- #1
Luqman Saleem
- 18
- 3
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.
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?
$$H=∑_n c^†_nc_{n+1}+H.C.+V∑_n cos(2πβn)c^†_nc_n$$
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?