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Hyperbolic Nodal Band Structures and Knot Invariants

by Marcus Stålhammar, Lukas Rødland, Gregory Arone, Jan Carl Budich, Emil J. Bergholtz

Submission summary

As Contributors: Emil Bergholtz · Marcus Stålhammar
Arxiv Link: (pdf)
Date accepted: 2019-07-17
Date submitted: 2019-07-05 02:00
Submitted by: Stålhammar, Marcus
Submitted to: SciPost Physics
Academic field: Physics
  • Algebraic Topology
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


We extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian realm where the knotted nodal lines are generic and thus stable towards any small perturbation. We show that these nodal structures, taking the forms of Turk's head knots, appear in both continuum- and lattice models with relatively short-ranged hopping that is within experimental reach. To determine the topology of the nodal structures, we devise an efficient algorithm for computing the Alexander polynomial, linking numbers and higher order Milnor invariants based on an approximate and well controlled parameterisation of the knot.

Ontology / Topics

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Band structure

Published as SciPost Phys. 7, 019 (2019)

Author comments upon resubmission

The authors are thankful for the obtained reports, contributing with constructive feedback on our first version of the manuscript. We have now adressed all the critique that was raised in the reports, and provided corresponding changes in the updated version of the manuscript.

List of changes

* Addition of a new subsection, Sec. 2.3, including further more exotic examples of hyperbolic nodal structures, Turk's head knots, and a discussion on transitions between knots with accompanying figures.
* Revised beginning of Sec. 3.
* Added more physical discussions in several sections.
* Treating two additional examples of Brunnian links in Sec. 3.4.
* Highlighting in a clearer manner what is new and how this work differs from and complements other works.
* Correcting various typos.
* Added several references.

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