Convergence Issues in the minimization of free energy

[hellolori]

Dear developers,

I am using SSCHA to calculate the phonon spectrum of a two-dimensional charge density wave material at 0 Kelvin. However, I found that the minimization of the free energy does not converge, even after generating 12 populations of new configurations. After visualizing the data during the calculation process (as shown in the following figure), I noticed that the system's free energy, FC gradient, and structure gradient all decrease to a certain value, but then start to oscillate continuously. I use 6x6x1 supercells with 108 atoms for the calculations. Below are the input parameters for minimizing the free energy in each generation.

• generation of configurations

import cellconstructor as CC, cellconstructor.Phonons
import sscha, sscha.Ensemble
import numpy as np

dyn = CC.Phonons.Phonons("final_sscha_dyn_population11_", nqirr=7)
#
#[ apply here the needed changes to dyn ]
dyn.Symmetrize()
dyn.ForcePositiveDefinite()
#
# Prepare the ensemble
temperature = 0.0 #  K
ensemble = sscha.Ensemble.Ensemble(dyn, temperature)
#
# Generate the ensemble
number_of_configurations = 1000
ensemble.generate(number_of_configurations)
#
# Save the ensemble into a directory
save_directory = "data"
population_id = 12
ensemble.save(save_directory, population_id)

• the input for the minimization of free energy

import sscha, sscha.Ensemble, sscha.SchaMinimizer
import sscha.Utilities
import cellconstructor as CC, cellconstructor.Phonons
#
POPULATION = 12
#
dyn = CC.Phonons.Phonons("final_sscha_dyn_population11_", 7)
dyn.Symmetrize()
dyn.ForcePositiveDefinite()
temperature = 0.0 #  K
ensemble = sscha.Ensemble.Ensemble(dyn, temperature)
ensemble.load("data", population = POPULATION, N = 1000)
#
minim = sscha.SchaMinimizer.SSCHA_Minimizer(ensemble)
minim.meaningful_factor=1e-5
minim.set_minimization_step(0.01)
minim.init()
#
# Save the minimization details
ioinfo = sscha.Utilities.IOInfo()
ioinfo.SetupSaving("minim_{}".format(POPULATION))
#
minim.run(custom_function_post = ioinfo.CFP_SaveAll)
minim.finalize()
minim.dyn.save_qe("final_sscha_dyn_population{}_".format(POPULATION))

Could you please give me some advice on this issue?

[mesonepigreco]

Hi, from your plot, this simulation appears to be well-converged. The system undergoes significant changes in the first few steps, after which the auxiliary frequencies converge to a constant value, while the gradients have nearly reached zero. Why do you think it is not converged?

[mesonepigreco]

The numerical noise you observe is probably due to a very small frequency (almost zero), which is numerically unstable. This is an indication that the phase you are simulating is highly unstable. I suspect that your system will have imaginary phonons if you compute the free energy Hessian. Try to use a higher temperature, and the free energy minimization should be more stable and proceed smoothly.

[hellolori]

Thank you very much for your reply. I suspect that the calculation has not converged because the free-energy minimizations always end with the message "you are out of the statistical sampling", even though new configuration populations are continually being generated. Based on your reply, I now realize that this may be due to numerical noise caused by imaginary phonons. Indeed, when I increase the temperature, the free-energy minimizations converge much better.