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- How is changing Hamiltonian related to the periodicity of the Brillouin zone

I am trying to understand how is topology used to characterize materials. So I understand that to calculate the Berry phase you will parameterize your Hamiltonian and change this parameter in some way and return to the initial value. What I do not understand is what does this changing of parameter correspond to? Are we changing the system somehow? I understand that the Hamiltonian describes the system so if we change the Hamiltonian then we are changing the system?

If we are really physically changing the system, why do everybody talk about the periodicity of the Brillouin zone and this satisfies the close loop evolution of the Hamiltonian? The Hamiltonian gives the band structure, shouldn't it change when we change the Hamiltonian? Why does the periodicity = this change?

Greatly appreciate if someone can help me with this

If we are really physically changing the system, why do everybody talk about the periodicity of the Brillouin zone and this satisfies the close loop evolution of the Hamiltonian? The Hamiltonian gives the band structure, shouldn't it change when we change the Hamiltonian? Why does the periodicity = this change?

Greatly appreciate if someone can help me with this