I am trying to understand Physics behind the Weyl Fermion in Condensed Matter Systems.
Electrons show Weyl fermionic behaviour in the vicinity of so called 'Diabolical Points' in the band structure. If I understand it correctly, these are accidental touchings between successive energy bands of a system. The paper I am referring to (Scientific Reports 5, Article number: 7816 (2015)) states that "Diabolical Points were made prominent by Berry (4,5), who showed that a system accrues a phase when it evolves adiabatically through a closed path in parameter space enclosing the DP: the Berry phase, or more precisely, a topological Berry phase (6)."
As someone looking at this problem from the point of view of experimental condensed matter physics, I find it difficult to understand the physical picture associated with the statement "system evolves adiabatically through a closed path in the parameter space". Is it the Hamiltonian of a system that is evolving with time? If so, what are the physical properties of the system that are changing? In fact, if I am given a compound which is said to contain such Weyl fermion like electronic excitations, then what does it mean to say this system is evolving adiabatically in time? What exactly is changing? And what does it mean to say that the system has acquired a phase? How does band structure and other characteristics change when the system has acquired this phase?
Please forgive my naivete, but I am finding it very difficult to unite the quantum mechanical and condensed matter picture in my head.
Thanks in advance!