I What Exactly is the Anapole Moment?

[kelly0303]

Hello! Can someone explain to me what is the anapole moment? I read several papers, and I understand its mathematical expression and its implications (parity violation for example), but I am not sure I understand physically what it is. For example I think of an electric dipole moment as 2 charges at a certain distance from each other. How should I think of the anapole moment? Also I am not sure I understand where it does come from. I read in several places that it is different from the normal magnetic and electric moments, so I am not sure where else can it come from. Lastly, how can it violate parity, if it is an electromagnetic phenomena. Isn't QED parity conserving? I would really appreciate a layman explanation as the stuff I read in papers didn't give me a clear understanding. Thank you!

 

[Cryo]

Many questions... Have you had a look at

Electromagnetic toroidal excitations in matter and free space
N. Papasimakis, V. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev
Nat. Mater. 15, 263 (2016) (https://rdcu.be/6I2l)

and

Optical anapoles
V. Savinov, N. Papasimakis, D.P. Tsai & N.I. Zheludev
Comms. phys. 2, 69 (2019) (http://www.nanophotonics.org.uk/niz/publications/Savinov-2019-OA.pdf)

It is full of pictures.

Anapole in the static regime is equivalent to toroidal dipole - currents flowing along the minor loops of a torus. In the dynamic regime anapole is a superposition of electric and toroidal dipoles. The key thing about the anapole is that it is the elementary radiationless source for Maxwell's equations. I.e. it is a point-like charge-current excitation that produces no electromagnetic fields outside the finite region in space that contains that excitation. More general theorems regarding the non-radiating excitations have been established by Devaney and Wolf https://journals.aps.org/prd/abstract/10.1103/PhysRevD.8.1044.

Anapoles, or more specifically toroidal multipoles, arise as a result of multipole decomposition of current density. The point that is often missed is that multipole decomposition of currents has more terms then multipole decomposition of electromagnetic fields. This is covered in slightly more detail in "Optical Anapoles".

Parity violation. Anapoles transform like electric dipoles under spatial inversion, but as magnetic dipoles under time-reversal. Therefore if you apply both space inversion and time reversal at the same time, both symmetries of Maxwell's equations, then electric and magnetic dipoles in your system will change sign, but anapoles (the static toroidal dipoles) will not. I think parity violation flows from there, but I am not a particle physicist so I will leave it here.

Hope this helps

 

[kelly0303]

Many questions... Have you had a look at

Electromagnetic toroidal excitations in matter and free space
N. Papasimakis, V. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev
Nat. Mater. 15, 263 (2016) (https://rdcu.be/6I2l)

and

Optical anapoles
V. Savinov, N. Papasimakis, D.P. Tsai & N.I. Zheludev
Comms. phys. 2, 69 (2019) (http://www.nanophotonics.org.uk/niz/publications/Savinov-2019-OA.pdf)

It is full of pictures.

Anapole in the static regime is equivalent to toroidal dipole - currents flowing along the minor loops of a torus. In the dynamic regime anapole is a superposition of electric and toroidal dipoles. The key thing about the anapole is that it is the elementary radiationless source for Maxwell's equations. I.e. it is a point-like charge-current excitation that produces no electromagnetic fields outside the finite region in space that contains that excitation. More general theorems regarding the non-radiating excitations have been established by Devaney and Wolf https://journals.aps.org/prd/abstract/10.1103/PhysRevD.8.1044.

Anapoles, or more specifically toroidal multipoles, arise as a result of multipole decomposition of current density. The point that is often missed is that multipole decomposition of currents has more terms then multipole decomposition of electromagnetic fields. This is covered in slightly more detail in "Optical Anapoles".

Parity violation. Anapoles transform like electric dipoles under spatial inversion, but as magnetic dipoles under time-reversal. Therefore if you apply both space inversion and time reversal at the same time, both symmetries of Maxwell's equations, then electric and magnetic dipoles in your system will change sign, but anapoles (the static toroidal dipoles) will not. I think parity violation flows from there, but I am not a particle physicist so I will leave it here.

Hope this helps

Thank you so much for this! I understand this much better now. So the main point is that a charge-current configuration, in its most general form, must have an electric, magnetic and toroidal moment expansion, the anapole moment being the toroidal dipole moment. On the other hand the fields themselves (or potentials) have just an electric and magnetic multipole expansion. There are however 2 things I am still not sure I understand. Are these 3 moments enough to describe any charge-current distribution, or there can be more (for example a configuration in which the toroidal vector forms a closed loop, the same way the magnetic field does in the case of the anapole moment)? And, how can one come up with the toroidal (anapole) moment in the first place? By this I mean did someone just come up (with an educated guess?) with a current-charge configuration that couldn't be expanded using just the normal electric and magnetic moments and this lead to the introduction of new moments? Or was there a more mathematical approach (e.g. some terms were not accounted for in some mathematical expression)? Or something else? Thank you!

 

[Cryo]

Regarding the expansion. Electric, magnetic and toroidal multipoles provide complete basis, so any non-static charge-current configuration can be written in terms of these multipoles, but only if you avoid taking the zero size approximation. If you do take the zero-size approximation, as is usually done, you will also get the mean-radii corrections (of toroidal and magnetic kind). Confusing, but such is life. There have been several works on this. I prefer this one:

E. E. Radescu and G. Vaman, Phys. Rev. E 65, 046609 (2002) https://journals.aps.org/pre/abstract/10.1103/PhysRevE.65.046609

The closed loops of toroid are sometimes known as super-toroids. These fractal-like extensions can be expanded in terms of the three multipole families if you include the mean radii (Phys Rev A 98, 023858 (2018) https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.023858)

In the static case you also get the electric toroidal multipoles. Which is even more confusing, but there is a simple way to explain it in terms of group theory. Multipoles are always irreducible representations of the SO(3) group (rotations). But SO(3) is not the full symmetry group of Maxwell's equations - you also have spatial inversion (parity, P) and time reversal (T). Now, if you consider the combined PT group it will be Abelian and will have 4 elements (Identity, P, T, PT) - so there will be four irreducible representations. Now if you combine these represenations with the irreps of SO(3) you get :

electric multipoles (changes sign under P, not under T)
magnetic multipoles (changes sign under T, not under P)
toroidal multipoles (changes sign under P and under T)
electric toroidal multipoles (does not change sign)

In the dynamic case, time-reversal makes no sense, so magnetic multipoles and electric toroidal multipoles are undistinguishable. See Phys. Rev. B 98, 165110 (2018) and refs within.

Coming up with toroidal multipoles. They have been re-discovered several times. Zeldovich came up with anapole, or toroidal dipole, when looking for something that would explain the parity-violation experiment. Reading his original paper the logic was, I think, that the necessary term in the Lagrangian must not vanish under PT, but must vanish under P - toroidal dipole is the simplest algebraic term that fits the bill. Other re-discoveries were both 'geometrical' and algebraic, but mostly algebraic, because the concept of toroidal/anapole multipoles was first developed in the particle/nuclear physics community (traditionally strong grasp on tensor algebra and representation theory).