# Invariants of winding-numbers and steric obstruction in dynamics of flux lines

### Submission summary

 As Contributors: Olivier CEPAS Arxiv Link: https://arxiv.org/abs/1907.04701v1 (pdf) Date submitted: 2019-07-11 02:00 Submitted by: CEPAS, Olivier Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We classify the sectors of configurations that result from the dynamics of 2d crossing flux lines, which are the simplest degrees of freedom of the 3-coloring lattice model. We show that the dynamical obstruction is the consequence of two effects: (i) conservation laws described by a set of invariants that are polynomials of the winding numbers of the loop configuration, (ii) steric obstruction that prevents paths between configurations, for lack of free space. We argue that the invariants fully classify the configurations in five, chiral and achiral, sectors and no further obstruction in the limit of low-winding numbers.

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

#### Published as SciPost Phys. 7, 032 (2019)

Resubmission 1907.04701v2 on 26 August 2019

Submission 1907.04701v1 on 11 July 2019

## Reports on this Submission

### Anonymous Report 2 on 2019-8-21 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1907.04701v1, delivered 2019-08-21, doi: 10.21468/SciPost.Report.1125

### Strengths

The paper carefully studies the important statistical mechanics/graph theory 3-coloring problem and solves it by introducing a new framework. The authors find how dynamics are hindered. These findings may broadly relate to "dynamical facilitation" and slow dynamics in constrained systems as well as a plethora of other problems. The calculations are very detailed and well explained.

### Weaknesses

There are no major weaknesses that I can point out to.

### Report

I recommend the publication of this work. The new framework introduced by the authors is quite illuminating. The below comments are not, at all, conditions for acceptance but rather questions/suggestions that the authors may heed if they wish to. The current paper is already very comprehensive and well written.

(1) As noted by the authors, the model relates to kagome systems with constraints. It might be useful to further explicitly and very briefly note that the center of the links of the honeycomb lattice studied here form the sites of a kagome lattice. The 3-coloring problem as it relates to links on the honeycomb lattice maps becomes the 3-coloring problem of the vertices of the kagome lattice.

(2) There have been other frameworks for studying the (3-) coloring problems on the links of general graphs (including lattices such as the kagome). One may ascribe a (cubic) roots of unity to each link or vertex. For the 3-coloring problem on the kagome lattice, this relates to configurations in which cubic roots of unity are associated with individual lattice site such that for two neighboring lattice sites $x \neq y$ (or, equivalently (given that $x^3=y^3=1$), these satisfy $x^2+ xy+y^2 =0$). The allowed 3-colorings can be related to polynomials in the Grobner basis.

(3) How might the results qualitatively change if the lattice was not on a torus but rather on a manifold with genus number $g$? Would the sector multiplicities be changed by such a change of topology?

(4) Is there is an intuitive rationale for the observed uniform fraction scaling for the winding number $n_{K}$ such as that of $0.004L^{4}$ found by the authors for $L \gg 1$?

Very more minor items of style:

(5) The sentence "Note that this symmetry is true in terms of number of homotopy classes (Fig. 6), but also in terms of number of color configurations (table 1)" did not read very well at this end.

(6) The word "dynamics" is plural. In the paper, the authors very consistently used it in singular form. One may change that to plural throughout the text.

(7) A colon may appear immediately after a word instead of a space (e.g., "First example: permutation of colors" instead of "First example : permutation of colors" and other similar subtitles and text).

I must apologize for these rather pedantic stylistic remarks.

As noted above, the authors did a very good job of trying to be extremely detailed yet clear. This is an excellent work.

• validity: top
• significance: high
• originality: high
• clarity: top
• formatting: excellent
• grammar: excellent

### Author:  Olivier CEPAS  on 2019-08-23  [id 584]

(in reply to Report 2 on 2019-08-21)

We would like to thank the referee warmly for his/her report and encouragements. We provide some very partial answers to the interesting questions raised.
(1) We have included this precision in a separate note to keep this discussion brief.
(2) This is an interesting point of view which may be useful for further developments or simplifications.
(3) The question is difficult and we do not know an immediate generalization of the invariants for g>1, at present. Such a lattice could be defined by using k-gons with k>6 and the winding number vectors (a,b,c) generalized to vectors with 2g coordinates. While axb and a.b are in vectorial form, suggesting they could still be invariants, it is not clear how the third invariant could be generalized. It would be, of course, particularly interesting if the number of invariants depends on g.
(4) We have not investigated this point further because some of the sectors that enter this number, 0.004L⁴, may not be physical, i.e. may not correspond to acceptable homotopy classes of the original coloring problem, as mentioned in the text. We do not know in particular if there remains a finite fraction of L⁴ or not, when the unphysical part is removed. A way perhaps to understand this point further would be to determine which sectors are physical and which are unphysical.
(5) is corrected.
(6) Thank you for raising this point too, especially as we are confused by scientific usage and opposite views from different native speakers. As both ways are fine with us, we leave this point with the editor.
(7) is corrected.

### Anonymous Report 1 on 2019-8-15 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1907.04701v1, delivered 2019-08-15, doi: 10.21468/SciPost.Report.1114

### Report

In this manuscript, the authors study the connectivity between configurations of the 3-coloring model in terms of winding loop updates. By carefully understanding the corresponding winding number transformations, they are able to classify the sectors of configurations and identify the relevant invariants and conservations laws. They further uncover an interesting steric obstruction that prevents finite size systems from being ergodic due to the absence of paths between configurations. This work provides a significant advance in our understanding of the dynamical obstruction observed in this iconic system, which is of interest in many different contexts, from combinatorics and glassiness, to magnetic systems and superconducting arrays. The manuscript is very well organised and clearly written, and I am happy to recommend it for publication in SciPost.

### Requested changes

1) p.1: "is therefore or particular interest" -- typo

2) p.4: "form self-avoiding closed loops" -- may it be worth to point out to the unfamiliar reader that the loops are fully packed?

3) Fig.1 and 2 are not legible in black and white printing, and this could perhaps be easily fixed to spare the need to print in colour

4) Eq.(25) contains a typo; the middle term on the right hand side should be b+2kb rather than b-2kb, I think

5) many equations are missing a punctuation at the end -- starting from the equation between Eq.(27) and (28), which should be followed by a full stop

6) p.10: "While in the continuum limit, we would" -- no comma?

7) p.15: "...cannot be... neither...nor" is a double negative, which I presume is not intended to be

• validity: high
• significance: high
• originality: top
• clarity: high
• formatting: perfect
• grammar: excellent

### Author:  Olivier CEPAS  on 2019-08-23  [id 583]

(in reply to Report 1 on 2019-08-15)

We would like to thank the referee warmly for his/her report and very careful reading. We have included all the well-needed requested changes, points 1 to 7.