Anomalous CPRs in Josephson Junctions

• Anomalous current–phase relations are deviations from the standard sinusoidal supercurrent-phase dependence due to symmetry breaking and topological effects.
• They emerge from mechanisms such as spin–orbit coupling, magnetoelectric effects, and noncoplanar magnetizations, offering precise probes of underlying microscopic physics.
• Understanding these CPRs enables the design of advanced superconducting devices like phase batteries, rectifiers, and qubits by tuning supercurrent flows.

Anomalous Current–Phase Relations

Anomalous current–phase relations (CPRs) in Josephson junctions represent deviations from the standard sinusoidal dependence of the supercurrent on the superconducting phase difference. Such CPRs are direct manifestations of symmetry breaking (e.g., time-reversal, inversion, and spin-rotation), topological band structure, or nontrivial scattering in the weak link. The occurrence of these anomalous CPRs leads to rich physical phenomena, including phase shifts away from 0 or π (φ₀-junction behavior), appearance of higher harmonics, fractional periodicity (e.g., 4π-periodic effects in topological systems), and critical-current anomalies at topological phase transitions. These signatures provide incisive probes of underlying microscopic physics, including magnetoelectric couplings, Majorana bound states (MBSs), and unconventional pairing. This article presents the theoretical foundations, mechanisms, and experimental consequences of anomalous CPRs, as established across a diverse set of systems.

1. General Structure and Symmetry Analysis of Anomalous CPRs

The conventional Josephson relation is $I(\phi) = I_c\sin\phi$, where $I_c$ is the critical current and $\phi$ is the phase difference across the junction. Microscopically, the current arises from the phase sensitivity of the Andreev bound states (ABSs), leading to a $2\pi$-periodic CPR in standard $s$-wave weak-link junctions.

Breaking time-reversal symmetry (TRS), inversion symmetry (IS), or introducing spin-active components can yield generic anomalous CPRs of the form

$$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$

where $\phi_0 = \arctan(I_{\mathrm{an}}/I_0)$ defines a spontaneous phase shift and $I(\phi=0)\neq 0$ for $\phi_0\notin \{0,\pi\}$. Higher harmonics and fractional periodicities are possible in the presence of strong junction transparency, topologically nontrivial states, or mirror symmetry constraints.

Symmetry constraints determine which terms can appear in the CPR:

• TRS: Forbids $\cos\phi$ term; ensures $I_c$0
• Inversion symmetry: If present, $I_c$1 must be odd in $I_c$2
• Topological protection and mirror symmetry: Enforces relations such as $I_c$3, leading to suppression of odd harmonics and favoring higher-order terms (e.g., $I_c$4) (Yamakage et al., 2012)

The full CPR can thus contain multiple harmonics and phase shifts:

$I_c$5

2. Mechanisms Generating Anomalous Current–Phase Relations

2.1 Spin-Orbit Coupling and Magnetoelectric Effects

In diffusive and ballistic Josephson junctions, intrinsic spin–orbit coupling (SOC) in conjunction with exchange or Zeeman fields gives rise to the so-called φ₀-junction behavior. The effective Josephson CPR shifts as:

$I_c$6

with $I_c$7, where $I_c$8 is a Zeeman or exchange field and $I_c$9 the equilibrium spin-current induced by SOC (Konschelle et al., 2015, Hasan et al., 2022, Hijano et al., 2021). This shift is the superconducting analog of the spin-galvanic or inverse Edelstein effect. The anomalous phase is generically present provided both inversion and TRS are broken, and persists in a wide class of systems, including those with Rashba or Dresselhaus SOC, and in ballistic setups (Hasan et al., 2022). The explicit value of $\phi$0 is nonuniversal and depends on material parameters such as SOC strength, Zeeman field orientation, and junction length.

2.2 Ferromagnetic and Spin-Active Systems

Noncoplanar magnetizations in multilayered SFS junctions can induce a finite current at zero phase bias, $\phi$1, via long-range triplet proximity effects and spin-dependent Andreev reflections. These effects require noncollinear exchange fields and spin-filtering at the interfaces to break the magnetization inversion symmetry of the quasiclassical description (Silaev et al., 2017, Liu et al., 2010, Choi, 2017). The CPR acquires a $\phi$2 harmonic,

$\phi$3

with the anomalous term $\phi$4 scaling with the magnetic chirality, e.g., $\phi$5 for S/FI/F/FI/S structures. The phase shift $\phi$6 can be electrically or magnetically tuned.

2.3 Topological and Majorana Physics

In proximitized nanowire Josephson junctions, the presence of MBSs at the junction yields a $\phi$7-periodic CPR,

$\phi$8

where $\phi$9 scales linearly with the junction transparency, in contrast to the quadratic dependence in the trivial phase (Huang et al., 2017). This fractional Josephson effect is protected by fermion parity and reflects coherent single-electron tunneling via the MBS channel. Experimentally, the direct observation of this $2\pi$0 periodicity is hindered by quasiparticle poisoning; however, a robust, abrupt enhancement of the critical current—a sharp step at the topological quantum phase transition (TQPT)—is a universal signature (Huang et al., 2017).

In topological insulator (TI) Josephson junctions, the CPR contains both standard $2\pi$1 components and $2\pi$2-periodic (Majorana) contributions. The node-lifting in SQUID and Fraunhofer patterns arises from these anomalous CPR harmonics, serving as a signature of topological supercurrent channels (Kurter et al., 2013).

2.4 Mirror and Crystalline Symmetry Protection

In certain topological superconductors, mirror symmetry enforces selection rules on the CPR. An example is the $2\pi$3-wave/STI$2\pi$4 interface, where the CPR contains only even harmonics; specifically, $2\pi$5 due to cancellation of the $2\pi$6 term by mirror parity (Yamakage et al., 2012). This constraint is robust against disorder and provides a symmetry-protected route to anomalous CPRs.

2.5 Geometry, Nonaligned Junctions, and Non-Equilibrium Effects

Purely geometrical factors—for example, nonaligned or planar Josephson junctions subject to perpendicular magnetic fields—can induce phase offsets in the CPR, yielding a tunable $2\pi$7-junction (ground state at arbitrary $2\pi$8). These effects are unrelated to magnetic or spin-orbit phenomena and result purely from orbital current flow and the spatial distribution of pair potentials (Alidoust et al., 2012). Non-equilibrium populations in SNS junctions (e.g., with voltage-biased proximized arms) can also result in an anomalous phase shift by introducing electron–hole asymmetry, even in the absence of magnetism or SOC (Margineda et al., 2021).

2.6 Driven/Floquet and Quantum Dot Systems

Time-periodic drives, such as phase-shifted microwave gating of double quantum dots, can induce anomalous CPRs via Floquet engineering. These driven Josephson junctions realize tunable $2\pi$9-junctions, with $s$0 set by drive amplitude, frequency, and phase, leading to rectification and nonreciprocal Josephson transport (Ortega-Taberner et al., 2022).

3. Manifestations: Harmonics, Fractional Periodicity, and Spectroscopy

Table 1: Characteristic CPRs in Key Systems

System/Mechanism	Generic CPR	Nontrivial Features
SOC + Zeeman (diffusive)	$s$1	φ₀ ∝ $s$2 (Konschelle et al., 2015Hasan et al., 2022)
SFS, noncoplanar mags	$s$3	$s$4 from chirality (Silaev et al., 2017Liu et al., 2010)
Majorana (nanowire)	$s$5	$s$6-periodicity, step in $s$7 (Huang et al., 2017)
STI/s-wave (with mirror)	$s$8	Mirror-protected zero of $s$9 term (Yamakage et al., 2012)
TI weak link	$$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$0	Node-lifting in $$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$1 (Kurter et al., 2013)
Planar (nonaligned geom)	$$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$2	Geometry-tunable φ₀ (Alidoust et al., 2012)
Driven double quantum dot	$$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$3	Floquet-induced φ₀ (Ortega-Taberner et al., 2022)

Higher harmonics in the CPR (e.g., $$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$4, $$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$5) arise naturally in high-transparency or topological junctions. Anomalous CPRs produce distortions in the interference patterns (e.g., Fraunhofer or SQUID lobes), with the critical current modulated nontrivially by magnetic flux, field angles, gate voltages, and drive parameters (Barash, 2010, Kurter et al., 2013, Alidoust et al., 2012).

4. Critical Current Anomalies and Topological Transitions

In topological Josephson junctions, the passage through a TQPT is signaled by a discontinuous step in the critical current $$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$6 as a function of the tuning field (e.g., Zeeman energy or chemical potential). In Rashba nanowires, the critical current jumps from $$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$7 (trivial) to $$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$8 (topological, with MBSs), yielding a step of height $$I(\phi) = I_0\sin\phi + I_{\mathrm{an}}\cos\phi \equiv I_c\sin(\phi + \phi_0),$$9, observable even when $\phi_0 = \arctan(I_{\mathrm{an}}/I_0)$0 periodicity is smeared by poisoning (Huang et al., 2017). This provides a direct, robust experimental signature for the emergence of topological superconductivity and Majorana physics.

For TI-based weak links, the survival of node-lifted Fraunhofer patterns and their insensitivity to chemical potential/gate tuning further indicate the presence of protected low-energy ABSs or MBSs (Kurter et al., 2013).

5. Experimental Probes and Device Implications

Routinely used techniques to detect anomalous CPRs include:

• Phase-sensitive interferometry (SQUID, gradiometric and diffraction patterns, Andreev interferometry) to resolve phase shifts, harmonic content, or node-lifting
• Tuning of Zeeman/exchange/parity gate voltages, chemical potential, or microwave phase difference to map out phase diagrams
• Observation of voltage-tunable φ₀ in nonequilibrium Andreev interferometers (Margineda et al., 2021), or direct measurement of spontaneous currents in closed-loop devices

The control of anomalous CPRs has direct applications as “phase batteries” (φ₀-junctions), rectifiers, or tunable couplers in superconducting electronics. Device innovations include programmable Josephson diodes, non-reciprocal circuit elements, and qubits stabilized by engineered φ-junctions (Konschelle et al., 2015, Hijano et al., 2021, Ortega-Taberner et al., 2022).

6. Representative Theoretical Models

Microscopic approaches include:

• SU(2) gauge-covariant quasiclassical transport (modified Usadel/Eilenberger equations for SOC and magnetic textures) (Konschelle et al., 2015, Hijano et al., 2021, Hasan et al., 2022)
• Bogoliubov–de Gennes formalism for spectral properties in clean topological and multi-band systems (Huang et al., 2017, Lu et al., 2024, Yamakage et al., 2012)
• Multi-channel and multi-harmonic Fourier expansions, including explicit disorder and transparency dependence (Barash, 2010)
• Floquet–Keldysh and scattering-matrix approaches for driven circuits and quantum dot systems (Ortega-Taberner et al., 2022)

In all cases, symmetry analysis, parameter tuning, and boundary condition engineering (spin-filter barriers, magnetic configuration, gating, geometry) are essential for predicting and controlling the anomalous CPRs.

7. Outlook and Emerging Paradigms

Recent advances extend the reach of anomalous CPRs to new classes of magnetic materials (e.g., altermagnets), hybrid systems (topological insulator/semiconductor/superconductor interfaces), and driven systems with time-periodic or nonequilibrium protocols (Lu et al., 2024, Ortega-Taberner et al., 2022). The field is progressing towards comprehensive phenomenological classification of anomalous Josephson effects, as well as quantitative material-specific modeling with predictive power for experimental design and topological quantum computation proposals.

Anomalous current–phase relations provide both fundamental insight into symmetry and topology in superconducting hybrid devices and routes for engineering controllable, nontrivial quantum states in superconducting electronics. Their detection and manipulation remain a frontier in mesoscopic and topological superconductivity.