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ThreeDimensional Quantum Hall Effect in Topological Amorphous Metals
by JiongHao Wang, Yong Xu
Submission summary
Authors (as registered SciPost users):  JiongHao Wang 
Submission information  

Preprint Link:  https://arxiv.org/abs/2309.05990v2 (pdf) 
Date submitted:  20231107 04:10 
Submitted by:  Wang, JiongHao 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Weyl semimetals have been theoretically predicted to become topological metals with anomalous Hall conductivity in amorphous systems. However, measuring the anomalous Hall conductivity in realistic materials, particularly those with multiple pairs of Weyl points, is a significant challenge. If a system respects timereversal symmetry, then the anomalous Hall conductivity even vanishes. As such, it remains an open question how to probe the Weyl band like topology in amorphous materials. Here, we theoretically demonstrate that, under magnetic fields, a topological metal slab in amorphous systems exhibits threedimensional quantum Hall effect, even in timereversal invariant systems, thereby providing a feasible approach to exploring Weyl band like topology in amorphous materials. We unveil the topological origin of the quantized Hall conductance by calculating the Bott index. The index is carried by broadened Landau levels with bulk states spatially localized except at critical transition energies. The topological property also results in edge states localized at distinct hinges on two opposite surfaces.
Current status:
Reports on this Submission
Strengths
 potentially novel way of detecting topology in amorphous topological metals
 clearly written manuscript
Weaknesses
 the more interesting case of topological semimetal phase protected by the time reversal symmetry is less explored
 it is not clear whether the results are generic to all timereversal symmetry protected topological phases
Report
The authors show that the amorphous topological metallic systems may exhibit a 3D quantum Hall effect in presence of the magnetic field. Since this phenomenon can be observed experimentally, the work offers a way to detect/confirm topological features in amorphous metallic systems that do not admit momentumspace topological invariants. This is particularly relevant for Weyl semimetals protected by the timereversal symmetry for which most of the realspace approaches to calculating the topological invariant fail, with the exception of the spectral localizer (see SchulzBaldes & Stoiber EPL 136 27001 (2021) and J. Math. Phys. 64, 081901 (2023); Cerjan & Loring PRB 106 064109 (2022); Dixon et al. PRL 131 213801 (2023); Franca & Grushin arXiv: 2306.17117).
I find the manuscript clearly written with interesting results. Provided the authors answer my comments, I would be happy to recommend this work for publication in Scipost Physics Core.
Requested changes
1. I do not agree with the authors that ARPES cannot be used to probe amorphous systems as it was in fact used in their Ref. 70 to demonstrate experimentally the existence of a topological phase in amorphous Bi2Se3. This is because in amorphous systems, we can still have well defined plane waves of momentum k describing the outgoing electron. The ARPES measures the overlap of these plane waves with the eigenstates of the Hamiltonian, implying that sharp spectral features in ARPES of amorphous systems indicate the presence of states.
2. In the last line of page 3, the authors set a value for parameter $A$ that I could not find previously defined in the text.
Since their Ref. 8 uses a parameter A in the Hamiltonian, and they use $\gamma$ I wonder shouldn't $A$ be replaced with $\gamma$? This should also explain why they never set the value of parameter $\gamma$ in the manuscript.
3. In Figs. 2(c) and (d), it is very unusual to see that the bulk states remain so visible even after averaging over 100 configurations. Could the authors explain this?
4. Concerning section 5 that focuses on the timereversal symmetry protected Weyl semimetals, it is not clear to me whether the results the authors obtain are specific to this model or are generic to this class of systems. It would significantly add to the value of the manuscript if the authors could provide a discussion on this.
5. In Fig. 6(b), we see that the Hall conductances for crystalline and amorphous systems have a very similar dependence on $E_F$, in comparison with the timereversal broken case. What would be the reason for this? In addition, I find that having a calculation similar to Fig. 3(a) would enrich the manuscript.
6. Could the authors calculate the Bott index for Hamiltonian Eq. (6) in presence of magnetic field? It would be a great to show the bulkboundary correspondence works for this case.
7. Finally, I wonder whether about the interplay between disorder strength and the magnetic field. In the presence of disorder, does the nonzero Hall conductance appear for any magnetic field strength?
Strengths
1  Novel findings, with an approach motivated by experimental relevance
2  Multiple methods applied to verify and explain results
3  Generally clear writing and presentation
Weaknesses
1 The cases with and without timereversal symmetry could be compared and contrasted more; results are often obtained only for one or the other with no explanation for the choice.
2  Potentially relevant aspects regarding parameters, in particular the magnitude of the magnetic field, left vague or unexplored.
Report
The main finding of the work appears to be that the presence of magnetic fields can allow for a detection of Weyl bands in amorphous materials even when timereversal invariance (in the absence of a magnetic field) causes the conductance from those bands to ordinarily cancel out. With recent experiments finding evidence for a 3D quantum Hall effects in crystalline system, and the challenges with measuring amorphous anomalous Hall conductance mentioned in the manuscript's introduction, the findings are quite relevant for current research in the field. The authors find that topology (as calculated by Bott index) and observables (Hall conductivity) match well, and explain the welldefined quantization by considering the localization length of bulk wavefunctions.
Two Hamiltonians are considered, first without and later with timereversal symmetry. The latter seems potentially more significant as a finding, as the case with broken TRS, although without a magnetic field, was already considered in Ref. [54].
However, many of the results, e.g. the localization lengths and scaling, as well as the tilted magnetic field in the appendix, are only obtained for the TRbreaking case. Even if I do find it plausible the results would not be too dissimilar, an explicit discussion of this is absolutely warranted.
As for the TRbreaking case, given that [54] found a quantized Hall conductance with no external magnetic field, the claim that parameters are chosen "with no loss of generality" is not obvious to me. Intuitively, while the Hamiltonian is different, could not a large enough TRbreaking term even here yield the observed results regardless of any external magnetic field? An elaboration on the effects of the magnitude of B on the observed results would do much to clarify this. This would also add scientific novelty to this part (in taking more of distance from [54]).
There are also some other details that could, in my opinion, improve the manuscript:
 It is specified that the sample used be thin along y. Whether this was a choice made for computational expedience, or something that is significant to the results, could be mentioned.
 The LDOS figures show high values in certain locations in the bulk of the system. While this would be expected for individual realizations, it is not immediately obvious why this feature is seen in a sampleaveraged quantity, where fluctuations due to random sites should be equally distributed over the bulk. Is it a finitesize effect due to the location of the leads? This could also be mentioned in the text.
 Given the current status of experimental research into the 3D quantum Hall effect, it could be of interest to consider whether there are any known materials that may be used for studying the case presented here, and if so how relevant parameters in those compare to the model Hamiltonians here.
I do hold that there may be results here of sufficient novelty and relevance to the scientific community to warrant publication, especially with regards to the TRinvariant Hamiltonian. However, the comparison between the separate parts of the work and between this and previous works  chiefly [54]  should be elaborated on in the text. In its current format these aspects are left much too vague.
Requested changes
1  Add a discussion of to what extent results apply to case with / without timereversal symmetry
2  Elaborate on the effects of the magnitude of B on the results.