Physical significance of toroidal moments

Question

The electric field of a toroidal dipole moment $\vec t$ is the same as the electric field of a electric dipole moment $\vec p$, except that it is scaled with the factor $ik$. The $i$ makes the toroidal dipole moment odd under time inversion, whereas the electric dipole moment is even under time inversion.

However, I am wondering where the deeper sense is to introduce the toroidal moments at all. One could just accept that sometimes the electric dipole moment is scaled with some complex factor. Are there other reasons why the toroidal moments are needed in the multipole expansion?

Answer 1

It is precisely the difference in behaviour under time inversion that uniquely separates the polar/electric toroidal moments from the electric moments.

Under normal time inversion, any static charge distribution (which gives rise to the electric multipoles) should remain invariant (and so positive parity). It is then clear that despite the electric dipole having the same far field behaviour as the toroidal dipole, they must originate from distinct charge/current distributions which must be accounted for in a multipole expansion.

The similarity to the electric dipole, and its generally weak behaviour in natural systems is part of the reason why this multipole family has been neglected, or treated as corrections to the electric dipole in most literature/textbooks. However, since Zel’Dovich's first description of the toroidal moments in parity non-conserving weak interactions, we now know that these moments can play an important role beyond simple mathematical constructs [1].

In practical terms, there are some physical systems that can have non-negligible toroidal moments. It is known that fundamental particles that are CPT self-conjugate can only have toroidal moments [2].

Since the nine years this question was posted, there have been proposals for using toroidal qubits to suppress noise channels in quantum computing [3], proposals to observe electric/magnetic-forbidden transitions in atoms [4], and a whole slew of metamaterial, optical, and other papers diving into toroidal electrodynamics [5].

A good reference (which I cannot recommend enough) for anyone interested in classical toroidal electrodynamics and the multipole expansion would be the book by Stefan Nanz [6]. For a more detailed answer to your question, chapters 1 and 2 should cover the fundamentals.

References

1. Electromagnetic interaction with parity violation
2. Electromagnetic properties of generalized Majorana particles (10.1103/PhysRevLett.62.852)
3. Toroidal qubits: naturally-decoupled quiet artificial atoms (10.1038/srep16934)
4. Toroidal optical transitions in hydrogen-like atoms (10.1126/sciadv.abq6751)
5. The Rise of Toroidal Electrodynamics and Spectroscopy (10.1021/acsphotonics.2c01953)
6. Toroidal Multipole Moments in Classical Electrodynamics (10.1007/978-3-658-12549-3)