Quantum robustness and phase transitions of the 3D Toric Code in a field

1- The introduction offers a very useful entry to the literature on the 2d and 3d toric code, as well as related models and implementations.
2- The properties of the unperturbed 3d toric code on a cubic lattice are well exposed, in a manner accessible to the newcomer.
3- The phase diagram under an external field is identified consistently through a variety of methods (exactly in certain limits, and by variational methods).
4- Throughout the manuscript, the authors try to provide a physical intuition of the mechanisms at play, as well as comparisons with the better-known 2d toric code.

1- The technicalities related to the variational determination of the ground state are hard to follow.

In the paper, the authors study the 3d toric code on a cubic lattice, which displays a topologically ordered ground state, and focus on the phase transitions towards a trivial state induced by an external uniform magnetic field. After a detailed introduction which clearly motivates their study, the authors describe the model and its ground state properties. Then, applying a combination of methods (p-CUT, exact dualities and variational computations), they reconstruct the phase diagram of the toric code under a uniform magnetic field in an arbitrary direction. Throughout this study, they provide a detailed description of the physical mechanisms induced by the external field, and compare them with the 2d case. Finally, a long discussion summarizes the main results of the paper.

Overall, I find the paper extremely well written, and accessible to the newcomer to the field. Although the technicalities associated with pCUT and the variational calculations are difficult to follow, their outputs are clearly summarized in physical terms. The phase diagram is convincingly reconstructed via various independent means. For these reasons, I recommend the publication of the manuscript.

1- Eq. (6) should be explained a bit more. The authors should explain why the ground-state degeneracy is given by $2^{N_{spins}} / 2^{N_{constraints}}$.
2- I think that Eq. (8) contains a typo. First, I would advise the authors not to use m as a variable inside the summation, as it brings confusion with the superscript m in $P^m_{xy}$. Then, I think that a $b_z$ is missing in front of the term $(n_z + 1/2)$.
3- Eq. (9) contains a typo : the last term is $b_z = (0, 0, 1)$.
4- p.7, first line: I would suggest to add "In the loop-soup picture of the ground state, these operators measure the parity..."
5- Just after Eq. (11): "with some fixed $(n_y, n_z) \in Z$..."
6- Footnote 3: "...can also be viewed, in the light of quantum codes, as the..." (with comas)
7- After Eq. (18): is really the ground state energy equal to 4N ? I would say that $E_0 = -(1/2)N_{stars} -(1/2)N_{plaquettes} = -N/2 - 3N/2 = -2N$, am I correct?
8- Eq. (19): what means the prefactor $1 / (2j)$ in front of the last term?
9- Eq. (21) and in other places, the authors use the symbol "=" with a "!" on top of it: I have never seen this symbol and some explanation would be welcome.
10- After Eq. (33): "...the two limiting cases $\alpha=\beta=1$ and $\alpha=\beta=0$ are exactly..." and "For $\alpha=\beta=1$, the normalization..."