## SciPost Submission Page

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

### by O. Cépas, P. M. Akhmetiev

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

### Submission summary

As Contributors: | Olivier CEPAS |

Arxiv Link: | https://arxiv.org/abs/1907.04701v2 |

Date accepted: | 2019-09-06 |

Date submitted: | 2019-08-26 |

Submitted by: | CEPAS, Olivier |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | 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.Published as SciPost Phys. 7, 032 (2019)

### Author comments upon resubmission

We would like to resubmit our manuscript to Scipost. We include all the corrections asked by the two referees. Regarding the questions (2), (3), (4) of the second referee, we have given brief and very partial answers as we feel they are interesting remarks but problems by themselves.

Sincerely yours,

the authors.

### List of changes

Following the first referee's comments.

1) typo corrected.

2) Page 4. We have added that "All loops are fully-packed.".

3) We have updated Figs. 1 and 2 to make them readable in black and white.

4) typo b-2kb-> b+2kb corrected.

5) ponctuation corrected.

6) comma suppressed.

7) double negation suppressed.

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Following the second referee's comments.

1) We have added a separate note on page 2, note [11], "The sites of the kagome lattice are the centers of the edges of the hexagonal lattice."

5) We have rewritten this sentence, page 19. Now it reads "This symmetry implies an equal number...".

6) Singular or plural "dynamics" is left open to proof correction or editor decision.

7) positions of the colons corrected.