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Hyperbolic Nodal Band Structures and Knot Invariants
by Marcus Stålhammar, Lukas Rødland, Gregory Arone, Jan Carl Budich, Emil J. Bergholtz
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|Emil Bergholtz · Lukas Rødland · Marcus Stålhammar
We extend the list of known band structure topologies to include 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, including the figure-eight knot and the Borromean rings, 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.
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- Cite as: Anonymous, Report on arXiv:1905.05858v1, delivered 2019-06-24, doi: 10.21468/SciPost.Report.1034
1. This manuscript provides a pedagogical introduction to knot theories is detailed in the context of nodal line semimetals.
2. Alongside the pedagogical introduction of the mathematical knot invariants, algorithms for the numerical calculation of the nodal line/knot and its invariants, are provided.
3. The topics is contemporary and the reported text provides a good entry point for researcher from a wide range of fields.
1. It is not clear while reading the paper: what are the novel aspects that are brought forward by the current work?
1. Can the authors elaborate more on the possible new physics that these structures might entail?
2. What would protect the different link invariants from having a small perturbation deform between the different knot topologies? What would be the physical signatures of such a transition?
3. There is not much physics discussed in the work, e.g., the physical implications and signatures of the new nodal structures as well as their robustness to disorder.
In this manuscript, a pedagogical introduction to knot theories is detailed in the context of nodal line semimetals. The two-band spectra of nodal line semimetals are parametrized in a general way for both Hermitian and non-Hermitian systems. This allows for various knot configurations of nodal lines and a general scheme for deriving tight-binding and continuum models that realize these knotted spectra is proposed. Alongside the pedagogical introduction of the mathematical knot invariants that help discriminating between the different knotted spectra, algorithms for the numerical calculation of the nodal line/knot and its invariants, are provided. These algorithms are demonstrated and compared on a number of examples, such as the figure-eight knot and Borromean rings.
Nodal line semimetals are one of the more recent members of the topological materials family. Topology of band structures remains a very interesting field of study with new realizations and implications showing up in a wide range of fields. In this context, providing a pedagogical and hands-on introduction to knot theory is very useful and the authors do a very good job about it. At the same time, I have some reservations before I can recommend publication:
1. It is not clear while reading the paper: what are the novel aspects that are brought forward by the current work? For a lay reader, it appears that all of the various parts are well-known in various fields. It would be, therefore, useful to have clear statements on what are imported methods from mathematics and computer science and what was newly developed for this work.
2. Whereas nodal line semimetals can be realized/found nowadays, from the current work I do not see the physical implications/importance of seeking out more complicated nodal link/knot structures. Can the authors elaborate more on the possible new physics that these structures might entail?
3. One of the main goals of analyzing the topology of different structures, is that there must be an obstruction in moving between the different topologies. This obstruction then manifests with some physical implication, e.g., a topological phase transition between different topological insulators. What would protect the different link invariants from having a small perturbation deform between the different knot topologies? What would be the physical signatures of such a transition?
4. The nontrivial topology of simple nodal line semimetals can be understood using a bulk winding invariant. Can the authors comments on the generalization of this characterization to the more complex spectral arrangements of nodal knots?
5. Correspondingly, in simple nodal line semimetals, topological boundary effects appear. What would be the expected boundary modes of the more complex knotted structures? On this note, I think that readers of this more mathematical-type work, would benefit from a short update on the physics state-of-the-art the is devoted to analyzing these effects, see e.g., Phys. Rev. Lett. 121, 166802 (2018) and references therein.
6. Last, the topology of structures is meaningful only when considering disorder. What would disorder do to the knotted nodal line structures? How robust are the presented algorithms to various disorder distributions?
Properly incorporate answers in the main text to the six point that are detailed in the report.
- Cite as: Anonymous, Report on arXiv:1905.05858v1, delivered 2019-06-15, doi: 10.21468/SciPost.Report.1021
1. This manuscript is clear and well-written, and is of pedagogical value particularly in Sect. 3, where mathematical results on the Milnor invariant are explained to a physics audience that is probably new to such concepts.
2. This work is self-contained in that the whole story, from model construction and approach to knot characterization, are all explained in sequential order.
1. This work is of limit novelty, since most of the results are not new at all. In Section 2, the microscopic models, or at least close variants of them, have already appeared in previous works ie. Refs 11 and 15 by most of the same authors. Section 3, while pedagogical, are mostly not new, being established mathematical results from knot theory.
2. Even the approximation approach on truncating j_max, which did not appear in any previous works by the same authors, is not new. For instance, an analogous truncation approach for realizing arbitrary knots appeared in
Bode, Benjamin, and Mark R. Dennis. "Constructing a polynomial whose nodal set is any prescribed knot or link." arXiv preprint arXiv:1612.06328 (2016).
As explained in the strengths and weakness remarks, this manuscript on the construction and characterizations of nodal knots is of limited novelty, even though it is well-written and of pedagogical value.
If I were to pick one aspect that is most interesting and original, it will be the determination of the Milnor invariant from the Conway polynomial.
1. The authors should consider also citing some other existing/contemporary theoretical and experimental works on physical knots:
Bode, Benjamin, and Mark R. Dennis. "Constructing a polynomial whose nodal set is any prescribed knot or link." arXiv preprint arXiv:1612.06328 (2016). - please also contrast the approach used in your manuscript with this work.
Sugic, Danica, and Mark R. Dennis. "Singular knot bundle in light." JOSA A 35, no. 12 (2018): 1987-1999.
Li, Linhu, Ching Hua Lee, and Jiangbin Gong. "Boundary states of 4D topological matter: Emergence and full 3D-imaging of nodal Seifert surfaces." arXiv preprint arXiv:1905.07069 (2019).
Larocque, Hugo, Danica Sugic, Dominic Mortimer, Alexander J. Taylor, Robert Fickler, Robert W. Boyd, Mark R. Dennis, and Ebrahim Karimi. "Reconstructing the topology of optical polarization knots." Nature Physics 14, no. 11 (2018): 1079.
Zhang, Yi. "Cyclotron orbit knot and tunable-field quantum Hall effect." arXiv preprint arXiv:1905.02192 (2019). - which gives knotted analogies to cyclotron orbits
2. In the note added pertaining to Ref 50, Ref 50 pertains to Hermitian and not non-Hermitian knots.
3. It will be desirable to provide a more extensive discussion on how the approach used here deviates/complements the approaches of Refs 6 and 22, beyond the level of the relationship between d_R and d_I.