Chern Vector Protected Three-dimensional Quantized Hall Effect

Hailong Li; Hongfang Liu; Hua Jiang; Shu-guang Cheng; X. C. Xie; Zhi-Qiang Zhang

REVIEW 2 major objections 5 minor 38 references

Chern Vector Protected Three-dimensional Quantized Hall Effect

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For a three-dimensional system with Chern vector $\mathbf{C}=(0,m,n)$, the sum of the two-terminal conductances is shown to be topologically protected, and in Hall-bar geometry it becomes the quantized Hall conductances…

desk verdict A plausible and genuinely new transport fingerprint for 3D Chern vectors, but the topological protection of the central sum rule is asserted rather than proved in the supplementary. read the letter →

arxiv 2501.14493 v1 pith:OWQ4RQVP submitted 2025-01-24 cond-mat.mes-hall

Zhi-Qiang Zhang , Shu-Guang Cheng , Hongfang Liu , Hailong Li , Hua Jiang , X. C. Xie This is my paper · ORCID

classification cond-mat.mes-hall

keywords Chernvectorthree-dimensionalquantizedHalleffectchiralsurfacestatesLandauer-Büttikerformalismconductance3Dinsulatortopologicallyprotectedtransportsample-sizedependence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a new kind of quantized Hall effect for three-dimensional systems whose topological character is described by a Chern vector $\mathbf{C}=(0,m,n)$ rather than a single Chern number. In such a system, chiral surface states on two different pairs of faces ($xy$- and $xz$-boundaries) can scatter into each other, so the individual two-terminal conductances are no longer protected. The paper's central claim is that the sum of the two conductances, $G_x^{(xz)}+G_y^{(xy)}$, is locked to the topological integers and sample dimensions: $(mL_y+nL_z)e^{2}/h$. In a multi-terminal Hall bar this sum appears directly as quantized Hall conductances $G_{xy}=-G_{yx}=G_{xz}=-G_{zx}=(mL_y+nL_z)e^{2}/h$. A reader should care because this predicts a measurable electronic response whose value depends on two sample dimensions at once, a distinctive fingerprint of three-dimensional topology, and it identifies what the Chern vector protects even when individual edge modes are scattered.

What carries the argument

The central object is the Chern vector $\mathbf{C}=(C_{yz},C_{xz},C_{xy})$, the vector of Chern numbers computed on the three coordinate planes of the three-dimensional Brillouin zone. The argument is carried by a two-block tight-binding model whose first block is a three-dimensional Chern insulator with $\mathbf{C}=(0,0,1)$ (red chiral modes on $xy$ faces) and whose second block has $\mathbf{C}=(0,1,0)$ (blue modes on $xz$ faces), coupled by the hopping $T_c$; the composite has $\mathbf{C}=(0,1,1)$, generalizing to $\mathbf{C}=(0,m,n)$. The load-bearing identity is the sum rule $G_x^{(xz)}+G_y^{(xy)}=C_{xz}L_y+C_{xy}L_z$ (Eq. 2): every backscattering event that removes a red channel from the $y$-direction conductance is argued to reappear one-for-one as an added transmission channel in the $x$-direction conductance, through the chiral network connecting front, back, and bottom surfaces. This preserved sum is what the multi-terminal Landauer–Büttiker calculation converts into the quantized Hall conductances of Eqs. (5) and (6).

What would settle it

A numerical or experimental test that would settle the claim: compute or measure $G_x^{(xz)}+G_y^{(xy)}$ as a function of energy across the whole gap, including the 'white' energy interval flagged in the paper's footnote 38; if the sum drops below $(mL_y+nL_z)e^{2}/h$ there, the neglect of trivial surface states is not benign. Separately, extracting the junction scattering matrix at the corners where the chiral networks meet would show whether the one-for-one channel matching asserted in Eq. (S8) is exact or approximate; any finite reflection at the junctions would make the protected sum drift with the coupling strength $t_c$.

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Extended reading notes

Core claim

For a clean or disordered three-dimensional insulator with Chern vector $\mathbf{C}=(0,m,n)$, the paper establishes that although the two-terminal conductances along the $y$ and $x$ directions are individually altered by inter-block coupling (which backscatters chiral modes between opposite surfaces), their sum is topologically protected. The protected quantity is $G_x^{(xz)}+G_y^{(xy)}=(C_{xz}L_y+C_{xy}L_z)e^{2}/h=(mL_y+nL_z)e^{2}/h$, verified numerically over a finite energy window and against disorder. In six-terminal Hall-bar geometries, the Landauer–Büttiker equations turn this protected sum into four quantized Hall conductances $G_{xy}=G_{xz}=-G_{yx}=-G_{zx}=(mL_y+nL_z)e^{2}/h$. Because the value depends on both $L_y$ and $L_z$, the effect is genuinely three-dimensional and serves as an experimental fingerprint distinguishing it from the layer-stacking case $\mathbf{C}=(0,0,n)$, whose Hall conductance depends on only one dimension.

Load-bearing premise

The protection of the sum rests on the claim that every chiral channel scattered away from the y-direction conductance reappears without loss as x-direction transmission, a perfect one-for-one match of the red and blue chiral channels in the limit of many sites, and on the assumption that the small energy interval containing trivial surface states can simply be neglected (footnote 38 of the paper).

Editorial extensions

If this is right

Hall-bar measurements on a three-dimensional sample with $\mathbf{C}=(0,m,n)$ should show quantized Hall conductances $G_{xy}=-G_{yx}=(mL_y+nL_z)e^{2}/h$, stable against Anderson disorder for $W$ up to several times the hopping.
The double dependence on the two sample dimensions $L_y$ and $L_z$ is a distinctive fingerprint: changing either dimension changes the quantized value by an integer multiple of $e^{2}/h$, unlike the $\mathbf{C}=(0,0,n)$ case where only $L_z$ matters.
Reported three-dimensional quantum Hall plateaus with $G_{xy}=\kappa L_z$ do not force the interpretation $\mathbf{C}=(0,0,\kappa)$; the same plateau could come from a Chern vector $(0,m,n)$ with $\kappa=(mL_y+nL_z)/L_z$, so sample-geometry checks are needed to identify the topology.
When a three-dimensional Chern insulator is coupled to metallic bands in a magnetic field, finite-size energy gaps can replace bulk band gaps, so the effective coefficient $m$ can take fractional values in $[1,2]$ while the quantized Hall response $\pm(mL_y+nL_z)e^{2}/h$ is preserved.
In photonic or acoustic crystals, where Hall voltages are not directly measurable, the same quantization can be observed indirectly through the specific backscattering and transmission trajectories of the chiral surface states that the sum rule relies on.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural reading is that the Chern vector protects a channel-count conservation law rather than individual channels: the total signed number of current-carrying chiral modes crossing the sample is invariant under surface–surface coupling even though individual surface conductances are not. This suggests a search for an operator statement, a global index or trace identity, that would turn Eq. (2)
The double-size dependence $(mL_y+nL_z)$ hints that the protected quantity is a bulk response with a boundary interpretation, effectively a 3D Hall charge set by the cross-sectional perimeter; measuring its dependence on both dimensions could map the full Chern vector $(C_{yz},C_{xz},C_{xy})$ of a material rather than a single component.
A testable extension: engineer junctions with different corner geometries in an acoustic or photonic realization and track wavepacket transmission through the chiral network; the sum rule predicts the total network transmission is geometry-insensitive while individual branches are not, which would isolate the paper's one-for-one channel-matching assumption experimentally.
The finite-size-gap regime where $m$ becomes fractional connects the proposal to thin-film quantum Hall physics: the Chern vector components may be measurable as effective fillings once sample thickness enters as a tunable parameter, making the proposed Hall bars a practical tool for extracting $(m,n)$ from transport data.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proposes that a 3D Chern insulator with Chern vector (0,m,n) has a topologically protected sum of two-terminal conductances, G_x^(xz)+G_y^(xy) = (m L_y + n L_z) e^2/h, and that this shows up as quantized Hall conductances with double sample-size dependence. That is a genuinely new statement; the cited prior work mostly discussed surface-state geometry or single-component stacks. The tight-binding and NEGF numerics are careful: several sample sizes, couplings, disorder strengths, and the plateau at the predicted value is convincing in the chosen energy window. The Hall-bar Landauer-Büttiker treatment is also consistent and well documented.

Soft spots: the analytical proof in SM.IV rests on an asserted perfect one-to-one matching between blue and red chiral channels in the N->infinity limit. The bulk Chern numbers being invariant under t_c does not fix the surface S-matrix at the junctions; if any reflection occurs, the sum rule would acquire corrections of order R. The authors state the matching rather than derive it from the Hamiltonian. Footnote [38] also concedes a 'white energy interval' with trivial surface states that is 'roughly neglected' because it is tiny. A topological protection claim cannot rest on 'roughly' and 'tiny' without a quantitative bound. These gaps are real but addressable. The numerics are finite-size and parameter-specific; they do not prove exactness.

The disorder robustness and the careful treatment of which transmission matrix elements matter strengthen the case. The experimental candidates are plausible, though the metallic-band construction uses finite-size gaps and a fractional effective m, which is another soft spot.

Overall, this is a worthwhile proposal with a plausible but not fully established central claim. It deserves peer review, because the question is important and the numerics are reproducible in principle. A referee should ask for either a rigorous derivation of the channel-matching step or a clear statement of the conditions under which it holds, plus a quantitative bound on the neglected white interval. I would not cite it as a settled result yet, but I would watch for the revised version.

Referee Report

2 major / 5 minor

Summary. The paper proposes that three-dimensional systems with Chern vector C=(0,m,n) exhibit a topologically protected quantized Hall effect. The central claim is that the sum of two-terminal conductances along the x and y directions, G_x^(xz)+G_y^(xy), equals (m L_y + n L_z)e^2/h, and that this sum can be measured in a Hall-bar geometry as quantized Hall conductances G_xy=-G_yx=(m L_y + n L_z)e^2/h. The authors support this with a tight-binding model consisting of two coupled 3D Chern insulators with Chern vectors (0,0,1) and (0,1,0), and with non-equilibrium Green's function transport calculations showing quantization of the sum in specific energy intervals and under disorder. They also propose photonic/acoustic and condensed-matter candidates for experimental realization, including a model with metallic bands where finite-size effects are argued to yield effective Chern vectors.

Significance. If the central claim holds, the paper establishes a new transport fingerprint of the Chern vector: a quantized Hall response that depends on two sample dimensions, L_y and L_z, rather than a single dimension as in the conventional 3D quantum Hall effect. This would be a genuinely novel extension of the quantum Hall paradigm and would connect recent experimental realizations of Chern vectors in photonic and acoustic crystals to transport observables. The paper's strengths include explicit tight-binding and NEGF calculations, numerical validation over a range of disorder strengths and inter-block couplings, and falsifiable predictions (the double sample-size dependence and the specific Hall-bar conductances). The numerical data in Figs. 2 and 3 and in the Supplementary Materials are reproducible from the described model and support the formula in selected energy windows. However, the analytical argument for topological protection is incomplete, as detailed below.

major comments (2)

[SM.IV, Eq. (S8)] The proof that G_x^(xz)+G_y^(xy) is quantized rests on the assertion that the blue chiral channels on the bottom surface 'perfectly match' the red chiral channels on the front and back surfaces in the N→∞ limit, and that the deficit in G_y^(xy) reappears one-for-one as an increment in G_x^(xz). This perfect channel-matching step is stated rather than derived from the Hamiltonian or from the TKNN invariants. The bulk Chern numbers computed in SM.I remain invariant under t_c, but that invariance fixes only the net number of chiral channels in each plane; it does not determine the S-matrix connecting the surface networks at the junctions. If the junction has a reflection probability R>0, a scattered channel contributes only (1-R) to G_x^(xz), so the sum would acquire a correction of order R. The numerical plates in Fig. 2 show quantization for specific finite systems (L=10, t_c=0.1t) and do not rule out such corrections. A rigorous derivation of the matching condition, or a quantitative bound on the corrections in the thermodynamic limit, is required to support the topological protection claim.
[Main text, footnote [38]] The quantization window is restricted by excluding a 'white energy interval' that the authors concede contains trivial surface states along the y direction. The footnote states that this interval is 'rather tiny' and can be 'roughly neglected.' For a statement of topological protection, a size-based neglect without a quantitative bound is insufficient: a topologically protected quantity should either be quantized everywhere in the gap or the interval should be proven to vanish in the thermodynamic limit. As it stands, Eq. (2) holds in an unspecified subset of the spectrum, and the manuscript does not provide a criterion for identifying that subset in a general sample. This should be addressed by giving a precise definition of the protected energy window and a rigorous argument for the irrelevance of the trivial-surface-state interval.

minor comments (5)

[Abstract and main text] There are several typographical errors, including 'propse' in the abstract, 'copuling' after Eq. (1), and 'Noevertheless' in footnote [38]; these should be corrected.
[Fig. 2(g) and (h)] The 'pink energy interval' in which G_x^(xz)+G_y^(xy) is claimed to be quantized is not precisely defined in the figure or the text; please specify the energy range and show clearly which parts of the curves lie in the protected interval.
[Candidate II, near Eq. (7)] The statement that the Chern vector coefficient m becomes fractional (m∈[1,2]) in the finite-size gaps is confusing, since the Chern vector is defined via integer TKNN invariants; please clarify whether the quantization in that energy window is a finite-size effect rather than a topological invariant, and whether the formula (m L_y + n L_z) with integer m,n still applies.
[General] The numerical evidence would be strengthened by a scaling analysis showing that the deviation of G_x^(xz)+G_y^(xy) from its quantized value decreases with increasing system size L; currently the quantization is demonstrated only at L=10.
[SM.V, Eqs. (S10) and (S13)] The derivation of the Hall conductances assumes that certain transmission coefficients are exactly zero; while the numerical plots show they are small, the degree of approximation should be quantified, for example by giving typical values of the neglected terms across the claimed quantization window.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central (mLy+nLz) conductance rule is fixed by explicit Chern-vector inputs and is independently tested numerically; the SM.IV channel-match argument and footnote [38] are rigor gaps rather than circular reductions.

full rationale

The central prediction, Eq. (2), is not obtained by fitting a parameter to the target conductance. The model in Eq. (1) is explicitly constructed from two blocks with layer Chern numbers Cxy=1 and Cxz=1, so the claimed response (mLy+nLz) uses only the input topological integers and sample dimensions. The numerical data in Figs. 2(g)-(h), 3, and S7 test the sum rule and Hall conductances across Fermi energy, disorder, coupling tc, and sample size without adjusting any free parameter to match the quantized value; this is independent evidence rather than a renamed fit. The author self-citations ([19], [20], [25]) are background references on 3D quantum Hall physics and are not load-bearing for the derivation of Eq. (2). The derivation in SM.IV is heuristic: the statement that 'the N→∞ blue chiral channels on the bottom surface perfectly match with the (Lz e^2/h - G_y^(xy))h/e^2 red chiral channels' is asserted rather than derived from the TKNN invariants, so the analytic proof has a missing justification. However, this is a rigor gap, not a circular reduction, because the premise is a microscopic channel-matching claim and not simply the conclusion restated, and the conclusion is independently verified numerically. Footnote [38] explicitly concedes that the 'white energy interval' contains trivial surface states and is 'roughly neglected'; this limits the energy window of the claimed protection but again is not a circularity. Overall, no equation reduces to its own input by construction, so the paper does not exhibit significant circularity; the low score reflects only the minor non-load-bearing self-citations and the admitted analytical gaps in SM.IV and footnote [38].

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard topological and transport background plus one ad hoc channel-matching assumption. No new physical entities are introduced. The main free inputs are model Hamiltonian parameters chosen by hand to realize Chern vector C=(0,1,1) or C=(0,1,2); none are fitted to the target conductances.

free parameters (3)

Hopping amplitudes A, M, t1, t2 = A=t, M=2t, t1=-0.3t, t2=-0.1t
Set by hand to open a bulk gap and give Chern vector C=(0,1,1); no experimental fit is performed.
Inter-block coupling tc = 0.1t, with scans up to 0.2t
Chosen to couple the two Chern-insulator blocks; the quantization claim is tested for these values.
Candidate-II metallic-band parameters and flux = tx=ty=t, tz=0.1t, epsilon0=-2t, phi=0.25pi
Chosen so that metallic bands plus the 3D Chern insulator produce Chern vectors (0,1,1) and (0,1,2); not fitted to data.

assumptions (4)

domain assumption TKNN relation between Chern numbers and Hall conductances, with G_alpha_beta=C_alpha_beta L_gamma e^2/h for decoupled single-Chern-component blocks
Used to assign G_y^(xy)=Cxy Lz and G_x^(xz)=Cxz Ly in the tc=0 limit and to motivate the protected combination.
domain assumption Bulk-boundary correspondence for 3D Chern vectors: nonzero Cxy and Cxz produce chiral surface modes on the x-y and x-z boundaries respectively
The whole transport picture depends on the existence and chirality of these surface modes.
domain assumption Landauer-Buttiker scattering formalism with ideal metallic leads, and neglibibility of omitted transmission coefficients in the Hall-bar equations
The six-terminal results in SM.V set several transmission matrix elements to zero based on smallness in the large-sample limit.
ad hoc to paper Perfect channel matching: the deficit in G_y^(xy) reappears exactly as an increment in G_x^(xz) because blue channels on the bottom surface perfectly match red channels on the front and back surfaces in the N to infinity limit
This is the load-bearing analytical step in SM.IV, Eq. (S8); it is asserted rather than derived from the Hamiltonian and underlies the central sum rule.

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Pith. "Pith review of Chern Vector Protected Three-dimensional Quantized Hall Effect." pith.science (2026). https://pith.science/paper/OWQ4RQVP

@misc{pith2026250114493,
  author       = {Pith},
  title        = {Pith review of: Chern Vector Protected Three-dimensional Quantized Hall Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OWQ4RQVP}},
  note         = {Machine review of arXiv:2501.14493}
}

abstract

Recently, Chern vector with arbitrary formula $\textbf{C}\!=\!(\mathcal{C}_{yz},\mathcal{C}_{xz},\mathcal{C}_{xy})$ in three-dimensional systems has been experimentally realized [\B{Nature 609, 925 (2022)}]. Motivated by these progresses, we propose the Chern vector $\textbf{C}\!=\!(0,m,n)$-protected quantized Hall effect in three-dimensional systems. By examining samples with Chern vector $\textbf{C}\!=\!(0,m,n)$ and dimensions $L_y$ and $L_z$ along the $y$- and $z$-directions, we demonstrate a topologically protected two-terminal response. This response can be reformulated as the sum of the transmission coefficients along the $x$- and $y$-directions, given by $(mL_y\!+\!nL_z)$. When applied to Hall bar setups, this topological mechanism gives rise to quantized Hall conductances, such as $G_{xy}$ and $G_{xz}$, which are expressed by $\pm(mL_y\!+\!nL_z)$. These Hall conductances exhibit a clear dependency on sample dimensions, illuminating the intrinsic three-dimensional nature. Finally, we propse potential candidates for experimental realization. Our findings not only deepen the understanding of the topological nature of Chern vectors but also enlighten the exploration of their transport properties.

Figures

Figures reproduced from arXiv: 2501.14493 by the authors.

**Figure 2.** FIG. 2: (a) Schematic distribution of the chiral surface states [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3: (a) The schematic plot of Hall bar with current along [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: (a) and (d): Schematic band spectra along [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Chern Vector Protected Three-dimensional Quantized Hall effects

Actually, the white energy interval possess the trivial sur- face states along the y direction. These states originate from the stacking of chiral modes along the y direction. Noevertheless, the white energy interval is rather tiny, which can be roughly neglected. Supplementar...

1982 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [38]

Evers, and A

F. Evers, and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008)

work page 2008

[2] [1]

M. Z. Hasan and C. L. Kane, Colloquium: topological insulators, Rev. Mode. Phys. 82, 3045 (2010)

work page 2010

[3] [2]

X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011)

2011

[4] [3]

S. Q. Shen, Topological insulators , Berlin, Springer (2012)

work page 2012

[5] [4]

V. K. Klitzing, G. Dorda and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Con- stant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494 (1980)

work page 1980

[6] [5]

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1981

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1982

[8] [7]

B. I. Halperin, Quantized Hall conductance, current- carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25, 2185 (1982)

1982