Reduction of a sum to the first Brillouin zone in a band structure calculation

Question

this might be a "standard trick" for many solid state physicists, however it's one that I'm not familiar with so maybe you can help me. Here's the Problem:

Suppose we're given a Hamiltonian of the form $H=\sum_{k} \epsilon_{k} c^{\dagger}_{k}c_{k} + \sum_{k} U c^{\dagger}_{k+Q} c_{k}$. Here U is some complex (!) number, k is a 2 dimensional wavevector and Q=$(\pi,\pi)$. Furthermore we impose $\epsilon_k = - \epsilon_{k+Q}$. The first Brillouin Zone is the set $\{ (k_x,k_y) ; |k_x|+|k_y|<\pi \} $ in 2D-k-space.

Now define the 2D-vector $\Psi_k = (c_k,c_{k+Q})$. Then the Hamiltonian be written as: $H= \sum_{k}' \Psi_k^{\dagger} A_k \Psi_k $ with the k-dependent 2x2 Matrix $A_k$ defined by:

$$ \left[ \begin{array}{ c c } \epsilon_k & U \\ U^* & \epsilon_{k+Q} \end{array} \right] $$

The prime (') in the sum denotes that it has to be taken over wavevectors in the frist Brillouin zone only!

Now multiplying this quadratic form out "in reverse" I obtain something like: $H=\sum_{k} \epsilon_{k} c^{\dagger}_{k}c_{k} + \sum_{k} U c^{\dagger}_{k+Q} c_{k} + \sum_{k} \epsilon_{k+Q} c^{\dagger}_{k+Q}c_{k+Q} + \sum_{k} U c^{\dagger}_{k} c_{k+Q} $

It's not clear to me why the third and fourth term are supposed to vanish in case I'm restricting my sum to the frist Brillouin zone.

I hope someone can help. It should be rather technical, but still important I think.

Thanks in advance.

Answer 1

The terms don't disappear--- they are the same as previous terms when you shift the dummy summation variable $k\rightarrow k+Q$.

Answer 2

I will formulate as an answer, because the discussion in comments is too long: during the following discussion I will be able to add more details here.

Third term indeed disappears, but by slightly different reasoning. Obviously $$ \varepsilon_k c^+_{k\sigma}c_{k\sigma} = \varepsilon_{k+Q}c^+_{k+q\sigma} c_{k+Q\sigma} $$ due to the fact that these states are equal.

It is not that evident but probably true that $$ c^+_{k\sigma} W c_{k+Q\sigma} = c^+_{k+Q\sigma} W c_{k\sigma} $$ as probability go "there" and "back" usually more or less equal.

However, it is not general rule or something. There could be interaction which makes them non-equal.

Probably, it can be derived from momentum conservation (thus assuming momentum concerving interaction), I am not sure.

I suggest consider this fact a part of a model.