Boundary-Bulk Interplay in Nonlinear Topological Transport

Annie G. Wang; Binghai Yan; Bomin Zhang; Chao-Xing Liu; Cui-Zu Chang; Deyi Zhuo; Han Tay; Huu-Thong Le; Ling-Jie Zhou; Xiaoda Liu

REVIEW 3 major objections 5 minor 17 references

Boundary-Bulk Interplay in Nonlinear Topological Transport

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Nonlinear transport in topological sandwiches is dominated by boundary modes, not bulk, the authors show.

desk verdict A clean symmetry-derived lead-voltage relation and solid experiments, wrapped around a boundary-dominance claim that needs one quantitative check. read the letter →

arxiv 2512.07017 v1 pith:Z2LN64GF submitted 2025-12-07 cond-mat.mes-hall cond-mat.mtrl-sci

Deyi Zhuo , Xiaoda Liu , Huu-Thong Le , Annie G. Wang , Han Tay , Bomin Zhang , Ling-Jie Zhou , Binghai Yan

show 2 more authors

Chao-Xing Liu Cui-Zu Chang

This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci MSC 82C7082D20 PACS 73.50.Fq73.43.-f

keywords nonlineartransportquantumanomalousHalleffectaxioninsulatorboundarymodesC2zsymmetrysecondharmonicLandauer-Büttikermagnetictopological

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

This paper is trying to establish that in magnetic topological insulator sandwiches tuned near the quantum anomalous Hall (QAH) and axion insulator states, the giant nonlinear (second-harmonic) electrical response comes primarily from boundary modes—chiral edge channels and hinge/surface states—rather than from the bulk's quantum geometry. The authors find that the signal is largest when the sample is slightly doped away from the perfectly quantized state, depends on which electrodes you measure, and flips sign with magnetization direction and carrier type. They support this with a symmetry argument using the electrode geometry and with numerical simulations. If correct, it means that in nearly insulating topological materials, the dominant nonlinear transport is a boundary effect, and the electrode layout is an intrinsic part of the physics.

What carries the argument

The central object is the second-order conductance tensor Gij,k(2), defined for electrode labels rather than spatial coordinates, which incorporates both the crystal symmetry and the geometry of the six-terminal Hall bar. From Neumann's principle, a C2z rotation of the electrode configuration constrains the second-harmonic voltages to satisfy Vxx,L^2ω = −Vxx,R^2ω = −Vyx^2ω. The numerical simulations use a nonlinear Landauer-Büttiker formalism where the second-order conductance is built from derivatives of the transmission matrix with respect to energy and electrode voltage, capturing the local boundary-channel response assisted by bulk carriers.

What would settle it

A measurement on a device with a deliberately broken electrode C2z symmetry (e.g., an asymmetric Hall bar) that still shows Vxx,L^2ω = −Vxx,R^2ω = −Vyx^2ω would contradict the symmetry explanation; conversely, observing a large bulk-like nonlinear Hall response with the same magnitude on both sides of the sample in a symmetric device would indicate a bulk contribution not captured by the boundary-bulk decomposition.

Watch

Extended reading notes

Core claim

The paper demonstrates that in MBE-grown V/Cr-doped (Bi,Sb)2Te3 sandwiches, the second-harmonic nonlinear Hall and longitudinal voltages are governed by nonlinear boundary transport assisted by bulk carriers. The central evidence is the electrode-dependent sign and amplitude of the second-harmonic voltages, the vanishing of signals exactly at charge neutrality, and the derivation of a universal C2z symmetry relation Vxx,L^2ω = −Vxx,R^2ω = −Vyx^2ω that forces the left and right longitudinal voltages to be opposite and the Hall voltage to match the right longitudinal voltage. Numerical transport calculations with a two-surface-state model reproduce these features, including sign changes upon m

Load-bearing premise

The attribution of the electrode-asymmetric second-harmonic signal to boundary transport rests on the assumption that the bulk's second-order conductivity is suppressed by the approximate C2z symmetry in the slightly doped regime, so that any residual bulk nonlinearity stays negligible compared to the boundary contribution.

Editorial extensions

If this is right

The sign of the nonlinear boundary transport can be used as an electrical readout of the Néel order direction in axion insulators.
Electrode geometry becomes a design knob: the same physical sample shows different nonlinear responses depending on which leads are used, which is absent in nonlinear optics.
The giant nonreciprocal coefficient γ (~1.22×10−9 m2/A) reported here is orders of magnitude larger than in previous nonlinear transport studies, suggesting practical rectification applications.
The universal C2z relation provides a model-independent diagnostic to separate boundary-dominated from bulk-dominated nonlinear transport in any six-terminal device with C2z symmetry.
Nearly bulk-insulating conditions are required to observe the boundary-dominated nonlinear response, guiding future material design toward cleaner topological insulators.

Reading between the lines

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

The C2z relation is a symmetry identity that holds regardless of whether the microscopic transport is ballistic or diffusive; this makes it a robust experimental fingerprint, but it also means that observing the relation alone does not prove that the signal is boundary-driven—it only proves that the bulk's own second-order conductivity is C2z-suppressed.
The necessity of doping slightly away from charge neutrality suggests a generic mechanism: boundary channels need bulk carriers to inject or extract charge to generate a finite second-harmonic voltage, which might apply to other topological states (e.g., higher-order hinge modes) beyond the specific sandwich studied.
A testable prediction is that the sign of Vxx,A^2ω should reverse across the sample width if the device is mirrored, and that the relation Vxx,L^2ω = −Vxx,R^2ω should hold exactly even when the linear Hall voltage is not quantized, which could be checked on any six-terminal magnetic TI device.
Because the pressed indium contacts are not tested for ohmicity at the second harmonic, residual contact rectification could mimic or distort the boundary signal; a control experiment with different contact materials or four-terminal voltage sensing would sharpen the interpretation.

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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

Two things to know. First, the paper gives a genuinely useful symmetry result: for a six-terminal Hall bar with approximate C2z symmetry, the second-harmonic longitudinal voltages on the left and right plus the Hall voltage satisfy Vxx,L = −Vxx,R = −Vyx. This is derived from Neumann's principle applied to an electrode-labeled conductance G(2), not fitted, and it matches the observed sign pattern. Second, the claim that the giant nonlinear response is boundary-dominated is plausible and well supported in places, but the key inference has a specific hole: the bulk second-order conductivity is assumed suppressed by C2z, and the paper does not quantify how much a disorder-activated, C2z-broken bulk term could leak into the antisymmetric signal VA used to identify boundary transport. The nonzero VS in the axion states shows some bulk nonlinearity is present, so the separation is not self-evident.

What is actually new: the electrode-based G(2) formalism, the universal lead-voltage relation, and a systematic MBE experiment crossing QAH and axion regimes with electrode-configuration dependence. The experimental controls are solid: quadratic current scaling, frequency independence, vanishing near Tc, gate-oscillation exclusion, and four samples. The KWANT simulation reproduces the qualitative behavior, and the model parameters are stated. The symmetry derivation is clean and internally consistent.

Soft spots, in order of softness. Main one: the missing bound on disorder-activated bulk contribution in the gating window where γ is quoted (Vg−Vg0 = −5 V). In the axion state σxx ≈ 0.22 e²/h, so the sample is not in the ideal quantized limit; a disorder potential locally breaks the two-fold rotation, and an inhomogeneous bulk term can produce electrode-dependent voltages that feed VA. The Methods defends approximate C2z by saying higher-order k·p terms are irrelevant near QAH/axion states, but that does not control quenched disorder. A quantitative estimate would settle it. Minor: the γ comparison with prior work is normalization-sensitive; the model's axion regime diverges while the measured state conducts; no error bars on the headline numbers; contact rectification is not explicitly tested; derivation details are in an SI that isn't in the arXiv text. None of these is fatal.

Verdict: the paper deserves a serious referee. If the boundary-dominance point survives a quantitative check, it is a meaningful advance. Even if not, the symmetry diagnostic is worth having. I would bring it to a reading group and cite it once revised.

Referee Report

3 major / 5 minor

Summary. The manuscript reports nonlinear second-harmonic transport in MBE-grown magnetic topological insulator sandwiches (V-doped and Cr-doped (Bi,Sb)2Te3 layers) in the quantum anomalous Hall (QAH) and axion insulator regimes. It claims that the observed giant nonlinear longitudinal and Hall voltages are dominated by boundary modes—chiral edges, in-band hinge/surface states—assisted by bulk carriers, rather than by the bulk quantum-geometric conductivity alone. A universal relation Vxx,L^2ω = −Vxx,R^2ω = −Vyx^2ω is derived from a C2z symmetry (Neumann's principle) applied to the six-terminal conductance tensor. The experimental data show opposite signs on the left and right longitudinal voltage pairs, approximate equality of Vxx,R^2ω and Vyx^2ω, quadratic current scaling, frequency independence, and gate/magnetization dependence. Nonlinear Landauer-Büttiker simulations with a two-surface Dirac model reproduce the sign structure and the qualitative gate dependence.

Significance. If the boundary-dominance interpretation is correct, this is an important advance: it extends nonlinear transport beyond bulk quantum geometry, introduces electrode-geometry symmetry as a diagnostic, and identifies boundary states as the key origin of large nonlinear responses in nearly bulk-insulating topological materials. The paper has notable strengths: the C2z symmetry derivation is elegant and the Landauer-Büttiker treatment is physically transparent; the experimental controls include current scaling, frequency independence, temperature dependence, multiple samples, and public data deposition. The main risk is that the clean separation between boundary and bulk transport relies on an exact C2z symmetry that the data themselves call into question, and the manuscript does not yet provide a quantitative bound on disorder-activated bulk contributions to the antisymmetric signal.

major comments (3)

[Methods, Eq. (4)] The derivation of the universal voltage relation is incomplete as written. From ∑_j G^(1)_ij (1−C2z)_jk V_k^{2ω}=0 for all i, the text concludes that (1−C2z)V^{2ω}=0 and hence V_i^{2ω} is C2z-invariant. But G^(1) is singular, as the paper itself later notes, so this conclusion does not follow without an explicit statement about the nullspace. The missing step is that the nullspace of G^(1) consists only of the uniform voltage vector, and that (1−C2z)V^{2ω} cannot be uniform (or is otherwise excluded). Please provide this argument and state the assumptions (e.g., connected sample, no isolated terminal clusters). This is load-bearing because the relation Vxx,L=−Vxx,R=−Vyx is the central symmetry claim.
[Figs. 2b, 2d and V_A/V_S decomposition] The paper uses exact C2z symmetry to identify the antisymmetric combination V_A with boundary transport, but simultaneously reports a sizable V_S in the axion states. A nonzero V_S is itself a C2z-breaking response, and assigning it to quantum-metric bulk transport while using C2z to define V_A is internally strained. In the axion state σxx≈0.22 e²/h, so the sample is not in the ideal quantized limit; quenched disorder can locally break C2z and activate a bulk σ^(2) that is not guaranteed to be left-right symmetric. Such a bulk channel would leak directly into V_A, the observable used to infer boundary dominance and to quote γ≈1.22×10⁻⁹ m²/A at (Vg−Vg0)=−5 V. Please supply a quantitative estimate of this leakage—for example, a disorder-averaged nonlinear Landauer-Büttiker simulation with C2z-breaking disorder, or an experimental scaling argument that controls the disorder-activated bulk
[Contact nonlinearity] The pressed indium contacts are stated to be ohmic, but no direct test of contact rectification is shown. A nonlinear contact would produce second-harmonic voltages that scale quadratically with current, are frequency-independent, and can be electrode-asymmetric, mimicking the reported boundary signal. The grounding-configuration measurement (Extended Data Fig. 3) does not exclude this possibility because both contacts could rectify. A control measurement on a non-topological Hall bar, or a four-terminal contact test, would substantially strengthen the claim that the antisymmetric signal originates in the sample rather than at the contacts.

minor comments (5)

[Abstract/Discussion] The phrase 'universal relation ... allows us to distinguish nonlinear boundary transport from bulk contributions' is too strong. The relation holds only under exact C2z symmetry, and the V_A/V_S separation assumes the bulk contribution is purely electrode-symmetric. Please qualify the statement accordingly.
[Experimental data] The claimed near-equality Vxx,R^2ω ≈ Vyx^2ω is shown graphically but not quantified. Provide the relative deviation (e.g., (Vxx,R−Vyx)/(|Vxx,R|+|Vyx|)) at representative fields and gate voltages, especially in regimes where V_S is large.
[Extended Data Fig. 5] The parabolic fits to the current dependence of V^2ω are shown without residuals or confidence intervals. Report the extracted exponent and uncertainties to support the quadratic scaling claim quantitatively.
[γ definition] The definition of γ = √2 V_A/(R0·i²) should specify R0 and the effective cross-section/current density used, so that the comparison with prior reports in Refs. 28, 42, and 48 is meaningful.
[Fig. 4 / simulations] The text says the simulations show 'identical behaviors' of Vxx,R^2ω and Vyx^2ω; given the approximate nature of the model and the absence of V_S reproduction, 'qualitatively similar' would be more accurate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the universal lead-voltage relation is derived from Neumann's principle with no fitted parameters, and the numerical simulation uses independently fixed model parameters.

full rationale

The paper's central derivation is the C2z symmetry constraint on the six-terminal second-order conductance. From Neumann's principle and the antisymmetric current configuration, the Methods derive Vxx,L^2ω = -Vxx,R^2ω = -Vyx^2ω (Methods, Eq. 4 and following). This is a group-theoretic result with no adjustable constants and is not fitted to the experimental curves. The nonlinear Landauer-Büttiker simulation uses fixed parameters (vF=2, m0=0.1, B=-1, Mt=±1.2, Mb=±0.6) that are not tuned to match the measured amplitudes; the comparison is qualitative (signs, relative magnitudes), so it is not a fitted-input-called-prediction. The V_S/V_A decomposition is algebraically defined and then interpreted: V_A is called the boundary contribution, and its observed dominance is used to claim boundary-mode dominance. Although this labeling could appear self-definitional, the paper supplies an independent physical basis for it: under the assumed approximate C2z symmetry, the bulk in-plane second-order conductivity vanishes, so the electrode-asymmetric second-harmonic response is attributed to boundary modes. That is an assumption—and a legitimate source of correctness risk if disorder breaks C2z—but it is not a circular reduction. Self-citations (e.g., Refs. 5, 19, 20) are used for background and prior experimental demonstrations of QAH/axion states, not as the load-bearing justification for the new symmetry result. No uniqueness theorem, ansatz, or prior result of the same authors is invoked to forbid alternatives or to define the key prediction into existence. Therefore the derivation chain is self-contained with respect to the paper's central claims.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central quantitative claims (boundary dominance, γ magnitude) rest less on fitted parameters than on interpretive assumptions: approximate C2z symmetry, the vanishing of bulk σ(2) under that symmetry, and the assignment of electrode-antisymmetric signals to boundary states. The simulation parameters are free but unfitted; the valence-band fix is ad hoc. No new entities (particles, forces, dimensions) are postulated; the in-band hinge states are prior-published objects (refs 6, 7).

free parameters (4)

Two-surface Hamiltonian parameters = vF=2; m0=0.1; B=-1; Mt=±1.2; Mb=±0.6; lattice a=1
Set the qualitative band structure and computed lead-voltage patterns in Fig. 4; chosen in arbitrary units, not fitted to the measured curves. Different values would change magnitudes and could alter the p-doped sign structure (Eq. 5, Methods).
Ad hoc parabolic bulk valence band (Fig. S10) = not stated in main text
Added after observing that the two-surface model misses the p-doped enhancement of V^2ω; the parameters are not reported in the main text, making this a data-motivated model fix.
Effective cross-section for current density i in γ = √2 V_A/(R0·i²) = not stated; bulk area used implicitly
The reported γ ≈ 1.22e-9 m² A⁻¹ and the order-of-magnitude comparisons against refs 28, 48-50 depend on this normalization, which is ambiguous for boundary-dominated transport.
Vg0 (charge-neutral reference gate voltage) = 5 V for Sample S1
Defined as the ρxx maximum in the axion regime (Extended Data Fig. 1); every (Vg−Vg0) claim is relative to this calibration choice, and the 'slightly away from quantized states' conclusion is expressed against it.

assumptions (5)

domain assumption Approximate C2z symmetry of the Hall-bar device, crystal, magnetization, and field configuration
Used to derive the universal relation Vxx,L = −Vxx,R = −Vyx and to assert suppression of bulk nonlinearity; the paper labels it an approximation/quasi-symmetry (Methods: 'higher-order terms that break two-fold rotation symmetry are irrelevant').
standard math Bulk second-order conductivity σ(2) vanishes in a C2z- (or P-) symmetric system
Cited from refs 22 and 55; the inference that residual nonlinear signal is boundary-related depends on this suppression holding in the real, disordered sample.
standard math Neumann's principle applies to the electrode-indexed conductance tensor G(2)
The symmetry derivation (Eqs. 3-4) rests on transforming G(2) under C2z as a third-rank tensor with electrode labels; this extends crystalline symmetry analysis to multiprobe geometry.
domain assumption Gauge-invariant nonlinear Landauer-Büttiker expansion (Eqs. 6-8) from Christen-Büttiker
The numerical V^2ω inherits the validity of this mesoscopic transport framework, including the split between ∂E T and ∂V T terms required for gauge invariance under global energy shifts.
ad hoc to paper Two-surface Dirac model with exchange gaps (Eq. 5) captures the sandwich's low-energy physics
A simplified effective description with surface magnetizations as inputs; omits bulk bands (patched later by an added parabolic band) and higher-order C2z-breaking terms.

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Cite this review

Pith. "Pith review of Boundary-Bulk Interplay in Nonlinear Topological Transport." pith.science (2026). https://pith.science/paper/Z2LN64GF

@misc{pith2026251207017,
  author       = {Pith},
  title        = {Pith review of: Boundary-Bulk Interplay in Nonlinear Topological Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2LN64GF}},
  note         = {Machine review of arXiv:2512.07017}
}

read the original abstract

Nonlinear transport has emerged as a powerful approach to probe the quantum geometry of electronic wavefunctions, such as Berry curvature and quantum metric, in topological materials. While nonlinear responses governed by bulk quantum geometry and band topology are well understood, the role of boundary modes (e.g., edge, surface, and hinge states) in nonlinear transport of topological materials remains largely unexplored. In this work, we demonstrate boundary-bulk interplay in nonlinear transport, including second-harmonic Hall and nonreciprocal longitudinal responses, in molecular beam epitaxy-grown magnetic topological insulator heterostructures. We find that the nonlinear transport is maximized when the sample is tuned slightly away from the well-quantized states, including the quantum anomalous Hall and axion insulator states. The sign and amplitude of the nonlinear transport depend on electrode configuration, magnetic order, and carrier type, establishing boundary mode transport as the dominant contributor. These findings, supported by symmetry analysis and nonlinear Landauer-B\"uttiker formalism, demonstrate that nonlinear transport in topological materials is governed by the interplay between boundary and bulk states. We further derive a universal relation between different lead voltages from electrode geometry symmetry, which allows us to distinguish nonlinear boundary transport from bulk contributions. Our work highlights the critical role of electrodes in nonlinear transport, which is absent in nonlinear optics, and establishes boundary modes as a key origin of the giant nonlinear response in nearly bulk-insulating topological materials. This insight opens new opportunities for engineering nonlinear transport through boundary-bulk interplay in future device applications of topological materials.

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Works this paper leans on

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

1 Boundary-Bulk Interplay in Nonlinear Topological Transport Deyi Zhuo1,2, Xiaoda Liu1,2, Huu-Thong Le1,2, Annie G. Wang1, Han Tay1, Bomin Zhang1, Ling- Jie Zhou1, Binghai Yan1, Chao-Xing Liu1, and Cui-Zu Chang1 1Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA 2These authors contributed equally: Deyi Zhuo, Xiaoda L...

2008

[2] [2]

Cava, R. J. & Hasan, M. Z. Observation of a large -gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398-402 (2009). 12 Chen, Y . L., Analytis, J. G., Chu, J. H., Liu, Z. K., Mo, S. K., Qi, X. L., Zhang, H. J., Lu, D. H., Dai, X., Fang, Z., Zhang, S. C., Fisher, I. R., Hussain, Z. & Shen, Z. X. Experimental Realization...

2009

[3] [3]

L., Ji, Z

Wang, L. L., Ji, Z. Q., Feng, Y ., Ji, S., Chen, X., Jia, J., Dai, X., Fang, Z., Zhang, S. C., He, K., Wang, Y ., Lu, L., Ma, X. C. & Xue, Q. K. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167-170 (2013). 14 Checkelsky, J. G., Yoshimi, R., Tsukazaki, A., Takahashi, K. S., Kozuka, Y ., Falson, J.,

2013

[4] [4]

& Tokura, Y

Kawasaki, M. & Tokura, Y . Trajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator. Nat. Phys. 10, 731-736 (2014). 25 15 Chang, C. Z., Zhao, W., Kim, D. Y ., Zhang, H., Assaf, B. A., Heiman, D., Zhang, S. C., Liu, C., Chan, M. H. & Moodera, J. S. High-precision realization of robust quantum anomalous Ha...

2014

[5] [5]

Zhang, S. C. & Wang, K. L. Metal-to-insulator switching in quantum anomalous Hall states. Nat. Commun. 6, 8474 (2015). 17 Mogi, M., Yoshimi, R., Tsukazaki, A., Yasuda, K., Kozuka, Y ., Takahashi, K. S., Kawasaki, M. & Tokura, Y . Magnetic modulation doping in topological insulators toward higher - temperature quantum anomalous Hall effect. Appl. Phys. Let...

2015

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