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Topological linear response of hyperbolic Chern insulators
by Canon Sun, Anffany Chen, Tomáš Bzdušek, Joseph Maciejko
Submission summary
| Authors (as registered SciPost users): | Canon Sun |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202406_00052v1 (pdf) |
| Date submitted: | 2024-06-21 16:59 |
| Submitted by: | Sun, Canon |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to $-e^2C_{ij}/h$, where $C_{ij}$ is the first Chern number. Through a flux-threading argument, we provide an interpretation of the Chern number as a topological invariant in hyperbolic band theory. We demonstrate that, although it receives contributions from both Abelian and non-Abelian Bloch states, the Chern number can be calculated solely from Abelian states, resulting in a tremendous simplification of the topological band theory. Finally, we verify our results numerically by computing various Chern numbers in the hyperbolic Haldane model.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
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Strengths
- Novelty
- Thoroughness
- presentation
Weaknesses
- applicability to physical systems beyond synthetic ones
Report
In this work the authors generalize the Hall response to the hyperbolic lattice case. The paper reads well and deserves publication after appended comments/queries are addressed.
1.I am a bit confused about how the authors go from eq 1 to eq 3. I thought translations are generally defined as the torsion free subgroup. This has Abelian and non-Abelian sectors. Here only the Abelian case is important. I think this can be made clearer from Eq. 1 to Eq.4. I.e making this more self contained before coming to the specific Hamiltonian.
2. Given Fig. 2, is there a connection to be made with homogies as flux threading?
3. The authors say “where $C(K)_{ij}$ is the first Chern number associated with the irrep K and is quantized to integer values. I understand this from a general -Euclidean- point of view. But maybe the authors can elaborate. In the end we take the Abelian sector and are integrating over a closed manifold the eq 35 using a standard procure shows it a character but are all sectors [Abelian and non-Abelian] always orthogonal?
smaller comments;
- What do the authors mean with “Hyperbolic matter is a novel form of synthetic matter” as this is clearly a lattice not a form of matter directly?
-I would strongly recommend the Abelian part in an updated title as this is important compared to the Euclidean cases
-With respect to Ref 59-64 it would be fair to acknowledge that all quantum Hall [Class A] crystalline responses as mapped to K-theory were pointed out in Physical Review X 7 (4), 041069 (2017), see also relation to irreps.
Requested changes
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Recommendation
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The authors of the present work consider the problem of calculating the characterization of the bands in a hyperbolic lattice in terms of Chern numbers. This is a rather important task given the recent developments regarding hyperbolic crystallography. By applying the theory of representations of finite groups to the translational group on the hyperbolic lattice the authors show that the Chern number characterizing a set of bands separated by a gap is given by a band transforming under a trivial representation. In other words, the Chern number of a set of bands is completely determined by this topological invariant for the band in the trivial representation. This is the central result of the paper, which is presented in eq. 34. All other bands, even though they contribute to this topological invariant, have a contribution that is proportional to the corresponding trivial representation. The correspondence can be established, as the authors show, by a smooth mapping in the space of the irreducible representations of the hyperbolic translational group, which is a crucial step in the proof. Besides, the authors present a rather detailed account of the generalization of the usual Fourier transform in terms of the irreps of the translational group on a hyperbolic space and the generalization of the usual Kubo formula to the case of the hyperbolic lattices, where in general translations do not commute. Finally, the obtained form of the Chern number is applied to calculate the Hall conductivity in Haldane model on the hyperbolic {8,3} lattice, which is already discussed in Ref. 13 (Urwyler et al., Phys. Rev. Lett. 129, 246402 (2022)).
In my opinion, the paper presents a very important result that will spur further research on topological band theory in hyperbolic lattices. In particular, it will motivate further generalizations of the topological invariants in the tenfold periodic table to the hyperbolic matter and motivate experimental efforts to verify these theoretical predictions.
In light of this, I recommend the paper for publication in SciPost in its current form.
Recommendation
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