# Difference between thermal hall and phonon hall effect

I know of thermal hall effect which refers to a charge-neutral excitations exhibit hall effect that transport heat: for example, a heat current along x-direction generates a temperature gradient along y-direction when a system is in a magnetic field.

I recently saw an abundance of other hall effects, among which there are phonon hall effect. I don't quite understand what the difference is except that thermal hall effect, the carrier could be any (neutral) excitations but the phonon hall effect, the carrier is phonon.

So is thermal hall effect just more general or there is something I missed?

Edits/Follow-up

Based on yu-v's answer, I have some follow-up questions clarifying these hall effects:

1) for spin-liquids, seems to me usually people refer to "spinon" thermal hall effect (THE) and disregards any other carriers. But how do we know the cause of THE for sure? Do they have different temperature dependence or other observable difference?

2) Related to the first question, would the usual Wiedemann-Franz (WF) law still apply to phonon hall effect? I suppose there could also be additional terms on top of WF law?

So is thermal hall effect just more general

Basically yes, but they are used in different and more specific context. In general, any (charge) Hall effect will also include thermal Hall effect, as the charge carriers carry with them energy. So thermal and electric conductivity are related (Wiedemann-Franz law). So if you just take a system with a charge Hall effect and apply a temperature gradient you will get a thermal Hall effect carried at least by the same charge carriers (e.g. electrons).

The phonon Hall effect was referred to as something that happens when only phonons are available for energy transfer, and then the only measurable Hall effect is the thermal one. From a theoretical point of view, it requires some breaking of the chiral symmetry in the phonon system. As acoustic phonons do not couple directly to the magnetic field, you either need to look at optical phonons for this, or to somehow couple the acoustic phonons to some symmetry breaking term. From an experimental point of view, the signatures are very weak and measuring a thermal Hall effect is quite challenging.

Recently, other types of exclusively-thermal Hall effects which do not involve phonons came into focus, as chiral spin-liquids do not have any electric Hall signature but should show a thermal Hall signature. As this type of systems attracts a lot of attention right now, there are some recent works on thermal Hall effect regardless of phonons.

So basically, the term "phonon Hall effect" is used exclusively when analyzing transport of phonons, while "thermal Hall effect" is more general, and pertains to a larger class of systems and energy carriers.

1) at least for quantum spin-liquids, the point is that the thermal Hall effect carried by the spinons should be quantized (and fractionally so for things like Kitaev spin liquids). And this is probably the strongest signature for such a QSL. This was (supposedly) measured two years ago by Kasahara et. al., which led to a lot of interest. However, theoretical explanations of this experiment also relied on phonons to carry the energy and heat (See here and here). Experimentally, at low enough temperatures the contributions of the phonons will be negligible, while the contribution of the gapless edge mode will still be quantized. So there is a point in studying just the spinons contributions in QSL.

2) No. Wiedemann-Franz law only relates to charge carriers, as it relates amount of energy and amount of electrical charge transported. You can say that for phonons it applies in a trivial manner (you multiply the thermal conductivity by zero charge, and get the electrical conductivity). When you have both types of carriers in your system and they both contribute, you have to calculate their contributions separately. However, mostly there will be some separation of scales and one will be a correction to the other. In addition to that, there is further complication as the quantum anomaly that gives the Quantum Hall Effect (any of the such) does not apply to the quantum thermal Hall effect, and WF law may break in such topological systems. See here for an example, and the beautiful theory paper by Mike Stone, who I think also contributes on this website.

3) See for example here, how small the phonon effect is. It is usually masked by contributions from electrons, and one needs an insulator to see it, and even then it is tiny.

4) by definition, acoustic phonons don't couple to magnetic field, and therefore the most obvious time-reversal symmetry mechanism does not effect them directly (higher order mechanisms will couple them to the magnetic field, by they will be smaller in size). Optical phonons couple to the magnetic field by they are massive so at low energy can be neglected.

• thanks for the quick answer. I have some follow-up questions: – Histoscienology May 14 '20 at 16:17
• 1) for spin-liquids, seems to me usually people refer to "spinon" thermal hall effect (THE) and disregards any other carriers. But how do we know the cause of THE for sure? Do they have different temperature dependence or other observable difference? 2) Related to the first question, would the usual Wiedemann-Franz law still apply to phonon hall effect? 3) Why do you say the experimental signatures of phonon hall effect is very weak? 4) Could you give a reference for the magnetic field coupling (or lackof) to optical (acoustic) phonons? – Histoscienology May 14 '20 at 16:23
• ok that's a lot of follow up questions :) I will try to edit my answer – user245141 May 14 '20 at 16:28
• thank you so much for the detailed answer! I'm reading the references and hopefully will understand this better. I'm still confused about phonon coupling the magnetic field... I know acoustic phonon doesn't couple to photons because of energy/momentum conservation in basic scattering theories. But we are talking about a static magnetic field and this should be different, right? – Histoscienology May 15 '20 at 14:09
• yes. as acoustic phonons move in-phase they don't have a dipole moment, and therefore lack direct coupling to electromagnetic potential. That is why optical phonons are called optical - because they do couple to that. There is a recent work that shows that in fact acoustic phonons do have a weak direct coupling to the magnetic field, because screening of the nuclei charge by electrons is not perfect: journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.255901 – user245141 May 15 '20 at 14:59