# Second quantization: periodicity of annihilation and creation operators in momentum space, originally on a lattice

I have a Hamiltonian $$H$$ on a periodic lattice, which is expressed as, say:

$$H = \sum_{n} (A_n a^\dagger_n a_n + B_n a^\dagger_{n+1} a_n + h.c.)$$

where $$A_n$$ and $$B_n$$ are periodic in space (over the lattice) with a period of $$\beta$$. Now, in order to get the momentum space representation of the Hamiltonian, I express the annihilation (and creation) operators as:

$$a_n = \sum_{k=-\infty}^{\infty}e^{-ikn}a_k$$

Instead of having the sum over $$k$$ running from $$-\infty$$ to $$+\infty$$, I would like to have the sum running only over the Brillouin zone. I am not sure how to go about doing that. Any help would be appreciated!

• I should have mentioned in my question that $n$ has infinite limits. Then, the Fourier transform from $n$ to $k$ would give infinite limits too. Could you please explain what you mean by relating it to DFT? As I understand, the Fourier transform from $a_k$ to $a_n$ (an vice versa) is discrete FT (as included in the question) and I am not sure how else one would use DFT. May 26, 2020 at 6:06

As the size of your system is infinite, then the range of $$k$$ becomes continuous. The correct form is $$a_n = \int_{-\pi}^{\pi}\! \frac{dk}{\sqrt{2\pi}}e^{-i k n} a(k)$$ where we implicitly took the lattice spacing to be $$1$$, which makes $$k$$ dimensionless. Note that the commutation relations that we want to preserve are $$[a^{\dagger}_n, a_m]=\delta_{m,n}$$, and this integral works. Note, however, that the commutation relations of $$[a^{\dagger}(k), a(q)] \propto \delta(k-q)$$.
In general, I would recommend to introduce dimensions and starting with a finite lattice site, taking the limit $$N\to\infty$$ while keeping lattice spacing $$\alpha$$ fixed, to get the feeling of how the sums become integrals and how, in fact, $$a(k)$$ must get dimensions of square root of inverse length.
• If I follow your definition of the annihilation operator $a_n$ in terms of $a_k$ (which I understand and I realize why my question is erroneous in that part) and substitute it in $H$, say I get terms like $\int dk \int dk' f(k,k') \, \delta(k-k'+\pi)$, where $k,k'$ are momenta. What would happen to this integral? I should naively get $f(k,k+\pi)$ but $k+\pi$ could lie outside the Brillouin zone. Also, since $k$ is periodic by $2\pi$, should $a_{k+\pi} = a_{k-\pi}$? I am concerned about the periodicity of $a_k$ mainly. May 26, 2020 at 8:40
• at least for uniform hopping you're not supposed to get that... I see that you have periodicity $\beta$, which means that you need to define a unit cell and take the FT with respect to that (Bloch theorem)