Topological linear response of hyperbolic Chern insulators

We thank the referees for their thorough review and insightful comments. Below we address each point raised by Referee 2.

1. Clarification from Eq. (1) to Eq. (4):

We appreciate the referee's suggestion for clarity. To clarify, there is a distinction between the periodic cluster $G$, defined as a torsion-free normal subgroup, and the U(1) Peierls substitution, which is due to an external electric field. To go from Eq. (1) to (3), we first define a tight-binding model with nearest-neighbor hopping on lattice sites $G$. The Abelian U(1) Peierls substitution is then applied to this tight-binding Hamiltonian.

It is important to note that after the Peierls substitution, the translation group remains non-Abelian. We did not select only the Abelian part of the translation group. In the revised manuscript, we have added a sentence after Eq. (5) to emphasize this fact.

1. Connection Between Homology and Flux Threading:

There is indeed a connection between homology and flux threading. Homology characterizes the number of "holes" in the surface on which the periodic cluster resides. Flux threading through these holes can be understood by introducing a flat gauge field $A$. Although the gauge field is flat (i.e., its exterior derivative vanishes), it belongs to a non-trivial cohomological class (i.e., it is not the exterior derivative of something). We have added a brief discussion at the end of the second paragraph on page 5 to elaborate on the connection.

More formally, it is indeed homology rather than homotopy which is relevant for flux threading, since the flux $\varphi_\alpha$ through the noncontractible cycle $\mathfrak{g}\alpha\in\Gamma\text{PBC}\cong\pi_1(\Sigma_h)$ (here, $\Sigma_h$ denotes the genus-$h$ surface in Fig.~2 and $\pi_1$ its fundamental group) only depends on the abelianization $\Gamma_\text{PBC}/[\Gamma_\text{PBC},\Gamma_\text{PBC}]$, which is isomorphic to the first homology group $H_1(\Sigma_h,\mathbb{Z})$. Indeed, $\Lambda_j(\mathfrak{g}_\alpha)$ remains unchanged if $\mathfrak{g}_\alpha$ is multiplied by an element of the commutator subgroup $[\Gamma_\text{PBC},\Gamma_\text{PBC}]$. The flux is given by integrating the gauge field one-form $A$ (which belongs to the cohomology group $H^1(\Sigma_h,\mathbb{Z})$) over a closed homology cycle in $H_1(\Sigma_h,\mathbb{Z})$.

1. Elaboration on the Quantization of Chern Numbers:

The integer nature of $C^{(K)}_{ij}$ arises because it is the integral of the Berry curvature over a closed manifold. This is true also for non-Abelian states. To clarify, we did not selectively take the Abelian sector; rather, all states, including those in both Abelian and non-Abelian sectors, are included. The simplification Eq. (35) only occurs due to a relation between the Chern numbers. The orthogonality of the Abelian and non-Abelian sectors follows from the fact that states transforming in different irreps are necessarily orthogonal.

Smaller Comments:

In describing "hyperbolic matter as a novel form of synthetic matter," our intention is to refer to matter that exists on a hyperbolic lattice. For example, in this work, we are specifically discussing a Chern insulator living on a hyperbolic lattice.

Title Suggestion: We would like to respectfully point out that the Hall conductivity receives contributions from both Abelian and non-Abelian states. The significant simplification in Eq. (35) arises because those contributions are related to each other, not because we only take account of Abelian states. The existence of a relation between Abelian and non-Abelian Chern numbers is nontrivial and indeed one of the main results of our work. Therefore, we prefer not to specify "Abelian" in the title, as such an inclusion would result in a misleading representation of our main result.

Citation: We thank the referee for bringing our attention to Physical Review X 7 (4), 041069 (2017). We have added the suggested citation to the paper.

- Page 5, at the second of the second paragraph. Added "Mathematically, the periodic cluster resides on a surface with non-trivial homology, which quantifies the number of "holes" the surface possesses. Flat connections corresponding to fluxes threaded through these holes belong to non-trivial cohomological classes."

- Page 5, immediately below Eq. (5). Added "We note that even though the U(1) Peierls phase $e^{-i\phi_j}$ itself is Abelian, the translation symmetry of the Hamiltonian (5) is still noncommutative, described by the non-Abelian group $G$. To make those..."

- Added code repository as Ref. [82]

- Added Ref. [65], J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane and R.-J. Slager, Topological classification of crystalline insulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017)