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!

  • $\begingroup$ 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. $\endgroup$ May 26, 2020 at 6:06

1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$ May 26, 2020 at 8:40
  • $\begingroup$ 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) $\endgroup$
    – user245141
    May 27, 2020 at 7:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.