Anomalous Josephson Effect (AJE)

Updated 24 April 2026

Anomalous Josephson Effect is a phenomenon where a finite supercurrent exists at zero phase difference due to the simultaneous breaking of inversion and time-reversal symmetries.
It is engineered via mechanisms such as spin–orbit coupling, magnetic exchange fields, and multichannel tunneling, which shift the current–phase relation.
Experimental setups using semiconductor junctions, topological insulators, and altermagnets demonstrate applications in superconducting diodes, phase batteries, and topological quantum circuits.

The anomalous Josephson effect (AJE) refers to the emergence of a finite supercurrent at zero phase difference in a Josephson junction when both time-reversal and inversion symmetries are broken. This produces a ground-state phase offset in the current–phase relation (CPR), commonly parameterized as $I(\phi)=I_c \sin(\phi+\phi_0)$ with $\phi_0\neq0$ the anomalous phase shift. Unlike the conventional Josephson effect—which requires a finite phase bias to support a supercurrent—AJE allows for zero-phase supercurrents, $\phi_0$ -junction physics, supercurrent rectification, and nonreciprocal critical currents. AJE arises from diverse physical mechanisms, including spin–orbit coupling (SOC), magnetic exchange fields, altermagnetism, symmetry-breaking interfacial scattering, or coherent circuit couplings, and underpins new device concepts in superconducting spintronics, phase batteries, topological qubits, and Josephson diodes.

1. Symmetry Principles and Microscopic Origins

The anomalous Josephson effect fundamentally requires simultaneous breaking of both inversion (I) and time-reversal (TR) symmetries. This condition is realized in various platforms:

SOC plus Zeeman Field: Combining Rashba SOC and a Zeeman field (in-plane for planar junctions, arbitrary orientation for nanowires) breaks both I and TR. Rashba SOC locks spin to momentum, while the Zeeman field introduces a preferred spin direction, yielding a finite $\phi_0$ (Hasan et al., 2022, Yokoyama et al., 2014, Monaghan et al., 2024). The simplest model for a planar Rashba–Zeeman JJ is the BdG Hamiltonian:

$H_{\text{BdG}}(\mathbf{k}) = \xi_{\mathbf{k}} \tau_z + \alpha(\sigma_x k_y - \sigma_y k_x) \tau_z + g\mu_B \mathbf{B}\cdot\boldsymbol{\sigma} + \Delta\tau_x,$

with the CPR shifted by $\phi_0\propto \alpha B L/v_F^2$ .

Altermagnets with Rashba SOC: In altermagnets, the Néel order breaks TR without net magnetization, and with Rashba SOC both I and TR are broken even in the absence of external field. The symmetry analysis shows that the AJE emerges only for Néel vectors away from crystal axes (Sahoo et al., 17 Sep 2025).
Multiband, Noncentrosymmetric, and Nematic Systems: Noncentrosymmetric superconductors possess mixed-parity (singlet–triplet) order parameters; when combined with a ferromagnetic barrier, each superconducting component experiences different phase shifts, producing a net anomalous current (Zhang et al., 2015). Similarly, nematic superconductivity with a two-component order parameter and anisotropic gradient couplings yields off-diagonal Meissner response and Hall-like Josephson currents (Akzyanov et al., 2022).
Extended Circuit/Circuit-QED Schemes: Nonlocal coherent coupling between Josephson junctions in hybrid SQUID devices can produce AJE via phase-sensitive cross terms, eliminating the need for material-induced symmetry breaking (Matsuo et al., 2023).
Quasiclassical and Disordered Systems: In diffusive JJs, standard quasiclassical Usadel equations intrinsically forbid AJE unless spin-dependent (magnetically active) boundary conditions or interface-induced inversion breaking are present, e.g., in systems with spin-filtering barriers or noncoplanar magnetization textures (Silaev et al., 2017).

2. Canonical Theoretical Models and Current–Phase Relations

The archetype $\phi_0$ -junction exhibits a CPR of the form: $I(\phi) = I_c \sin(\phi + \phi_0),$ with

$I(0) = I_c \sin\phi_0 \neq 0,$

reflecting a spontaneous supercurrent at zero phase bias. The anomalous phase $\phi_0$ is generically a function of SOC strength, Zeeman field magnitude and orientation, junction length, and material or device-specific parameters.

Key analytic results include:

Planar Rashba–Zeeman Junction:

$\phi_0\neq0$ 0

Diffusive Junctions with Rashba S' Layer:

$\phi_0\neq0$ 1

universal for all harmonics in the CPR; realized as shifts in $\phi_0\neq0$ 2, $\phi_0\neq0$ 3, etc. (Osin et al., 2023).

Coherently Coupled Planar SQUIDs:

$\phi_0\neq0$ 4

Spin-Josephson $\phi_0\neq0$ 5-junction (Excitonic condensates):

$\phi_0\neq0$ 6

where $\phi_0\neq0$ 7 is the in-plane misalignment angle of the polarization in adjacent spin superconductors (Zeng et al., 2023).

Topological Insulator Surface States:

$\phi_0\neq0$ 8

with “giant” $\phi_0\neq0$ 9 enhancements due to single Dirac contour and large $\phi_0$ 0-factor (Hüttner et al., 28 Jan 2026).

3. Experimental Realizations and Detection Techniques

Semiconductor Heterostructures and Nanowires: Devices based on InAs/Al hybrid JJs, proximitized semiconducting nanowires (e.g., InSb, InAs) with gate-tunable Rashba SOC, and in-plane vector magnet fields directly display tunable $\phi_0$ 1 (up to $\phi_0$ 2) and clear AJE signatures in SQUID/CPR readout (Mayer et al., 2019, Yokoyama et al., 2014, Nesterov et al., 2015, Matsuo et al., 2023).
Topological Insulators: Surface states of TIs (e.g., Bi $\phi_0$ 3Se $\phi_0$ 4, HgTe) exhibit AJE under in-plane magnetic field, with large $\phi_0$ 5 and gate-tunable control; the effect is sensitive to the spin-momentum locking angle and can be used to probe spin texture (Hüttner et al., 28 Jan 2026, Zhang et al., 2021).
Altermagnets and Multiterminal Diode Configurations: Four-terminal JJs with altermagnetic–Rashba central regions allow field-free, giant transverse AJE and unidirectional Josephson transport by geometrically tuning the Néel vector orientation (Sahoo et al., 17 Sep 2025).
RF-SQUID and Trijunction Devices: Implementation in SQUID loops with anomalous ( $\phi_0$ 6) junctions enables hysteretic responses, calibration-free phase readout, and topological manipulation of zero-energy Majorana states in multiterminal networks (Guarcello et al., 2020, Zhang et al., 2021).

Measurement protocols rely on:

Direct CPR mapping via phase-biased SQUIDs or asymmetric interferometers,
Extraction of switching flux shifts in hysteretic rf-SQUIDs,
Fraunhofer interference pattern analysis under magnetic field, and
3-terminal differential conductance spectroscopy for local minigap closure (Majorana trijunctions).

4. Engineering and Tunability of the Anomalous Phase Shift

AJE can be engineered and controlled through a variety of device and material parameters:

SOC Strength (α): Gate voltages in semiconductor 2DEGs allow more than an order-of-magnitude modulation in $\phi_0$ 7; the anomalous phase shift $\phi_0$ 8 scales accordingly, affording phase-battery functionality and programmable $\phi_0$ 9-junctions (Mayer et al., 2019).
External Magnetic Field & Orientation: In-plane field magnitude and direction set both the amplitude and sign of $\phi_0$ 0; for instance, in planar Rashba JJs, only the field component parallel to the SN interface contributes to the phase shift. Field rotation enables switching between longitudinal/transverse AJE and current diodicity (Hasan et al., 2022, Sahoo et al., 25 Mar 2025).
Altermagnetic Néel Vector Orientation: In multiterminal altermagnet-based JJs, rotating the intrinsic Néel vector modulates both the magnitude and direction of $\phi_0$ 1 and associated diode efficiency. Field-free control arises due to the intrinsic symmetry-breaking order (Sahoo et al., 17 Sep 2025).
Junction Circuitry: Nonlocal phase control in coherently coupled JJs (Andreev molecules) and multiterminal architectures provide avenues for on-chip, dissipationless phase batteries, logic elements, and “programmable” phase offsets unattainable in conventional setups (Matsuo et al., 2023).
Multiband and Multilayer Systems: Josephson diode effect emerges when two or more bands contribute different $\phi_0$ 2 and harmonic content to the total CPR, breaking global oddness and enabling unidirectional or highly nonreciprocal supercurrent flow (Osin et al., 2023, Minutillo et al., 2018).

5. Josephson Diode Effect and Nonreciprocal Supercurrents

A hallmark consequence of the AJE in systems with both I and TR breaking and at least two independent tunneling channels is the Josephson diode effect (JDE), characterized by critical current nonreciprocity: $\phi_0$ 3. The diode efficiency is defined as: $\phi_0$ 4 and can exceed $\phi_0$ 5 in optimized field-free altermagnet–Rashba structures (Sahoo et al., 17 Sep 2025). Such nonreciprocity enables dissipationless superconducting rectifiers, nonvolatile logic elements, and circuit-integrated supercurrent diodes.

In multilayer, multiband, or multiterminal JJs, the precise values and symmetry of the anomalous phase shifts (e.g., $\phi_0$ 6 for two channels) set the window for unidirectionality and diode effect (Osin et al., 2023, Sahoo et al., 25 Mar 2025, Minutillo et al., 2018).

6. Topological, Spintronic, and Correlation Effects

Topological Josephson Junctions: In regimes where the weak link is topological (e.g., QSHI edge, Majorana nanowire, Dirac surface), AJE can be enhanced, made $\phi_0$ 7-periodic, or used as a probe of underlying spin texture and topological phase transitions. In HgTe TIs, the “giant” AJE allows direct reconstruction of spin-momentum locking angles via field-angle–dependent $\phi_0$ 8 (Hüttner et al., 28 Jan 2026).
Nematic and Unconventional Order: In TIs with nematic superconductivity, unique component-mixing, off-diagonal Meissner response, and Josephson Hall effects emerge with the AJE even in the absence of magnetism (Akzyanov et al., 2022).
Spin Josephson Analogue: In excitonic spin superconductors, noncollinear polarizations yield an anomalous spin Josephson effect, i.e., a finite spin supercurrent at zero phase, driven by the misalignment angle (Zeng et al., 2023).
Dissipation-Enabled AJE: In topological regimes, $\phi_0$ 9-periodic AJE may be stabilized by two-particle dissipation, enabling direct observation of fractional Josephson oscillations and power-law scaling of current and noise spectra (Sticlet et al., 2018).

7. Applications, Device Concepts, and Outlook

AJE underpins a suite of superconducting device functionalities:

Phase Batteries and Programmable Phase Biases: $H_{\text{BdG}}(\mathbf{k}) = \xi_{\mathbf{k}} \tau_z + \alpha(\sigma_x k_y - \sigma_y k_x) \tau_z + g\mu_B \mathbf{B}\cdot\boldsymbol{\sigma} + \Delta\tau_x,$ 0-junctions act as self-biased phase sources for attached circuits (Matsuo et al., 2023, Mayer et al., 2019).
Josephson Diodes and Supercurrent Rectifiers: Nonreciprocal transport and logic (Sahoo et al., 17 Sep 2025, Osin et al., 2023, Minutillo et al., 2018).
Majorana Control and Topological Quantum Information: Tuning of Majorana zero modes and gate operations in multiterminal settings via continuous-phase AJE (Zhang et al., 2021, Hüttner et al., 28 Jan 2026).
Sensitive Probes of Spin Texture and Correlations: AJE magnitude and phase recover key spintronic and many-body properties otherwise hard to access by conventional means (Hüttner et al., 28 Jan 2026, Akzyanov et al., 2022, Hasan et al., 2022).

The theoretical framework predicts—and experiment confirms—that AJE can be realized, tuned, and exploited across clean, disordered, ballistic, diffusive, planar, quasi-1D, and multiterminal architectures, providing a robust platform for field-free, gate-controllable, and topologically nontrivial superconducting circuits.

References