Gauss law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories

1- pedagogical review
2- short enough
3- timely
4-original approach

1- lack of historical perspective with citation to relevant papers
2- too strong statement that are not supported by equivalent theorems

Overall I enjoyed reading these lecture notes, from my point of view they are a good addition to the available literature on the subject. I however think that the introduction should be expanded slightly in order to give a better account of the related works on the subject.

The justification on how one should gauge a tensor network state, or an Hamiltonian is introduced gently by starting from gauging a single system of two fermions, a nice pedagogical choice. They then proceed by showing that the idea of gauging a system by acting locally with unitary transformations on a globally invariant matter system results too restrictive.
Unless one is happy with extending the system including auxiliary matter degrees of freedom.

Here there are multiple way to do this, and one of the natural choices result into a tensor network state, the gauge invariant FPEPS. In this way tensor networks are presented as the natural tool for gauging the "extended" matter system locally.

At the end of the review however they show that the ideas do not need to rely on tensor networks, generalizing what has been done for bosons in the Fradkin and Shenker paper in the 80s. It is indeed possible to obtain a lattice gauge theory by acting on matter states with local unitaries.

The authors decide to present the opposite process, in section 4.3 where they show how to generalize the known duality that allows disentangling matter fields from gauge fields locally in order to disentangle the fermions from the gauge degrees of freedom via a local unitary transformations.

This seems more interesting since it allows to study fermionic gauge theories without fermions (though it is not at all clear that the resulting bosonic theory is simpler to characterize than the original one containing the fermions).

However the process of entangling fermions with gauge bosons could have been presented already (or at least meantioned) at the end of section 2.3, thus showing that a slight modification of what is attempted there (entangling gauge bosons with fermionic matter with local unitary transformations) is possible even without using tensor networks.

In the present form of the review, this message is some how hidden and it is easy to get the wrong picture that there is actually no way to locally entangle two different systems (one containing the matter and the other the gauge bosons) resulting in a usual gauge theory using local unitary transformation.

The authors try to clarify this point a the end of section 2.3 in the sentence about "modifying" rather than "transforming". I believe however that it would be appropriate to expand that sentence into a full paragraph and better explain the physical and conceptual consequences. As it is now, it sounds very cryptic and what modifying and transforming mean is not clear at all.

On a separate ground I suggest that the introduction should be extended.
It would be important to have an introduction that gives a better perspective of the field.

For example the use of tensor networks as a natural tool to force the the Gauss Law and extract the physical Hilbert space of a Gauge theory embedded into a larger tensor product Hilbert space has been originally discussed without relation (or with an implicit relation) to the problem of minimally coupling the matter to gauge fields.

In the high energy community, the identification of the gauge invariant Hilbert space has been done by finding dualities between gauge invariant systems and spin systems. A nice review on the subject is the one by Robert Savit Rev. Mod. Phys. 52, 453 (1980) that I guess deserves to be mentioned.

The necessity to embed the Hilbert space of gauge theories into a tensor product structure containing also to non-gauge invariant configurations has also been discussed in the context of lattice gauge theories when people started to be interested in measuring the entanglement entropy. (See for example Buividovich and Polikarpov Phys.Lett.B670:141-145,2008).

From the point of view of tensor networks, the first paper that has made connection with the above results and the possibility to use tensor network in order to describe states in the gauge invariant Hilbert space is the one by Tagliacozzo and Vidal published in Phys. Rev. B 83, 115127 (2011).

A discussion along these lines would only require a couple of extra paragraphs in the introduction and would give the reader a better overview of the field.

I thus believe that a small set of modifications will improve these already nice lecture notes.

1- Improve the introduction by adding the citations to the work described in the report where appropriate.
2- Define Dirac Gamma matrice after (2).
3- The notation in (12) is a bit unfortunate, consider replacing j by another letter (just a suggestion).
4- After (13) in locally gauge invariant locally and gauge actually mean the same thing, chose either one or the other.
5- In section 2.3 state explicitly that there are many different ways of "fixing the gauge" via enforcing the Gauss law. Eq 41 could seem the more natural one but this is not the only one, please mention it.
6- I am a bit confused with 42 since I would have expected that the sum on the psi^dagger psi part, not on the phases.
7- Review the three last paragraphs of section 2.3 to accommodate the observations made in the Report section and better explain what is meant.
8- First line of section 2.4 specify what this refers to.
9- From the discussion in the text I do not understand why two fermionic modes per auxiliary leg are necessary. I understand this allows to avoid adding extra tensors. Is it equivalent to add an extra link tensor with two fermionic modes encoding the electric field as (72) that is then contracted with a fermionic peps tensor with just one fermionic mode per auxiliary link?
10 First line of 4.2 change this of for that of.
10- In section 4.3 possibly mention that the opposite of this idea is a good way to locally entangle some matter fields with gauge fields resulting in a usual gauge theory (in any dimension).