An exact local mapping from clock-spins to fermions

1- The proposed mapping between general clock-spins and fermions is exact and local
2- The results of this work open to the possibility of employing existing techniques to tackle 1D interacting fermionic system to clock-spin models
3- Clock-spins enjoy unusual properties and are likely to rise to prominence as they can be realized in artificial systems or simulated at the quantum level

1- The proposed mapping implies a substantial growth of the Hilbert space, since most of the fermionic states are unphysical. It is not clear from the paper's discussion how this affects the numerical implementation and efficiency of the mapping.
2- The author discuss a n=3 clock-spin model with unusual features and manage to reproduce those within the fermion language. It is a pity that this exercise is not used to draw some general conclusions or to present a deeper explanation of the peculiarities of the even/odd effects in this model.

This manuscript details an exact, local mapping between clock-spins and fermions, developing a generalization of the Fedotov-Popov method to map spin-1/2 degrees of freedom into spinless fermions.
The paper is clearly written and motivated and its content seems correct to me. With the rise in interest toward synthetic materials, the understanding and exploitation of clock-spin degrees of freedom can become a growing field of activity for the community and thus I would favor publication of this work, once some minor revisions are implemented. However, at the moment I feel inclined in recommending its acceptance in a more specialized journal, such as SciPost Physics Core, since I am not sure that these results rise to the level of general interest aimed at by the flagship SciPost journal.

1- In the middle of page 2, not enough references are provided to vindicate the overwhelming amount of results for the quantum Ising chain. The authors should add additional citations, including some of the existing review articles.
2- Immediately after [10-12], the authors write that the non-locality of the Jordan-Wigner transformation has hindered its application to numerical approached in 1D. I feel that this sentence does not do justice to the great success of the JW when local interactions are considered. I think that the author should clarify that the issue arise when dealing with long-range interactions.
3- The authors sometime spell Fedetov and sometime Fedotov. Consistency should be restored through the manuscript.
4- After eq. (17) on page 6, it is written that a periodicity is implied in the number of species of fermions, so that f_{n+1} has to be identified with f_n. This is a crucial assumption for the whole construction which should be highlighted earlier in the construction of the mapping, preferably in Sec. 2.
5- I do not understand how the addition of that constant in eq. (43) does not affect the unphysical states as well, giving them a finite contribution to the partition function. The author should elaborate.
6- Before eq. (74) on page 16, the authors claim that they can drop the interaction piece of the Hamiltonian, since they are looking for the ground state only. Such statement is not correct in general and thus this sentence should be reformulated.
7- I find the solution and discussion in sec. 5 too rushed and I would like for the authors to add more details on the derivation and a discussion on what generates this weird even/odd effect in the fermionic language. It is clearly uncommon for the ground state degeneracy to depend on the parity of the chain length and even more for the order parameter to show such dependency...

- It is an interesting mapping to a fermionic system of clock models, a thing that was unexpected.

- I do not see anything physically interesting. I mean, in the similar context of Z_2 spin chains, the Jordan-Wigner mapping was able to map an apparently unmanageable system (the quantum Ising chain in transverse field) to solvable quadratic fermions [P. Pfeuty, Annals of Physics 57, 79 (1970)]. In this representation the model showed Majorana fermions at the boundaries in the symmetry-broken phase [A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001)], so that the magnetization transition in the spin representation was mapped to a topological transition in the fermionic one.

Nothing like that can I see here. The fermionic models that the authors obtain are not quadratic, so not solvable. Hence there is not the interesting point of making something complex simpler.

Is there at least some physical conclusion? For instance an interpretation of some Z_n symmetry breaking in terms of boundary topological modes in the fermionic representation (in analogy with the quantum Ising chain in transverse field)? What does their work add to the parafermions introduced in [1603.00095] in terms of physical interpretation and possibility to make analyses easier? The authors should answer to these questions before their work can be considered for publication in SciPost

In the present form the paper seems to me not suitable for publication in SciPost. The authors should consider some physical problem related to clock models (see ``Weak points'') and show that their method makes the analysis easier or gives new physical insight.