globally vanishing magnetic moment.

The addition of Rashba SOC, with the Hamiltonian term

$$H_{\mathrm{SOC}} = \alpha_x \sin(k_y a) - \alpha_y \sin(k_x a),$$

locks spin orientation to crystal momentum, mixing spin and orbital degrees of freedom. The coexistence of altermagnetic order and Rashba SOC produces a highly anisotropic, momentum-selective spin texture and breaks both inversion and (microscopic) time-reversal symmetries. As a direct consequence, the Josephson current becomes intrinsically direction-dependent: pairing amplitudes and Andreev processes differ substantially for different directions in momentum space, leading to the appearance of finite supercurrents at zero phase bias (anomalous Josephson current) and the possibility for the transverse (perpendicular) flow of Cooper pairs—a signature of the transverse AJE.

2. Four-Terminal Multiterminal Geometry: Realizing Transverse AJE

The prototypical device consists of a cross-shaped Josephson junction with a central altermagnetic region subject to Rashba SOC, contacted by four $s$-wave superconducting leads. The left and right leads are set to phases $+\varphi_S/2$ and $-\varphi_S/2$, establishing a longitudinal phase bias, while the top and bottom leads remain at zero phase.

This setup enables the following:

• Longitudinal phase bias ($\varphi_S$) induces a conventional (longitudinal) Josephson current ($J_x$).
• Crucially, because of the symmetry breaking, a transverse Josephson current ($J_y$) is generated in the top and bottom terminals—even though no direct phase bias is applied along that direction.

This effect is fundamentally different from the ordinary Hall effect or the classical Josephson effect, as the transverse supercurrent arises purely from equilibrium processes (no net voltage, no dissipation) and is encoded in the ground-state properties of the hybrid system.

3. Characteristics: Anomalous Phase and Nonreciprocal (Diode) Behavior

Finite Anomalous Phase Offset

The transverse current $J_y$ remains finite at zero phase difference between the top/bottom superconductors, reflecting an anomalous phase offset in the current-phase relation (CPR). This is the critical signature of the anomalous Josephson effect:

$J_y(\varphi_S = 0) \neq 0.$

Diode-like Nonreciprocity and Unidirectionality

The current-phase relation for the transverse direction exhibits strong nonreciprocity: the critical current for positive $J_y$ differs from that for negative $J_y$. The diode coefficient,

$$\gamma_d = \frac{2(J_d^{\max} + J_d^{\min})}{J_d^{\max} - J_d^{\min}},$$

quantifies this. For the transverse direction ($d = y$), a large $\gamma_y$ signals pronounced diode effect—in other words, supercurrent flow is heavily favored in one direction, and can even become strictly unidirectional for specific dynamical regimes.

Unidirectionality (i.e., $J_y$ flows only in one direction for all $\varphi_S$) is realized when the momentum-space asymmetry is maximized. This occurs, for example, by selecting the Néel vector orientation (the order parameter direction in the AM region) away from symmetry-protected angles, tuning the Rashba SOC strength, or increasing the altermagnetic splitting parameter $t_j$.

4. Tunability and Control

Néel Vector Rotation

A defining feature of the transverse AJE in altermagnets is the direct tunability of both the magnitude and direction of $J_y$ via simple rotation of the Néel vector in the $xy$ plane (parameterized by angle $\varphi$ relative to the $x$ axis). As $\varphi$ is varied,

• The magnitude of both the diode effect and anomalous current can be enhanced, suppressed, or sign-reversed.
• The efficiency (diode coefficient) can be continuously tuned, with some regimes yielding $\gamma_y$ values exceeding $1000\%$.

This controllability is due to the explicit dependence of the effective Hamiltonian on $(\cos\varphi, \sin\varphi)$ in both the altermagnetic term and the SOC, giving rise to a momentum-dependent anisotropy that is “steered” by the Néel vector orientation.

Absence of External Magnetic Field

The absence of a net magnetization and the full tunability without external fields distinguish altermagnet-based systems from conventional ferromagnets or spin–orbit-coupled metals. This field-free realization is not only technologically advantageous (suppressing magnetic noise and cross-talk in circuits) but also fundamental: the symmetry properties inherent to altermagnets are fully sufficient to enable robust nonreciprocal superconducting transport.

5. Implications for Superconducting Transport and Device Applications

Field-Free Nonreciprocal Devices: The field-free transverse AJE enables Josephson diodes—superconducting components that allow dissipationless current to flow preferentially in a single direction—without relying on applied fields or extrinsic magnetic elements. This is of particular importance for quantum circuits, where magnetic fields can be detrimental to coherence.

Logic and Interconnect Functionality: The strong tunability and the multiterminal design allow for current routing and rectification in complex superconducting networks. Devices built on this principle could serve as the backbone for low-dissipation logic elements, unidirectional current guides, and quantum interference devices.

Scalability: The four-terminal geometry is intrinsically suited for scalable superconducting architectures, as it extends naturally to more complicated circuits exploiting multiterminal phase control, enabling new classes of nonlocal and topological operations.

6. Theoretical Formulation: Hamiltonian and Diode Efficiency

The central AM region with Rashba SOC is described by the momentum-space Hamiltonian

$$H_{\mathbf{k}} = \xi_{\mathbf{k}0} + 2t_j \left[\cos(k_x a) - \cos(k_y a)\right]_{\varphi} + \alpha_x \sin(k_y a) - \alpha_y \sin(k_x a),$$

where the subscript $\varphi$ denotes dependence on the Néel vector orientation through $(\sigma_x \cos\varphi + \sigma_y \sin\varphi)$. This coupling structure is responsible for both the necessary symmetry breaking and the momentum selective occupation of states, leading directly to the emergence of robust, tunable, and highly nonreciprocal transverse Josephson currents.

The diode coefficient,

$$\gamma_d = \frac{2(J_d^{\text{max}} + J_d^{\text{min}})}{J_d^{\text{max}} - J_d^{\text{min}}},$$

serves as an experimental metric for nonreciprocity and can be controlled through band structure and SOC parameter tuning.

Controlled Parameter	Physical Effect	Device Outcome
Néel vector ($\varphi$)	Direction and sign of the transverse current	Current reversal, tuning
SOC strength ($\alpha$)	Magnitude of nonreciprocity, momentum anisotropy	Efficiency, unidirectionality
AM splitting ($t_j$)	Band splitting, momentum selectivity	Transverse current magnitude

7. Broader Context and Conclusions

The transverse AJE in altermagnets extends the landscape of Josephson physics into new regimes of symmetry-engineered nonreciprocal transport. The field-free realization in multiterminal devices positions altermagnets as foundational platforms for nontrivial superconducting behavior with unprecedented control. The underlying principle—cooperative symmetry breaking via momentum-space spin splitting and spin–orbit locking—suggests a fertile ground for further theoretical investigations and application in superconducting spintronics, quantum computing, and beyond (Sahoo et al., 17 Sep 2025).