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Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories
by Patrick Emonts, Erez Zohar
This is not the current version.
|As Contributors:||Patrick Emonts|
|Submitted by:||Emonts, Patrick|
|Submitted to:||SciPost Physics Lecture Notes|
|Subject area:||Quantum Physics|
In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced, by looking at the Gauss law from two different points of view: for the gauge field it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.
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Author comments upon resubmission
We found the comments and suggestions for modifications very insightful and have modified the paper accordingly as summarized in the list of changes.
List of changes
Below, we provide detailed answers to the referee's suggestions and explain the respective modifications.
1- Improve the introduction by adding the citations to the work described in the report where appropriate.
We have now added a few sentences on the topic, including the requested references in section 2.2 which addresses the structure of the Hilbert space.
Although it is not the introduction section, it is still an introductory part of the lecture notes, where we believe it fits best.
2- Define Dirac Gamma matrice after (2).
Agreed and done.
3- The notation in (12) is a bit unfortunate, consider replacing j by another letter (just a suggestion).
Agreed and done -- replaced by l.
4- After (13) in locally gauge invariant locally and gauge actually mean the same thing, chose either one or the other.
Indeed, the formulation locally gauge invariant is redundant, we modified the sentence to "[...] is gauge invariant, i.e. invariant under local transformations generated by the Gauss law operators [...]".
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.
We rephrased the passage next to the equation to clarify this issue.
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.
Indeed, the sum over x was missing and we added it. Nevertheless, the phases must be summed as well in order to obtain the right transformation.
This is the manifestation of the non-locality of the Gauss law solution in the transformation.
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.
We modified those paragraphs accordingly, aiming to clarify the difference between a unitary transformation and the process of minimal coupling.
8- First line of section 2.4 specify what this refers to.
We substituted this by "unitary gauging of separate building blocks".
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?
The construction with two fermionic modes per auxiliary leg is not necessary, but rather used as an example.
We made it clear in the text.
10- First line of 4.2 change this of for that of.
11- 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).
Here we did not make any change. As argued in the first sections of the lecture notes, disentangling gauge fields from matter is in general not possible unitarily (in more than 1+1d) because of the non-uniqueness of the Gauss law solution. For this reason, we do not see how the opposite process of entangling matter with gauge fields would be possible - unless some configurations of electric fields are arbitrarily chosen. For this reason we decided not to include this remark in the manuscript.
Submission & Refereeing History
Reports on this Submission
Report 1 by Luca Tagliacozzo on 2019-12-13 Invited Report
- Cite as: Luca Tagliacozzo, Report on arXiv:1807.01294v3, delivered 2019-12-13, doi: 10.21468/SciPost.Report.1390
Already mentioned in the previous report
Still point 2 of previous weakness partially present in the new version (see report)
I am happy with the reviewed version.
I still however disagree with the last statement of their reply
My observation was:
"1- 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)."
The authors' reply is
"Here we did not make any change. As argued in the first sections of the lecture notes, disentangling gauge fields from matter is in general not possible unitarily (in more than 1+1d) because of the non-uniqueness of the Gauss law solution. For this reason, we do not see how the opposite process of entangling matter with gauge fields would be possible - unless some configurations of electric fields are arbitrarily chosen. For this reason we decided not to include this remark in the manuscript."
We obviously disagree on this point.
In order to understand if it is a lexical disagreement or a disagreement on the physics I attach here a short note where I explicitly perform the calculations I was referring to in my comment.
They show that in any dimension we can locally disentangle the matter from the gauge fields using unitary transformations.
The final Hamiltonian is just the one of bosonic matter minimally coupled to gauge fields. This means that minimal coupling can be obtained using local unitary transformation in arbitrary dimension. The opposite to what the authors claim in their reply.
Isn't the result they present in section 4 achieving the same for fermionic matter?
Maybe we are trying to say the same thing using a different language?
I would appreciate if the authors could comment on this, and why my example works while they state that it is impossible to disentangled matter from gauge fields locally
1- Answer my question taking into account the attached notes