Kagome antiferromagnet

[prinsmag]

Hey everyone,

We are trying to calculate the magnon band structure of a simple kagome antiferromagnet. As a reference, we are following the example discussed in the SpinW tutorial:

https://spinw.org/tutorials/07tutorial

However, when we use the same Hamiltonian and parameters as given in the tutorial, we obtain negative magnon frequencies at the (\Gamma) point (nan as the freq value). Since we expect the results to match the SpinW calculation, we are wondering whether there is something we might be missing in our setup.

Has anyone encountered a similar issue or can suggest possible reasons for negative frequencies at (\Gamma) in this case?

Thanks in advance for any help.

Magnopy-kagomeferromagnet.ipynb

[adrybakov]

Hello, I will convert this issue into an Q&A discussion.

What version of Magnopy do you use (print(magnopy.__version__))?

[prinsmag]

Hey,
We are using 0.5.3.
Thanks

[adrybakov]

I can reproduce this behaviour.

The grand dynamical matrix is Hermitian, but not positive definite at Gamma, thus the magnon bands are not computable nor meaningful.

The question whether the bug is in Magnopy or in SpinW is open for me for now.

Best,
Andrey

Magnopy-kagome-ferromagnet-v2.ipynb

[prinsmag]

Hello
Thanks a lot for the reply.
We compared the methods used in SpinW and Magnopy, and found that spinW can either use Colpa method or the White's method (see screenshot). Then we further found that for using the Colpa method for this particular example (kagome AF), it solves the issue as "Warning: To make the Hamiltonian positive definite, a small omega_tol value was added to its diagonal! ".
Anyway this info can help us get the code running?
Thanks a lot again

[adrybakov]

Yes, Magnopy does the same trick:

magnopy/src/magnopy/_lswt.py

Lines 673 to 695 in 91baa1a

	# Diagonalize via Colpa's method
	try:
	E, G = solve_via_colpa(GDM)
	except ColpaFailed:
	# Try to diagonalize with suspected Goldstone mode
	try:
	E, G = solve_via_colpa(
	GDM + (1e-10) * np.eye(GDM.shape[0], dtype=float),
	)
	# Return NaNs if it still fails
	except ColpaFailed:
	# Try to diagonalize for the negative GDMs
	# Note: solve_via_colpa will return positive eigenvalues,
	# so we need to negate them back
	try:
	E, G = solve_via_colpa(-GDM)
	E = -E
	except ColpaFailed:
	return (
	[np.nan for _ in range(self.M)],
	np.nan,
	[[np.nan for _ in range(2 * self.M)] for _ in range(self.M)],
	)

However, it does not help in this example.

Colpa's and White's methods are expected to give the same results when the system is diagonalisable into the physically meaningful magnon states (i. e. when the grand dynamical matrix is hermitian and positive definite):

When the system is not diagonalisable, then Colpa's method fails (which is desired) and White's method gives incorrect solutions (as correctly written in the screenshot from SpinW's docs).

Do you have a peer-reviewed reference in mind, in which the Kagome AF studied and the Hamiltonian is clearly written down?

Best,
Andrey

[adrybakov]

Code used (note: only correct for this particular example):

g = np.diag([1., 1., 1., -1., -1., -1.])
white_omegas = []

for k in kp.points(relative=False):
    GDM = lswt_hp.GDM(k=k, relative=False)

    E, G = np.linalg.eig(g @ GDM)

    white_omegas.append(sorted(E, key=lambda x: x.real))

white_omegas = np.array(white_omegas)
white_omegas = white_omegas[:,3:] - white_omegas[:, [2,1,0]]

[prinsmag]

Hey, our ultimate goal is to calculate magnon dispersion for something like discussed in this report, but with the nice additions to H that magnopy has.
https://www.nature.com/articles/s41535-018-0137-9

But before that, I wanted to point out something else

handpicked_sd = [
    [1, 0, 0],
    [-0.5, 0.866, 0],
    [-0.5, -0.866, 0]
]

energy = magnopy.Energy(spinham=spinham)
optimized_sd = energy.optimize()

for i, sd in enumerate(optimized_sd):
    print(f"{spinham.magnetic_atoms.names[i]} : {sd[0]:>7.4f} {sd[1]:>7.4f} {sd[2]:>7.4f}")

print("Pairwise angles handpicked:")
print(f"{wulfric.geometry.get_angle(handpicked_sd[0], handpicked_sd[1]):.2f}")
print(f"{wulfric.geometry.get_angle(handpicked_sd[1], handpicked_sd[2]):.2f}")
print(f"{wulfric.geometry.get_angle(handpicked_sd[2], handpicked_sd[0]):.2f}")

print("Pairwise angles:")
print(f"{wulfric.geometry.get_angle(optimized_sd[0], optimized_sd[1]):.2f}")
print(f"{wulfric.geometry.get_angle(optimized_sd[1], optimized_sd[2]):.2f}")
print(f"{wulfric.geometry.get_angle(optimized_sd[2], optimized_sd[0]):.2f}")

This code gives me
─────┬─────────────┬──────────────────┬─────────────
step │ Energy │ delta Energy │ max torque
─────┴─────────────┴──────────────────┴─────────────
1 -5.4985280 3.682537823856 1.5033663
2 -5.8706888 0.372160814722 0.9270401
3 -5.9384448 0.067756015242 0.6746305
4 -5.9987515 0.060306746828 0.0998279
5 -5.9999489 0.001197407132 0.0185404
6 -5.9999977 0.000048721150 0.0040994
7 -6.0000000 0.000002300831 0.0005810
8 -6.0000000 0.000000042288 0.0000304
9 -6.0000000 0.000000000125 0.0000015
────────────────────────────────────────────────────
Mn1 : 0.9988 -0.0386 -0.0290
Mn2 : -0.4608 0.8576 0.2285
Mn3 : -0.5381 -0.8190 -0.1995
Pairwise angles handpicked:
120.00
120.00
120.00
Pairwise angles:
120.00
120.00
120.00

I am not able to see why there would be an energy difference if the spins were at the same angle to each other, before and after the energy minimization. I used only J1 and J2 =0

[adrybakov]

energy.optimize() starts from the random configuration of spin directions. See initial_guess in Energy.optimize()

EDIT: add link to docs

[prinsmag]

Hey there,
Positive news. We found the bug which was causing the not-positive definite issue. In the lswt.py, we replaced the

self.B2[nu][alphas[0], alphas[1]] = (
    0.5
    * np.sqrt(self.spins[alphas[0]] * self.spins[alphas[1]])
    * (
        np.conjugate(self.p[alphas[0]])
        @ parameter
        @ np.conjugate(self.p[alphas[1]])
    )
)

block with

self.B2[nu][alphas[0], alphas[1]] = (
    0.5
    * np.sqrt(self.spins[alphas[0]] * self.spins[alphas[1]])
    * (
        self.p[alphas[0]]
        @ parameter
        @ self.p[alphas[1]]
    )
)

after which, the problem is completely gone. I think the issue was related to a non-collinear spins having a phase between them. Please do let me know what you think. Although we will also go back and check if other simple examples work same or not.
Best regards

EDIT by @adrybakov: Fix is published in v0.5.4

[adrybakov]

Wow, indeed this is a bug and quite a crucial one. Many thanks for finding it!

I would prefer if you'd open a pull request with the exact change of code as in your previous comment, so you contribution gets recorder.

Would you like to do so?

Best,
Andrey

[adrybakov]

Yes, this bug does affect non-collinear spin configuration and a subset of anisotropic exchange interactions.

[prinsmag]

Hey. Glad it works. I will make a pull request for the same.
Best Regards

[adrybakov]

Merged and published in v0.5.4.

Thank you for spotting and fixing the bug.

Best,
Andrey