Energy spectrum from dispersion relation E(k)

[sherumann]

Hi. What I'm trying to do is to obtain the energy spectrum from the following dispersion relation:

$$E^4-A·E^3+B·E^2-C·E+D-F·E^2·cos(k·a_0)^2+G·E·cos(k·a_0)^2-H·cos(k·a_0)^2 = 0$$

where E is the energy, k is the wave vector and a0 the distance between adjacent neighbors in a 1-Dimensional lattice with a two-atom basis, with some weird on-site energies.

Given the following model parameters:

A = -48.37528081
B = +877.6426691
C = -7077.321036
D = +21403.79575
F = -0.00002232761528
G = +0.0005479196789
H = -0.003361487230

I keep getting imaginary energies! What I do is simply solve the equation for E given some values for k. Am I doing it wrong? Or the parameters must be wrong? Please help! :(

 

[kanato]

Where does that equation come from? That doesn't have the usual form of a tight-binding equation.

 

[sherumann]

Where does that equation come from? That doesn't have the usual form of a tight-binding equation.

It comes from the renormalization of a more complex structure in terms of some on-site energies. I yet don't understand very well were it comes from (I'm sure about the dispersion relation, though), all I want to know is if what I'm doing is right or not...

 

[kanato]

Well all you're doing is solving for E, right? It's an algebraic problem at this point, and the physics has been done. Since it's a 4th order polynomial, there are four solutions, which may be real or complex.

If everything is correct, and you are getting complex energies, then that would mean you have states with finite lifetimes. How are you solving for E? Many software packages can find roots of polynomials, have you checked with different ones?

 

[PRB147]

If there is a renormalization process, then the energy could be complex
the imaginary of the energy represent the lifetime of that particular state.