berry's phase is also said to be a 'spinor property'. why is it so important in solid state physics?
As far as I am concerned, Berry phase is a kind of geometric phase that has spatial information of a problem. You can see the book "Semiconductor nanostructures" by T. Ihn, page 235, for a simple example of it. As you can see in this example, the spatial angle of the Magnetic field will apear in the conductance of the ring. Also, I have seen that people use this phase in studying energy bands. I do not know further information about it. I hope this little example works for you.
Berry's phase in quantum physics appears as a general result of the adiabatic evolution of a Hamiltonian, and the proof is quite simple. To understand it better you can go through the derivation in pages 6 and 7 here: https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/dipl_diss/Durstberger_Diplomarbeit.pdf
You will also find various other examples in that work of its applications, for example to describe an electron in a magnetic field.
In solid-state physics you will find the Berry phase used many times. You can find it in topological properties of materials; for example the Topological Hall effect (https://sundoc.bibliothek.uni-halle.de/diss-online/07/07H039/t8.pdf ). In ferroelectrics, you can find it used to define the Berry's phase theory of polarization: https://arxiv.org/pdf/1202.1831.pdf
For many other uses of Berry's phase you can also read 'Topological Insulators and Topological Superconductors' by Bernevig.
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