Berry phase effects!?

In the last few days, I had a chance to think about some fundamental aspects of Berry phase and related transport phenomena. What triggered me is the ICMAB school on orbital currents, where I explained how the orbital-dependent emergent gauge field appears in the theoretical description. Also, at last, I am finalizing the manuscript on the orbital-dependent anomalous position, and as I go through all the equations line by line, I start to think lots of things.

To be honest, I think I don’t really understand the nature of Berry phase effects yet. I know the definition, mathematical properties (mostly about some rules under a gauge transform), how it is related to the Hall effect, why the Hall conductance is quantized in insulators in 2D, semi-classical equations of motion, how the Berry phase effect is included in the Kubo formalism, and how it is implemented in numerics. But I don’t really understand! (What do I mean by “understanding” anyway?)

I noticed that when I tried to write down concrete but intuitive mathematical expressions, all those knowledges are yet very loosely connected in my brain. When deriving equations from diffent angles, I end up finding a paradox between the equations from different approaches, and I felt like my whole research (and probably most people’s in my field) are being shaken at a very fundamental level! I just wonder, “Did anyone really understand properly”, “Is it just that we (everyone) pretend that we know the stuffs?”.

Hopefully, in the next year as soon as I find some free time, I plan to look at this more deeply and hopefully manage to write a paper in relation to the role of the orbital degree of freedom in multi-band systems. As usual, the best way may be to find concrete examples and solve these “exercises” without using any tricks and by relying only on the axioms of quantum mechanics.