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Consider a 2D lattice model like this enter image description here

Assuming the mass of atom and force constant is 1, we could easily calculate the dispersion relations of the system. As there are four atoms per unit cell, there should be eight eigenvalues for every wave-vector $k$. The dispersion relation is like below enter image description here We see that there are two zero energy bands or constant zero dispersion relations in the figure, now my question is:

What is the physical meaning or how to understand the zero energy bands?

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One mode looks like a small rigid rotation of the individual squares. That does not stretch any bonds to first order in the rotation angle --- so no restoring force and the rotations of different squares are independent. Perhaps the second is a small rotation of the 8-member octogons?

Did you not plot the normal modes when you solved for the frequencies?

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