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Phase Diagram and Conformal String Excitations of Square Ice using Gauge Invariant Tensor Networks
by Ferdinand Tschirsich, Simone Montangero, Marcello Dalmonte
This is not the current version.
|As Contributors:||Marcello Dalmonte · Ferdinand Tschirsich|
|Submitted by:||Tschirsich, Ferdinand|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
We investigate the ground state phase diagram of square ice -- a U(1) lattice gauge theory in two spatial dimensions -- using gauge invariant tensor network techniques. By correlation function, Wilson loop, and entanglement diagnostics, we characterize its phases and the transitions between them, finding good agreement with previous studies. We study the entanglement properties of string excitations on top of the ground state, and provide direct evidence of the fact that the latter are described by a conformal field theory. Our results pave the way to the application of tensor network methods to confining, two-dimensional lattice gauge theories, to investigate their phase diagrams and low-lying excitations.
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Submission & Refereeing History
- Report 2 submitted on 2019-02-15 17:35 by Anonymous
- Report 1 submitted on 2019-02-13 14:34 by Anonymous
Reports on this Submission
Anonymous Report 2 on 2018-11-8 Invited Report
1- analysis of a U(1) link model with MPS
2- interesting characterization of the string excitations in the RVB phase
1- Use of MPS rather than more powerful TN.
2- Presentation sometimes too technical and with cumbersome notation
3- Exponentially complex trick to get rid of doubling of the Hilbert space
4- The results obtained are not conclusive about several aspects due to the limitation of the ansatz chosen
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?
1- Modify the title, that as it is I find a bit misleading. I would suggest to substitute gauge invariant tensor network with symmetric MPS
Critically address the above comment 1-13
Anonymous Report 1 on 2018-10-8 Invited Report
1. The paper is one of the first ones to report numerical studies, using tensor networks, of lattice gauge theories in more than 1+1d.
2. The authors have applied several study methods and reached similar results.
3. Although not dealing directly with a scenario that would have suffered, had another method been used, from the sign-problem, the methods presented in this work are sign-problem free and hence this work serves as an important benchmark.
4. The paper combines both theory with numerics. Only after the theoretical background is set, the authors turn to the description of numerical methods and discussion of results, keeping them apart.
1. Some theoretical concepts in the beginning could have benefited from graphical and/or more detailed technical presentation (see below).
I find this work very interesting and important, mostly for the strong points listed above. As written by the authors, the field of tensor network studies of lattice gauge theories, while being relatively new, is growing very fast, offering various new computation methods that allow one to overcome problems encountered by the traditional Monte-Carlo methods of lattice gauge theories. This work presents a very nice and remarkable result for a two dimensional theory with one compact dimension, by converting a cylindrical system into a one dimensional MPS and employing TEBD techniques.
The authors present the model in question (spin ice in two space dimensions) in a very clear manner – both the Hamlitonian and symmetries, and the different phases it exhibits. They further introduce a mapping of the system (when put on a cylinder) to a 1d one, which allows one to construct an MPS ansatz (discussed as well) for the study of the model.
Although this part of the paper (description of the mapping to 1d and construction of the MPS) was very clear to me, I am afraid it might not be clear, in general, to readers not familiar with tensor networks, or lattice gauge theories. In my opinion, subsection 2.3, where the MPS is constructed, can be significantly improved if some figures are added, showing the system (cylinder) and how it is being blocked for the construction of a one dimensional MPS. It can also help if a graphic demonstration of the symmetric MPS construction described in the first paragraph of page 7 ("Here we follow…last MPS bond") is added.
This applies to the next subsection (2.4) – as well. Eq. (8) and the inline equations around it show how to rewrite the Hamiltonian terms in terms of the effective 1d system. A figure and a few more equations (perhaps such equations could be placed in an appendix) can really boost the clarity of discussion.
Then, the authors explain the numerical method and turn, in the following section, to a discussion of the relevant observables and presentation of results. I find these parts of the paper very clearly written, and the results, in my opinion, are clearly presented in the figures given. The combination of "CM-like" observables (section 3.1) and "HEP-like" ones (3.2,3.3) is also very nice and reflects the fact that quantum field theory underlies the same physics, independent of the frame of reference (physical community). In particular, I liked the discussion of relevance of space-space Wilson loops in a model that breaks Lorentz invariance, as the one described in this work, and the nice conclusion about it given at the end of the section (quantitative but not qualitative difference, hence it is still a valid order parameter).
The quantum informative perspective (Entanglement studies in section 4) completes the picture very well.
My last remark connects to the introduction and conclusion sections. The 2d system discussed in the paper has one compact dimension (cylindrical geometry): in such cases, 2d computations are possible by converting the system into an MPS and applying 1 dimension methods, as the authors did. It would be nice to include some comments in the conclusion section about the prospects to performing such calculations in 2d without a compact dimension – what are the possibilities? What are the obstacles? What might have to be changed?
"Conventional" HEP-like lattice gauge theories include fermionic matter, and thus studying them with tensor networks requires the use of fermionic tensor network states (such as fermionic PEPS). Such studies have been carried out by the MPQ collaboration in the last few years, for 2 space dimensions and more (N. J. Phys. 18, 043008 ; Ann. Phys. 363, 385-439 ; Ann. Phys. 374, 84-137 ; Phys. Rev. D 97, 034510). The last one, in particular, suggests a numerical method, combining tensor networks with Monte-Carlo, independent of the space dimension and boundary conditions, which is sign-problem free. I believe that these works should be mentioned along with the other LGT-TN works mentioned in the introduction, as well as in the context of 2d LGT-TN calculations.
1. Not necessary, but potentially useful, as explained in the report – adding figures and equations to sections 2.3 and 2.4 to improve clarity.
2. A comment in the conclusions on general 2d systems and higher dimensions.
3. Referring to the previous works on LGT in 2d (and more) with fermionic PEPS.