Isotropic 3D topological phases with broken time reversal symmetry

We thank the Referee for the positive evaluation of our manuscript and the insightful criticism, we address these points below in our reply for the detailed report. We will submit a revised manuscript shortly, we refer to changes therein throughout our reply.

First, a general comment: The author’s proposal of a scalar magnetic order parameter is known in the literature as the magnetic toroidal monopole (see for example the review by Hayami, Symmetry 16, 926 (2024), or Hayashida, Advanced Materials 37, 2414876 (2025)). Toroidal multipoles have an extra minus sign under inversion compared to regular multipoles. The normal electric monopole is the charge (P-even, T-even), the magnetic monopole is P-odd, T-odd, the toroidal electric monopole is P-odd, T-even (it represents scalar chirality) while the toroidal magnetic monopole is P-even, T-odd, and it is the magnetic scalar the authors are after. In the discussion the authors mention the magnetoelectric response P. E cross B, which is exactly the response expected for a material with magnetic toroidal monopole order. I believe the manuscript would connect with a broader readership by reviewing this nomenclature since there are materials with this type of order (which the authors could also review). In relation to this, the authors could mention the space groups of the crystalline tight binding models and the magnetic space group of the model with complex hoppings.

We thank the Referee for pointing out these relevant pieces of literature. We were not aware of these results, some of which were published after the first version of our manuscript was completed. We have cited the suggested papers, and clarified the relation of our results to the concept of magnetic toroidal monopoles.

The most important physics question to me is this: The authors show that the amorphous model with $C_M = 2$ does show two Dirac points, which suggests a Z classification. However, transport calculations in this case show localization, which suggests that the classification is Z2. The authors then suggest that a local, symmetry preserving perturbation should be able to open a spectral gap too. Does this mean the authors did search for such a perturbation and did not find it? If such a perturbation were to exist, wouldn’t this be resolved by computing the invariant (6) in the doubled model? The sentence “We constructed a bulk Z invariant […] indicating the presence of a protected ungappable surface Dirac cone for odd values” is very confusing. If the mirror Chern number is Z, it protects any number of Dirac cones. Otherwise the invariant is not Z. Or there is something in the classification of amorphous systems which does not work in the same way as crystalline systems, but then it should not be called a “bulk Z invariant”. In my opinion, the authors should use any of their proposed methods to compute invariants to do so for the doubled model, and comment on these paradox.

We thank the Referee for pointing out this apparent inconsistency in our results. However, we believe that there is no paradox here. We did compute the bulk invariant for the doubled model, and found the value 2, we added a plot showing this. In general, the presence of a nontrivial bulk topological phase does not imply the presence of protected surface modes. For exaple it is known that inversion symmetry alone can protect bulk topological phases with a Z classification, but these phases have fully gapped surfaces (see e.g. Phys. Rev. B 83, 245132, 2011; Phys. Rev. B 85, 165120, 2012).

We also clarified that the bulk Z invariant uses a continuum approximation with exact rotation symmetry, and in the presence of zeros of the disorder-averaged Green's function (which may arise if the self-energy has poles) its values may change without a bulk delocalization transition. We further added an argument based on the Symmetric approximant formalism for statistical topological matter (arXiv:2601.00784) to support the statement that the $Z_2$ part of the invariant is topologically protected, this, however, does not decide the question whether even values of our invariants correspond to distinct topological phases.

We attempted to explicitely construct a symmetric surface perturbation that opens a surface gap, but (possibly due to the system size and energy resolution limitations of the numerical methos used to compute the surface spectra) we were unsuccessful. We believe that the localization observed in the network model is a strong evidence that the surface states are not critical for even values of the invariant. It remains an open question whether the gapless (but localizable) surface spectrum is a topologically protected feature, or whether the even values correspond to a bulk topological phase without gapless surface states. Definitively deciding these questions would involve significant effort beyond the scope of this manuscript.

We added a plot showing the invariants as a function of mu for the doubled model.

We clarified the conclusion about the topological invariants and the protection of the surface states.

About the order of presentation: The microscopic origin of the phase dependent hopping from virtual hopping through a cluster of magnetic atoms does not appear very important for the rest of the text, and makes the symmetry analysis and the arguments about spin splitting harder to follow. The four band crystalline model is described in the section “microscopic implementation” but it is not microscopic but effective. I think a self contained description of four band models (continuum, lattice, amorphous) with their symmetries would be most easily understood, with a separate section, possibly an appendix, for the microscopic proposal to generate the phase dependent hopping.

We thank the referee for suggesting a better structure for our manuscript.

We expanded the discussion of symmetry constraints in both the continuum and the tight-binding case to make the manuscript more self-contained.

We have restructured the manuscript to first present the symmetry analysis and construction of the continuum models, then of the amorphous tight-binding models, with a separate section dedicated to deriving the effective hopping through a magnetic cluster.

In Section 2.1.1, it is stated that the continuum model has time-reversal-like symmetry if restricted to order $k^2$. Can the authors explicitly write down what this symmetry is with an equation S H S = H? And how do the cubic terms break it down? I am unable to follow the discussion of this effective time-reversal through the text, and I believe many readers will too.

We have added a more detailed explanation of the effective time-reversal-like symmetry in 2.1.1.

Later on: “The hopping phase is distance dependent due to the different distance dependence of the microscopic hopping amplitudes from the px and py,z orbitals. This ensures that the hopping phase cannot be removed by a global basis-transformation introducing a relative phase between the s and p wavefunctions”. Can the authors write what distance dependence they mean? And how does this ensure the breaking of T?

We have added a more detailed description of this point, spelling out that the overlaps t have different distance-dependence for realistic atomic orbitals.

Similarly, in Appendix D, the authors write “The TRS-breaking terms of our model are next-next-nearest neighbor terms, which leads to linear TRS-breaking terms intrinsically cancelling out and only cubic terms remaining”. This was also unclear to me. Here linear and cubic means expanding in k? And for some reason the t3 terms do not lead to $k^3$ terms, but the t4 terms do? Again equations would help.

We have clarified this argument. As the referee correctly guesses, we mean a series expansion in k around k=0.

At the end of section 2.1.2, the authors claim to “provide a minimal spin model” that realizes the magnetic texture needed to make the desired complex hopping. But the authors do not write any equation at all. I think if the authors deem this important, they should write down the model explicitly and explain their claims with equations. Otherwise, they may consider removing this paragraph.

We agree with the referee that this paragraph is not essential, and we removed it.

We thank the Referee for the positive evaluation of our manuscript, and for their insightful criticism, we address these points below in our reply to the detailed report. We will resubmit an updated version of the manuscript shortly, we refer to clarifications therein throughout our reply.

1) The authors should clarify which features of the model are truly minimal. Is it sufficient to have distance-dependent complex hoppings as in Eq. (5), possibly with a simple constraint? In particular, can a more general criterion for the scalar TRS breaking nature of these hoppings be given, beyond its origin from the scalar quantity $(\nabla\times M) \cdot r$ in the microscopic model? The authors state: "The assumption of spatial isotropy constrains the dependence of Hhopping on the hopping vector, satisfied for example by Hhopping s-p found in the previous section.", so they seem to have a broader criterion in mind.

If on the other hand any complex hopping where the phases cannot be gauged away suffices, this should be clearly stated, ideally early on, before one has to understand the intricacies of the microscopic model.

We thank the Referee for pointing out that our presentation of the model construction was not sufficiently clear.

We added the specific symmetry constraint spatial isotropy imposes on the hopping terms in section 2.2.

We also emphasized that the nontrivial distance-dependence of the hopping phase is crucial, so the scalar TRS breaking cannot be gauged away, we added a remark about this both at the point where the continuum model is constructed, and also where the tight-binding model is introduced.

2) Related: The microscopic model is completely discarded when it comes to the actual results in Section 2.2. How important is the specific realization of these hoppings via chiral magnetic clusters?

The realization by hopping through magnetic clusters serves as proof-of-concept for hoppings that satisfy the scalar TRS breaking condition. We change the order of sections 2.2 and 2.3 in order to improve the logical flow of the manuscript.

3) Reproducibility of the perturbative calculation: The paper merely states: "We use second-order quasi-degenerate perturbation theory (assisted by the Python software package Pymablock [12]) to obtain the effective hopping tsp between the s and p1/2 orbitals." What steps must a reader follow to reproduce these hoppings?

Pymablock implements standard quasi-degenerate perturbation theory found in most quantum mechanics textbooks. In order to improve reproducibility, we included the formula that is evaluated in order to reproduce the result.

4) The statement "different distance dependence of the microscopic hopping amplitudes from the px and py,z orbitals." should be clarified in the diagrams. It should also be distinguished from the distance dependence due to hopping to next-nearest or to third-nearest-neighbors. (App D states "third-nearest-neighbor s–p hopping [Fig. 1(b)]." but this Figure does not indicate these hoppings.)

We thank the referee for pointing out this unclear phrasing. Figure 1(b) does include the third-nearest-neighbor (next-next-nearest-neighbor) hoppings through the body diagonal in green. We unified the terminology and only use "third neighbor" in the manuscript to avoid confusion.

5) The paper states: "and f is a complex function of the hopping distance." Do the authors mean a specific function (if so, which one?), or would any function do for this purpose?

We clarified that any complex function with non-constant phase suffices, while also stating the specific complex function that appears in this result in terms of the overlap integrals t.

6) "Rather than simulating an amorphous system with two families of atoms and two degrees of freedom per atom, for simplicity and without loss of generality we simulate one type of atom with four degrees of freedom." I understand the points of simplicity and that this does not lose generality. However, this further deemphasize the value of the earlier considerations of the microscopic model. At many points onwards, it is difficult to find out which model is then discussed, especially since detailed figure descriptions are in the Appendix, and this contains yet other model variants, such as in Appendix D. For instance: At the start of 2.2 the authors introduce yet another, effective model, Heff. Is this now obtained from the 4x4 models of Appendix A, the amorphous realization (5) of the microscopic model, ...?

We added further details about the construction of the effective Hamiltonian in 3.1. We clarified that $H_{eff}$ obtained from the minimal tight-binding model in 2.2 has the same properties as the continuum model considered in 2.1. In essence, this calculation serves an a posteriori check that the tight-binding model we constructed indeed implements the continuum model that we intended. It also forms the basis of the topological classification in later sections, where we apply topological invariants defined for continuum models to the numerically computed effective Hamiltonian of the amorphous system.

We thank the referee for pointing out the difficulty for the reader to follow which models we consider. We added several clarifying sentences throughout the manuscript.

Main clarifications about the results:

7) All figures: Beyond stating plot details are in App B, it would be useful to state at least from which model it is obtained, and what can be learned from these results.

We have updated the figure captions.

8) Bulk invariants. The abstract states that a bulk invariant will be "constructed". Is the invariant used in 2.2.1 a new one, or in [10,13] or elsewhere (is the reference at the end for the definition of a compactified mirror-invariant plane, or for the invariant itself? In the latter case more standard citation convention would be to put the reference just before the equation, and the word "constructed" would not apply).

The invariant presented here is new, and the references are to earlier works using similar ideas and formalism. We have clarified the phrasing and placement of citations in this section.

9) In the discussion the authors state: "We found results consistent with critical scaling, deviations from which are likely due to finite-size effects." but these points are not discussed in detail in the results section.

This part of the discussion is a summary of the surface transport results in 2.2.3. There we provide a detailed analysis using standard finite-size scaling techniques of the results from the transport calculations, which confirm that the surface remains critical in the case of an odd invariant (odd number of surface Dirac cones) and localizes in the case of even invariant.

• We rephrased the sentence to clarify the results and the distinction between the single and doubled cases.

Due to these issues, a more detailed assessment of the specific results is difficult, as the paper is too vague, beyond which model has been applied, on which of the ingredients then are then relevant for the physics. This is not helped by the brevity of the discussion.

Minor points.

10) In the abstract the authors call the model "fully" isotropic, even though it is only isotropic on average/in the ensemble. This distinction is important and is, in fact, one of the main interesting aspects of these systems.

We thank the Referee for pointing out our misleading use of terminology. What we wanted to emphasize here, is a distinction from many previous works on amorphous topological phases, where the structure was random, but the Hamiltonian itself was not isotropic. We rephrased the manuscript to avoid the term "full" isotropy, and added the average qualifier wherever necessary for clarity.

11) The beginning of subsection 2.1.2 states "Based on the symmetry-allowed terms of the continuum model (1), we now construct a microscopic model that preserves isotropy while breaking TRS.", but this subsection only proceeds up to the rock salt implementation, while the statistically isotropic model is only constructed in 2.1.3. The main purpose of 2.1.2 seems to be the construction of scalar TRS breaking hoppings.

We thank the referee for pointing this out, we rephrased the introductory sentence of 2.1.2 (now 2.3) accordingly.

12) The references appear inconsistent. General context example: "This topological phase is analogous to crystalline mirror- Chern insulators" - it would be pertinent then to cite papers describing these (see also point 8 above). Why is [16] cited for the kernel polynomial method? If it contains anything of specific relevance for this paper, it should be mentioned. In contrast, only in Appendix C we learn that some key ideas about the amorphous models used here have already been developed in the earlier work [34] involving one of the authors. To fully describe its significance, the paper would reach further, e.g., seek context with works such as PRX 13, 031016 (2023).

We are thankful for suggesting a relevant reference from the SPT literature, we added it to the introduction. We cite reference [16] because it was the first use of KPM for in the computation of topological invariants (mirror Chern number through the local Chern marker) in disordered systems. The amorphous model in [34] only shares the underlying amorphous structure, the tight-binding Hamiltonian in this work is different and novel. We clarified these points in the manuscript.