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I'm confused

by Petar Mali
Tags: confused
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Petar Mali
#1
Jan25-11, 01:01 PM
P: 290
If I have cubic structure where plane is define by vector [tex]\rho[/tex] and in [tex]z[/tex] direction I have planes [tex]...m-2,m-1,m,m+1,m+2...[/tex]

and if I have for example

[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m+1,\vec{\rho}}[/tex]

how to go with that in K-space?

If I had


[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}[/tex]

I will say

[tex](m,\vec{\rho})=\vec{n}[/tex]

and then I will have


[tex]\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}}[/tex]

[tex]=\frac{1}{N}\sum_{\vec{k},\vec{q}}\hat{B}_{\vec{k}}\hat{B}_{\vec{q}}N\d elta_{\vec{k},-\vec{q}}=\sum_{\vec{k}}\hat{B}_{\vec{k}}\hat{B}_{-\vec{k}}[/tex]

But what to do in case with m+1. Thanks for your answer!
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shawl
#2
Jan26-11, 11:08 PM
P: 63
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
#3
Jan27-11, 01:17 AM
P: 416
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 [tex]\sum_q f(q) B_q B_{-q}[/tex] 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.


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