Hello, I have some questions about calculating the dynamic spin structure factor.Please help me.

First, in the script provided by HPhi (samples/tutorial_4.3/spinchain_example.py), the dynamic spin structure factor of the Heisenberg model is calculated. In lines 78 and 79 of this script, the data is passed to the pair.def file as follows:
f.write('%i 0 %i 0 0 %f %f\n' % (j, j, wr, wi))
f.write('%i 1 %i 1 0 %f %f\n' % (j, j, -wr, -wi))
However, why is it written this way instead of:
f.write('%i 0 %i 0 1 %f %f\n' % (j, j, wr, wi))
f.write('%i 1 %i 1 1 %f %f\n' % (j, j, -wr, -wi))
After all, 0: C_{i sigma1} C_{j sigma2}^\dagger and 1: C_{i sigma1}^\dagger C_{j sigma2}.

Second, when calculating the DSSF, what is the difference between the Lanczos method and the CG method? Also, how does the role of Lanczos_max differ between these two methods?

I noticed that when I set Lanczos_max = 5000, the Lanczos method performs exactly 5000 iterations, whereas the CG method does not—it seems to check for convergence and terminate early if satisfied.

[tmisawa]

On the 1st question, your expression is correct, c+c is more natural.
Since cc+=1-c+c, I believe the results do not change except for a constant term.

As for your second question, in the case of Lanczos,
the excitation spectrum is calculated using continued fraction expansion,
while in CG, the shifted Krylov subspace method is used.
The key difference is that the continued fraction method runs for a fixed number of iterations,
whereas the shifted Krylov method stops when a specified accuracy is reached.
So I think that's why their behaviors differ.

[tmisawa]

Additional comment to the 1st question: Since there is an e^{iqr} term, I think that the constant term also vanishes. That means you will obtain the same results whether you use c+c or cc+ for this case.

So which method do you recommend more for calculating the dynamic spin structure factor? If I use the Lanczos method for calculation, what should the Lanczos_max be set to? Setting it too high would waste resources, but would setting it too low affect the calculation accuracy?

[tmisawa]

I recommend using the CG method (shifted Krylov).

In the Lanczos method, determining how many iterations are needed depends on the convergence of the spectral shape. Therefore, you either need to run several calculations with different Lanczos_max values, or restart the calculation repeatedly.
This process can be quite time-consuming, so I recommend using the CG method instead.

Then, when calculating the dynamic spin structure factor for a new model, how should I determine the values of OmegaMax, delta omega, and OmegaIm(eta)? Does this relate to the ground state energy and the energy difference between the first excited state and the ground state?
Clearly, this is a question unrelated to HPhi, and you can not answer it.

[tmisawa]

When computing the dynamical spin structure factor,
the parameters OmegaMax, delta omega, and OmegaIm should be
chosen based on the energy scales of the system.
I note that OmegaOrg can be set to the ground state energy E_0,
allowing the spectrum to be referenced from omega = 0,
which corresponds to excitations above the ground state.
OmegaMax should be large enough to cover the target excitations.
Typically, in spin systems (energy scale is governed by the exchange interaction J),
OmegaMax is set to a few times J (e.g. OmegaMax~ 2J-5J).

The energy resolution delta omega is typically chosen in the range of 0.01J to 0.1J,
depending on the desired spectral sharpness and computational cost.
The imaginary part OmegaIm (or eta)
provides spectral broadening and is usually set to be a few times delta omega
to suppress numerical noise.
These parameters should be adjusted to balance spectral resolution and numerical stability.