Tags: Dynamical Spin Structure Factor

Integral don’t converge

I started plotting formulas from the previous post using Mathematica. It seems like I’m missing something. In particular, the integrals don’t seem to converge when I compute them.

Here (Evaluation of Dynamic spin structure factor for the spin-1/2 XXZ chain in a magnetic field, 2004) is a reference that seems to have more details and clarify some of the computations. More refernces:

Two-spinon dynamic structure factor of the one-dimensional s=1/2 Heisenberg antiferromagnet (1997)
Computation of dynamical correlation functions of Heisenberg chains in a field (2005)
The 4-spinon dynamical structure factor of the Heisenberg chain (2006)
Dynamical structure factor of the J1 − J2 Heisenberg model in one dimension: The variational Monte Carlo approach (2018)

Resolution

It turns out that I had the wrong initial state for the formula of the dynamical spin structure factor. I was taking a deterministic spin initial state. Instead, I should have been taking the ground state as the intial state.

Additionally, I was confused by the following integral formula

\[\int_{-\infty}^{\infty} e^{i x t} dt = \delta(x) = \begin{cases} \infty , & \quad x=0\\ 0 , &\quad x \neq 0 \end{cases}.\]

It turns out that the equality above is an equality as a distribution. Otherwise, the integral above doesn’t converge. That is why my results were not making sense.

Next Step

I will do the computation in detail where the integral idenity above is used.
Try the \(N=1\) case again.