Defining specific symmetry sectors

[RehMoritz]

Implementation of symmetry sectors such that states with specific quantum numbers can be modelled.
The LatticeSymmetry class now takes two arguments to initialize, namely the orbits that generate the transformation as well as the factors that give the prefactor in front of the wave function coefficient.
The previous behavior is obtained if one sets all of these prefactors to 1.

[laurinbrunner]

I suggest we try to keep a similar interface for the MCSampler and the ExactSampler, so I think it would be best to call the 'mu' parameter you added to the MCSampler 'logProbFactor' and make it behave in a similar way as it does in the ExactSampler.

Furthermore I don't see how changing the probability return for the MCSampler from 'None' to the actual probabilities works. Isn't the idea that the samples should already follow the wanted probability distribution, so the average is just the mean without the probabilities since they are already subsumed in the frequency of a configuration in the samples? This argument should be independent from the choice of mu, as it already works for POVMs where we basically get something corresponding to mu=1.

[RehMoritz]

I would like to keep these options separate as they fulfill different purposes.
The MC reweighting scheme (also discussed in some more detail in PR #41) accepts samples with the 'wrong' probability, i.e. $|\psi_\theta|^\mu$ instead of $|\psi_\theta|^2$. This requires reweighting physical observables with $|\psi_\theta|^{2-\mu}$ at a later point.
I agree that this would need to be adapted if one were to apply this to the POVM case aswell, but since there is currently no logProbFactor for MCMC anyways (i.e. no support for POVMs), I did not see the need.

If we wanted to extend the reweighting scheme to MCMC with POVMs (which I think would be a nice feature), we would simply need to have the acceptance probabilities to be $|P_\theta|^\mu$ instead of $|P_\theta|$. This would require reweighting physical observables with $|P_\theta|^{1-\mu}$ at a later point.
Generally, the reweighting factor would then be $|\chi_\theta|^{\frac{1}{\mathrm{logProbFactor}}-\mu}$ with $\chi$ either $P$ or $\psi$.

[laurinbrunner]

Ah thank you, now I have a better understanding of what you did. I agree that one should keep logProbFactor and mu different, but I think it would be best to make the MCMC sampler compatible with POVMs at this point.

[markusschmitt]

I changed the interface for the get_orbit_* functions a bit. Instead of the clumsy dictionaries, one can now simply pass the names of the desired symmetries and separately the quantum number prefactors if they differ from 1, for example:

get_orbit_1D(L, "reflection", reflection_factor=-1)