Stålhammar, Rudneva, Hansson and Wilczek (Phys. Rev. B 109, 064514 (2024)) showed that parity-violating superconductors can support an emergent, purely spatial Chern-Simons interaction proportional to A⃗⋅B⃗\vec A\cdot\vec B, together with a dimensionally reduced 'axion-wind' term, producing complex penetration depths, anapole moments, currents along vortices, and massless surface modes; they also gave microscopic derivations from a flux-biased Weyl superconductor and from the chiral anomaly. The same paper candidly identified what remains uncontrolled: the Wess-Zumino counterterm ambiguity in the microscopic derivations (the authors 'presently lack a sound theoretical argument excluding any Wess-Zumino term'), the exact equilibrium response content at nonzero chemical potential with intercone scattering, the boundary-value problems created by higher-derivative parity-violating terms, the nonlinear ground states in the regimes where the linearized analysis is unstable (4γ<β24\gamma<\beta^2 for the Chern-Simons superconductor, ε<0\varepsilon<0 for 'actively chiral' magnets), and concrete material realization.

Background

Chern-Simons (CS) terms are gauge-field couplings of the form ϵαβγAα∂βAγ\epsilon^{\alpha\beta\gamma}A_\alpha\partial_\beta A_\gamma whose change under a gauge transformation reduces to a surface term; in 2+1 dimensions they encode the topological electromagnetic response of quantum Hall states, with edge modes whose anomalies cancel that surface variation. Stålhammar, Rudneva, Hansson and Wilczek asked what new physics appears when the same algebraic structure emerges as a purely spatial interaction in an ordinary 3+1-dimensional material:

LCST=−β2 A⃗⋅B⃗,L_{CS}^T = -\frac{\beta}{2}\,\vec A\cdot\vec B,

with B⃗=∇×A⃗\vec B=\nabla\times\vec A the magnetic field and β\beta a material-dependent constant. Such a term violates parity but not time reversal, and it yields a current j⃗CS∝B⃗\vec j_{CS}\propto\vec B — current-field mixing, in contrast to the London diamagnetic current j⃗d∝A⃗\vec j_d\propto\vec A of a superconductor (the phenomenological current relation behind the Meissner effect, characterized by a penetration depth). The predicted consequences are striking: a superconducting sphere excludes flux while developing an anapole moment (a toroidal dipole moment Ti=110∫[ri(r⃗⋅j⃗)−2r2ji]d3xT_i=\frac{1}{10}\int\left[r_i(\vec r\cdot\vec j)-2r^2j_i\right]d^3x that couples to the curl of a magnetic field), a current-carrying wire acquires an azimuthal magnetic field, and vortices carry currents along their cores. Mathematically, solutions of the resulting generalized London equations follow from ordinary London solutions evaluated at a complex penetration depth.

Gauge invariance is the central subtlety. In a normal material in thermodynamic equilibrium the A⃗⋅B⃗\vec A\cdot\vec B term is excluded: the would-be equilibrium current j⃗∝B⃗\vec j\propto\vec B is the chiral magnetic effect (CME), which vanishes in equilibrium once the limits q⃗→0\vec q\to0 and ω→0\omega\to0 are taken in the correct order, as established in the Weyl-semimetal literature (Chen, Wu and Burkov; Chang and Yang). A Weyl semimetal is a crystal whose valence and conduction bands touch at isolated Weyl nodes of definite chirality; the CME debate concerned whether a node separation in momentum space (an axial vector b⃗\vec b) sources equilibrium currents. The authors argue that a superconductor escapes this no-go argument: condensation of an order parameter ⟨ϕ⟩=v\langle\phi\rangle=v turns the manifestly gauge-invariant density Re ϕ†iD⃗ϕ\mathrm{Re}\,\phi^\dagger i\vec D\phi (with covariant derivative D⃗=∇−iqA⃗\vec D=\nabla-iq\vec A) into qv2A⃗⋅B⃗qv^2\vec A\cdot\vec B in the London gauge (the superconducting gauge choice ∇⋅A⃗=0\nabla\cdot\vec A=0), and persistent currents can absorb the surface gauge variation. A second, dimensionally reduced CS term, 12(β⃗⋅A⃗×E⃗−A0β⃗⋅B⃗)\frac{1}{2}(\vec\beta\cdot\vec A\times\vec E - A_0\vec\beta\cdot\vec B) (with β⃗\vec\beta a material-dependent constant vector) — the 'axion-wind' term, so named because it is the form taken by the standard axion-electrodynamics coupling a E⃗⋅B⃗a\,\vec E\cdot\vec B when the axion field aa varies linearly in space but not in time — acts as a stack of independent 2+1-dimensional CS theories and requires massless surface modes whose anomalies cancel the surface terms, exactly as edge modes do in the quantum Hall effect.

Two microscopic routes were demonstrated. First, a flux-biased Weyl superconductor — a superconducting Weyl semimetal in a stack of alternating topological-insulator and conventional BCS (Bardeen-Cooper-Schrieffer) superconductor layers, with a flux-bias circuit that gaps out the particle-hole conjugate cones of one chirality — where a one-loop polarization calculation (the lowest-order quantum correction to the electromagnetic response) yields the A⃗⋅B⃗\vec A\cdot\vec B term together with a partner term ∼Ax∂zAy\sim A_x\partial_z A_y of fixed relative size that is not gauge invariant. Second, a chiral-anomaly derivation (the triangle anomaly is the non-conservation of chiral charge in parallel electric and magnetic fields) of the axion-wind terms, valid for the vacuum state. In the one-loop derivation the polarization tensor contains a logarithmically ultraviolet-divergent contribution; the authors subtract it on physical grounds (anomalies are infrared phenomena) and note agreement with a direct numerical calculation of the JzJ_z component in the eight-band parent model of O'Brien, Beenakker and Adagideli, where the bulk current is accompanied by a surface counterflow carried by Fermi arcs (arcs of surface states joining Weyl nodes of opposite chirality) that keeps the ground state free of net current. But, as they state, they 'presently lack a sound theoretical argument excluding any Wess-Zumino term' — a counterterm of the type Bardeen introduced to restore the gauge invariance of anomalous theories — so the superconducting effective action is not fixed by any principle. The paper also extends the analogy with optical activity to 'actively chiral' magnets, whose free energy density −L=12μB2+κ B⃗⋅(∇×B⃗)+λ2(∇×B⃗)2-\mathcal{L}=\frac{1}{2\mu}B^2+\kappa\,\vec B\cdot(\nabla\times\vec B)+\frac{\lambda}{2}(\nabla\times\vec B)^2 (with μ\mu here the magnetic permeability, not the chemical potential) contains the higher-derivative term κ B⃗⋅(∇×B⃗)\kappa\,\vec B\cdot(\nabla\times\vec B); this again produces complex penetration depths but changes the boundary-value problem, since the added derivatives demand extra boundary conditions and generate boundary layers, as in hydrodynamics. Two linearized instabilities are identified and left unresolved: the magnetic energy density 12B2+βA⃗⋅B⃗+γA2\frac{1}{2}B^2+\beta\vec A\cdot\vec B+\gamma A^2 is positive definite only if 4γ≥β24\gamma\ge\beta^2 (with γ\gamma the inverse-squared penetration depth), and for the chiral magnets the analogous condition ε<0\varepsilon<0, with ε≡1μ−κ2λ\varepsilon\equiv\frac{1}{\mu}-\frac{\kappa^2}{\lambda} the minimum value of the quadratic energy density, signals an instability toward fields that rotate periodically along one axis — 'optical activity frozen in time'. In both cases the true ground state requires stabilizing higher-order terms, such as (B2)2(B^2)^2, which the paper does not construct.

Problem Statement

Complete the theory of emergent Chern-Simons interactions in 3+1 dimensions in the places where the source analysis is explicitly uncontrolled. All strands serve one objective — turning the formal effective theory into a microscopically consistent, boundary-complete, and physically realizable one — and they are linked through the same effective action, whose content, fields, and boundary data must be fixed quantitatively and consistently: (1) Fix the effective action. Determine, from the microscopic theory, whether Wess-Zumino counterterms are allowed or excluded in the superconducting context; equivalently, put the ultraviolet subtraction of the logarithmically divergent polarization contribution on a principled footing and fix the allowed gauge-invariance-restoring counterterm structure, including the fate of the non-gauge-invariant partner term ∼Ax∂zAy\sim A_x\partial_z A_y. (2) Settle the equilibrium electromagnetic response. Determine the relative A⃗⋅B⃗\vec A\cdot\vec B and axion-wind content of the equilibrium effective action (with ω→0\omega\to0 taken before q⃗→0\vec q\to0) of the microscopic models at nonzero chemical potential, including the role of intercone scattering (scattering between the Weyl nodes) and relaxation, and the surface counterflows and massless boundary modes required for consistency — established in the prior literature only for the JzJ_z component. (3) Complete the boundary-value problem. Formulate well-posed boundary conditions and the matching to exterior regions for finite samples of media with parity-violating gradient terms, including the boundary-layer structure, the surface modes, and the presence of superconducting order-parameter phases. (4) Construct the nonlinear ground states. In the regimes where the linearized analysis is unstable (4γ<β24\gamma<\beta^2 for the Chern-Simons superconductor; ε<0\varepsilon<0 for actively chiral magnets), construct the ground states stabilized by higher-order terms and determine their measurable signatures. (5) Identify concrete material realizations — chirally purified or chiral-doped organic superconductors, non-centrosymmetric superconductors, flux-biased Weyl superconductor heterostructures, or chiral magnets and metamaterials — with parameter estimates showing that the Chern-Simons response is detectable, and specify the corresponding experimental signatures. The normal-state order-of-limits question and the JzJ_z surface counterflow are already settled and are not part of what remains open.

References

  1. M. Stålhammar, D. Rudneva, T. H. Hansson, F. Wilczek, 'Emergent Chern-Simons interactions in 3+1 dimensions', Phys. Rev. B 109, 064514 (2024), DOI 10.1103/PhysRevB.109.064514, arXiv:2309.10025, https://arxiv.org/abs/2309.10025
  2. T. E. O’Brien, C. W. J. Beenakker, İ. Adagideli, 'Superconductivity provides access to the chiral magnetic effect of an unpaired Weyl cone', Phys. Rev. Lett. 118, 207701 (2017), DOI 10.1103/PhysRevLett.118.207701, arXiv:1612.06848, https://arxiv.org/abs/1612.06848
  3. M. J. Pacholski, C. W. J. Beenakker, İ. Adagideli, 'Universal chiral magnetic effect in the vortex lattice of a Weyl superconductor', Annals of Physics 417, 168103 (2020), DOI 10.1016/j.aop.2020.168103, arXiv:1911.00312, https://arxiv.org/abs/1911.00312
  4. Kevin A. Madsen, Piet W. Brouwer, Maxim Breitkreiz, 'Equilibrium current in a Weyl-semimetal - superconductor heterostructure', Phys. Rev. B 104, 035109 (2021), DOI 10.1103/PhysRevB.104.035109, arXiv:2104.08195, https://arxiv.org/abs/2104.08195
  5. Y. Chen, Si Wu, A. A. Burkov, 'Axion response in Weyl semimetals', Phys. Rev. B 88, 125105 (2013), DOI 10.1103/PhysRevB.88.125105, arXiv:1306.5344, https://arxiv.org/abs/1306.5344
  6. Ming-Che Chang, Min-Fong Yang, 'Chiral magnetic effect in two-band lattice model of Weyl semimetal', Phys. Rev. B 91, 115203 (2015), DOI 10.1103/PhysRevB.91.115203, arXiv:1411.1479, https://arxiv.org/abs/1411.1479
  7. Yong Tao, 'Spatial Chern-Simons interactions and complex magnetic penetration depth', Annalen der Physik 537, 2400352 (2025), DOI 10.1002/andp.202400352, arXiv:1610.07414, https://arxiv.org/abs/1610.07414
  8. Vira Shyta, Jeroen van den Brink, Flavio S. Nogueira, 'Chiral Meissner state in time-reversal invariant Weyl superconductors', Phys. Rev. Research 6, 013240 (2024), DOI 10.1103/PhysRevResearch.6.013240, arXiv:2309.14262, https://arxiv.org/abs/2309.14262
  9. Pedro D. S. Silva, Mario J. Neves, Manoel M. Ferreira, 'Optical properties and energy propagation in a dielectric medium supporting magnetic current', Phys. Rev. B 109, 184439 (2024), DOI 10.1103/PhysRevB.109.184439, arXiv:2305.08153, https://arxiv.org/abs/2305.08153
  10. Alex Q. Costa, Pedro D. S. Silva, Manoel M. Ferreira, 'Double rotatory power reversal, continuous Kerr angle, and enhanced reflectance in bi-isotropic media with anomalous Hall current', Phys. Rev. B 112, 085140 (2025), DOI 10.1103/jkjf-7kkq, arXiv:2412.15338, https://arxiv.org/abs/2412.15338
  11. Pedro D. S. Silva, Alex Q. Costa, Ronald A. Pereira, Manoel M. Ferreira, 'Anomalous increasing super reflectance in chiral matter', arXiv:2507.18056 (2025), https://arxiv.org/abs/2507.18056
  12. L. H. C. Borges, A. F. Ferrari, P. H. O. da Silva, F. A. Barone, 'Field sources in a planar anisotropic CPT-odd gauge model', arXiv:2401.05216 (2024), https://arxiv.org/abs/2401.05216
  13. Thijs Douwes, Marcus Stålhammar, 'Twisted (co)homology of non-orientable Weyl semimetals', arXiv:2511.22303 (2025), https://arxiv.org/abs/2511.22303
  14. Bahar Jafari-Zadeh, Chenan Wei, Tigran A. Sedrakyan, 'Chiral vortex-line liquid of three-dimensional interacting Bose systems with moat dispersion', Phys. Rev. B 111, 245130 (2025), DOI 10.1103/n5lh-yzk7, arXiv:2504.01080, https://arxiv.org/abs/2504.01080
  15. Paul Leask, 'Anyon bound states and hybrid superconductivity', Phys. Rev. Lett. 137, 026003 (2026), DOI 10.1103/15fc-r786, arXiv:2510.04830, https://arxiv.org/abs/2510.04830
  16. Junyeong Ahn, Ashvin Vishwanath, 'Circular-polarization-selective perfect reflection from chiral superconductors', Nature Communications 16, 6493 (2025), DOI 10.1038/s41467-025-61658-5, arXiv:2502.01793, https://arxiv.org/abs/2502.01793
  17. Zhicheng Zhang, Kou-Han Ma, 'Majorana flat bands in the vortex line of superconducting Weyl semimetals', Phys. Rev. B 113, 104502 (2026), DOI 10.1103/lv9q-ks98, arXiv:2501.00410, https://arxiv.org/abs/2501.00410