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## Main Question or Discussion Point

I am trying to understand Aubry-Andre model. It has the following form

$$H=∑_n c^†_nc_{n+1}+H.C.+V∑_n cos(2πβn)c^†_nc_n$$

This reference (at the 3rd page) says that if ##\beta## is irrational (rational) then the period of potential is quasi-periodic incommensurate (periodic commensurate) with underlying lattice period.

Furthermore, this reference says that with irrational ##\beta## (they are taking the inverse of Golden Mean i.e. ##(\sqrt{5}−1)/2)## to avoid the unwanted boundary effects, we have to take the system of a size of any number from Fibonacci series.

$$H=∑_n c^†_nc_{n+1}+H.C.+V∑_n cos(2πβn)c^†_nc_n$$

This reference (at the 3rd page) says that if ##\beta## is irrational (rational) then the period of potential is quasi-periodic incommensurate (periodic commensurate) with underlying lattice period.

**Question 1:**What does incommensurate potential mean here?**Question 2:**How does irrational ##\beta## guarantee that potential is quasi-period incommensurate with underlying lattice?Furthermore, this reference says that with irrational ##\beta## (they are taking the inverse of Golden Mean i.e. ##(\sqrt{5}−1)/2)## to avoid the unwanted boundary effects, we have to take the system of a size of any number from Fibonacci series.

**Question 3:**How does the system of a size of any Fibonacci series' number avoid unwanted boundary effects?