I see that in many papers where optimization results presented, they do not mention phonons and their force convergence criterion is 0.01 eV/A. Can you publish a result like that? And is not this tolerance too high? Could the reason for using this tolerance be because of the non-measurable effect of lower tolerances on the resulting geometry? I also saw that in the papers where phonons were presented, lower tolerances were being used. Is it a convention in the community?
Phonon calculations tend to be very expensive to run. That being said, for gas phase molecules it is very common and expected that frequency calculations are run to ensure the molecule is not on a saddle point.
In general, you can publish anything if it makes it past peer review. Phonon calculations are something you would do if you fear you are on the saddle point but in my experience it is much harder to optimize to a saddle point in bulk. For this reason, I expect others feel the same and it is just not commonly performed. If you have the computational time and power though, I don't think anyone will ever ask "Why would you bother?".
It should be noted though that anytime entropy/zero point energy is mentioned that they likely have actually done a frequency calculation even if they are not referring to it explicitly.
In general it is not justified to published the geometry of a system without performing a phonon calculation. This is where you may end up in the potential energy surface depending on which type of calculation you perform:
- Geometry optimization. With a geometry optimization, you may end up at a local minimum or at a saddle point of the potential energy surface. You can end up at a saddle point if you perform a geometry optimization enforcing the initial symmetry of the system (a very common strategy), because enforcing symmetry reduces the dimensionality of the potential energy surface which may lead to the removal of important directions that lower the energy further. With only a geometry optimization there is no way of distinguishing between a saddle point or a minimum, and this is why you need phonons.
- Phonons. With phonons, you are calculating the Hessian about a stationary point of the potential energy surface, to which you got via a geometry optimization. If all eigenvalues of the Hessian are positive (corresponding to real and positive phonon frequencies, which are the square root of the eigenvalues), you then know that you are at a local minimum. If an eigenvalue of the Hessian is negative (imaginary phonon frequency), then you are at a saddle point. You should then distort the structure along the phonon eigenvector associated with the negative eigenvalue and you will find a lower-energy structure by performing a new geometry optimization. Combining geometry optimizations and phonons in this way can ensure that you end up at a local minimum.
- Structure prediction. With phonons you can ensure that you are at a local minimum of the potential energy surface. However, there is no way to ensure that you are at the global minimum. In fact, there is no general solution to the problem of locating the global minimum of the energy surface. However, structure prediction methods have shown to be rather good at finding them, so depending on how much is known about your material, it may be a good idea to perform a structure search.
Having said all this, when may it be justified to perform a geometry optimization without a phonon calculation? I would say that: (i) if the material is well-characterized experimentally, and (ii) the properties you are interested in are not directly related to phonons (e.g. optical properties); then I think most people would consider it OK to assume that the experimental structure is a reasonable guess, and only perform a geometry optimization before moving on to performing the additional calculations you are really interested in that are not connected to phonons.
For your questions about numerical tolerances, the guideline should always be that the quantity that you are interested in is converged to the required level. Phonons do typically require relatively stringent numerical tolerances, in particular a lower energy tolerance for the SCF cycle convergence because forces are not variational.