Topological linear response of hyperbolic Chern insulators

We thank the referee for the intriguing questions. In Ref. [1], the author considered electrons on a lattice under a uniform magnetic field. The translation operators have to be gauged because even though the magnetic field is translationally invariant, the vector potential is not. Indeed, in the presence of a nonzero flux per unit cell, it is impossible to choose a gauge that preserves the original translation symmetry.

On the other hand, in the case we studied, there is zero net flux per unit cell and thus we $are$ able to choose a translationally invariant gauge. As a result, the translation operators remain the same as those in the absence of gauge field. If we were to apply a uniform magnetic field, however, the translation operators would indeed have to be gauged, in a similar way to Ref. [1].

In fact, hyperbolic band theory (HBT) in the presence of a magnetic field has been studied in J.~Phys.: Condens. Matter 33, 485602 (2021). It would be interesting to study the Hall response of a hyperbolic lattice under uniform magnetic field [see also Phys. Rev. Lett. 128, 166402 (2022)] by combining HBT in a magnetic field and the methods developed here. However, this goes beyond the scope of the present paper where we focus on Chern insulators (i.e., models "without Landau levels" like Haldane's 1988 honeycomb lattice model).

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

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 $g_\alpha \in \Gamma_{\text{PBC}} \cong\pi_1(\Sigma_h)$ (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, $\Lambda_j(g_\alpha)$ remains unchanged if $g_\alpha$ 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.