How do you analyze interactions in K-space with vector $\rho$ and planes m ± 2?

[Petar Mali]

If I have cubic structure where plane is define by vector $$\rho$$ and in $$z$$ direction I have planes $$...m-2,m-1,m,m+1,m+2...$$

and if I have for example

$$\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m+1,\vec{\rho}}$$

how to go with that in K-space?

If I had$$\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}$$

I will say

$$(m,\vec{\rho})=\vec{n}$$

and then I will have$$\sum_{\vec{n}}\hat{B}_{\vec{n}}\hat{B}_{\vec{n}}=\sum_{\vec{n}}\frac{1}{\sqrt{N}}\sum_{\vec{k}}\hat{B}_{\vec{k}}e^{i\vec{k}\cdot\vec{n}}\frac{1}{\sqrt{N}}\sum_{\vec{q}}\hat{B}_{\vec{q}}e^{i\vec{q}\cdot\vec{n}}$$

$$=\frac{1}{N}\sum_{\vec{k},\vec{q}}\hat{B}_{\vec{k}}\hat{B}_{\vec{q}}N\delta_{\vec{k},-\vec{q}}=\sum_{\vec{k}}\hat{B}_{\vec{k}}\hat{B}_{-\vec{k}}$$

But what to do in case with m+1. Thanks for your answer!

 

[shawl]

I am not familiar with these equations, but I try to give some opinions for discussion.

The problem is that there will be an unexpected term, i.e. exp(i*q), if the B(m)B(m+1) are transformed in K-space.

As I learned, B(m)B(m+1) denotes the transition process between state m and state m+1.

The complete formula is usually written in the sum of V(m,m+1)B(m)B(m+1), where V(m,m+1) is the transition matrix element.

When the formula is transformed in K-space, V(m,m+1) are also transformed as V(m,k,m+1,q), or written in V(k,q) for the shortness.

And what i am thinking is that the unexpected term exp(i*q) will be absorbed in V(k,q).
That means you can do the transformation in the case of (m,m+1) just like what you did in the case of (m,m). The difference for the m and m+1 only appears in the transition matrix elements.

 

[kanato]

Yes, as shawl mentions you get an extra factor of exp(i*q). With the right symmetry in your lattice you will be able to combine the exponentials to end up with something like $$\sum_q f(q) B_q B_{-q}$$ where f(q) is some real function, probably composed of cosines. This is exactly the sort of thing you get in tight binding, except there you have a creation and annihilation operator on different sites.