I have read the paper and found it overall very interesting and well written.
Unfortunately the authors have chosen the simplest TN available, the MPS that in this context turns out to be of very limited help, in getting definite answers about relevant quantities such as the string tension and the related Wilson loop area law coefficient.
Nevertheless the paper is an interesting addition in the growing field of gauge theories studies with tensor networks.
I first have a side remark. I guess the title is a bit misleading since the present work does not use any of the advanced construction of gauge invariant tensor networks but rather uses a smart trick in which the gauge symmetry is enforced locally on an extended Hilbert space and then the Hilbert space is reduced by using global U(1) symmetries. This is technical but very different from the spirit of using gauge invariant tensor network on the original Hilbert space.
Before recommending it for publication I strongly suggest that the authors
take critically in consideration the observations below that should help them improving the presentation and clarifying some controversial aspects of their work.
1- The notation is very difficult to understand for example
s_1 ...s_4 is used to label spins on links below Eq 1 but just the line above the sigma are labelled with 1 to 4. Are they actually acting on the spins s_1 ....? Why not to use the same s_1... to label the sigmas
2- I strongly recommend to define the concept they use, for example
what is "pyrochlore lattice" ?
3- The sentence (or equivalent with an Ising model with the same term....) cannot be understood. Which term are they referring to?
4- The sentence at the end of sect 2.2 starting with
The nature.... monopole contribution, needs to be reformulated.
As it stand mentioning the continuum limit in order to justify confinement is completely misleading. How do the authors plan to construct the continuum limit of the present lattice model? Can they elaborate on this point?
5- Again the notation of Eq 4 is unfortunate, s_i where the spins on the links below Eq 1 and in Fig. 1, adding a bold font does not help the reader, can the authors use another letter of the alphabet?
6- The authors group in a single site all vertical links and the horizontal links on the left and on the right of them. This implies a double counting that is fixed by a new constraint defined in Eq 5.
While I would see how to solve the constraint by using a bunch of copy tensors, they decide to go along a different path. My understanding is that they enumerate the configurations of spins on the left and on the right and introduce as many blocks in the tensor as configuration on the horizontal links (an exponential number as a function of the transverse size). This seems to add a lot of overhead to the calculations, have they compared with the more traditional approach based on copy tensors?
7-In any case the explanation would strongly benefit from a schematic drawing of the encoding of their 2D lattice in a 1D TN.
8- In Figure 5 the bond dimension is cited as the source of the error bars, can the authors explain in which sense they are able to associate an error bar with a certain value of the bond dimension and cite the relevant references in which this technique have been used/ tested on known models? I have in mind the recent extrapolations methods based on either the correlation length of the state or the DMRG truncation error.
9- They claim that the critical point is extracted from the finite size scaling, can they be more precise? Is it extracted from the extrapolation of the crossings of appropriately re-scaled curves? Do they need to know a certain critical exponent? Do they use Binder cumulants?
10- The analysis about entanglement does not reflect the current results in the field where entanglement in gauge theories can be divided in two parts distillable entanglement and entanglement due to the symmetry constraints. Can the authors make contact with those results and explain which part of the entanglement they are dealing with?
11- The string entanglement is defined by subtracting the vacuum entropy from the entropy of the string configuration. Is this actually a genuine entropy measure?
It could still be that the string state is orthogonal to the ground state but has the same entanglement, why would they then associate zero entanglement to it?
12- The authors seem to apply a standard TEBD algorithm although they mention that the size of the symmetric blocks of the Hamiltonian exceeds 100GB, can they explain how they do this?
13- With respect to the string excitations being described by a free bosonic theory, I find this piece of the work possibly the most interesting one. Is there any understanding of it from the microscopic details of the model? Namely the fact that the strings are extended objects seems to coincide with the fact that the authors obtain a dimensional reduction of the problem from 2D to 1D. Why are the string bosonic, can they show it in terms of braiding and commutation relation?
Can they expand a bit this section by adding the relevant discussion in the references they mention?