Periodic

Quasiperiodic

Dielectric

$\epsilon \left(x,y\right)={\displaystyle \sum _{G_1,G_2}{\epsilon }_{G_1,G_2}{\text{e}}^{i\left(G_{1}x+G_{2}y\right)}}$

$y=\tau x$

$\epsilon \left(x,\tau x\right)={\displaystyle \sum _{G_1,G_2}{\epsilon }_{G_1,G_2}{\text{e}}^{i\left(G_1+G_2\tau \right)x}}$ (Projection of Periodic Case)

Eigenfunction Hz

${\text{e}}^{i\left(k_{1}x+k_{2}y\right)}$

${\text{e}}^{i\left(k_1+k_2\tau \right)x}$ (Again, Projection of Periodic Case)

Periodicity in Configuration/Physical Space

Yes

No

Periodicity in Reciprocal Space

Yes

Yes

Bloch Theorem

Applies

Geometric Periodicity

=>Eigenfunction Periodicity

Does Not Apply, (due to 3^rd point above) Chaos and renormalization on the circle

${\text{e}}^{i{G}_2\cdot slope\left(x+a\right)}={\text{e}}^{i{G}_2\cdot slope\cdot x}$ , Requires:

$\left({\text{e}}^{i{G}_2\cdot slope\cdot a}\right)={\left({\text{e}}^{i{G}_{2}a}\right)}^{slope}={\left(1\right)}^{slope}=1$

And the last step requires that

slope = integer or rational (with adjusted periodicity)

otherwise multivaluedness of roots of unity kicks in

and in our case slope = τ = irrational

But we still have a relation between Geometry and Physics(and in fact quasiperiodic periodic problem needs to be solved in conjunction with periodic problem two unknown frequencies require two equations see below):

Projective Periodicity of Geometry

=> Projective Periodicity of Eigenfunction

Temporal Dependence

$H\left(x,y;t\right)=H\left(x,y\right){\text{e}}^{-i\omega t}=H\left(x,y\right){\text{e}}^{-i\left({\omega }_1+{\omega }_2\right)t}$

$H\left(x;t\right)=H\left(x\right){\text{e}}^{-i\left({\omega }_1+{\omega }_2\tau \right)t}$

Mechanical Analogy

Isotropic Oscillator

Or Anisotropic Oscillator (commensurate case)

Anisotropic Oscillator(incommensurate case)