Intrinsic non-Hermitian topological phases

We thank the referee for the careful reading of our manuscript and for the constructive comments. Below we address each point in turn and describe the corresponding changes made in the revised version.

Referee’s comment:

Above Eq.(7), what does the phrase "for simplicity" mean? Does it mean that there are many cases out there beyond this work? From the equation uuuH=Hvvv, it is indeed an oversimplification that uuu and vvv are U1 phases.

Our reply: Thank you for the comment.

If the product u_k(gh)^\dag u_{hk}(g) u_k(h)^{phi_g} (and similarly for v_k) were not a U(1) phase, then the symmetry would not be described by the group G itself, but rather by an extended group \tilde G that is a nontrivial extension of G by another group as u_k(g) is not a projective representation of G. In this sense, allowing non-U(1) factors effectively amounts to enlarging the symmetry group.

Our assumption therefore means that the group G is already chosen so as to incorporate such possible extensions. To avoid confusion, we have clarified this point by revising the corresponding sentence in the manuscript and adding a footnote:

Previous: For simplicity, we assume that (6) imposes only trivial constraints.

Revised: We assume that the symmetry group $G$ is chosen appropriately such that (6) imposes only trivial constraints~\footnote{If the product $u_\bk(gh)^\dag u_{h\bk}(g) u_\bk(h)^{phi_g}$ (and similarly for v_\bk(g)) were not a U(1) phase, then the symmetry would not be described by the group $G$, but rather by a group $\tilde G$ that is a nontrivial extension of $G$ by another group as $u_\bk(g)$ is not a projective representation of $G$}.

Referee’s comment:

Regarding the definition of point gap, I am more accustomed to the definition that the complex spectrum of Hk diodes the complex plane into a few disconnected regions, and each of these regions is a point gap. This makes sense to me because in this case each "gap" can indeed shrink to a point or an arbitrarily small disk. The definition of point gaps in the paper is, in certain symmetry classes, subject to a change of the energy origin. Can the author comment on that?

Our reply: In the present work, a point gap is introduced in a way that is compatible with the basis change H_k -> U_1^\dagger H_k U_2, where U_1 and U_2 are independent unitary matrices. Under this transformation, the only reference energy that is universally compatible is the origin E_P = 0. Once v_k(g) is chosen to be either u_k(g) or -u_k(g), other choices of the reference point E_P may become possible, but they are subject to constraints imposed by the symmetry. For example, in the presence of PT symmetry satisfying H^ = H, the reference point must obey E_P^ = E_P in order to be compatible with the PT symmetry in the topological classification.

Referee’s comment:

I want to know how general the simplification in Eq.(20-21), this particular form of uk(g), would be? Are there important cases where this does not hold? In any of those cases, what results can we expect?

Our reply: While more general symmetry realizations can be considered at a mathematical level, we are not aware of concrete physical applications where such generalizations are essential. For this reason, the present work focuses on the simplest and physically relevant cases.

Referee’s comment:

Above Eq.(43), the quotient group is used for the topological classification. This clearly assumes a group structure among the topological classes given the same symmetry realization. In Hermitian physics, this is the case because the topological invariant z for H=H1 \oplus H2 is just z=z1+z2. Does the same apply to non-Hermitian cases, too?

Our reply: Even for Hermitian phases, the mathematical structure of a classification depends on the chosen framework, such as homotopy theory, vector bundles, or K-theory. In this work, we adopt the K-theory framework, where the classification forms an Abelian group. Physically, this corresponds to classifying relative differences between two Hamiltonians H0 and H1, a viewpoint that remains meaningful for point-gapped phases.

Referee’s comment:

Table 4 is the main result of the work. How do I physically interpret symmetries such as \eta {+-}, S? Are there some point-group symmetries of order 2? Why are these symmetries considered specially rather than discuss some general point-group symmetry?

Our reply: In this manuscript, we focus exclusively on internal symmetries, which do not act on spatial coordinates. Any spatial symmetry, including point-group symmetries, is therefore outside the scope of this work. The symmetry denoted by “S” corresponds physically to a genuine sublattice symmetry, which is approximately realized, for example, in graphene. The symmetry “\eta” represents pseudo-Hermiticity, which is realized, for instance, in the Dirac operator of lattice QCD.