Theory of off-diagonal disorder in multilayer topological insulator

A. Kudlis; Z.Z. Alisultanov

REVIEW 4 major objections 4 minor 75 references

Theory of off-diagonal disorder in multilayer topological insulator

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In a multilayer topological insulator, a single interlayer-tunneling defect creates an in-gap bound state that crosses zero energy only in the trivial phase, never in the topological phase, making the defect a local marker of bulk topology.

desk verdict The single-defect crossing rule is a genuine result, but the paper's finite-disorder DOS claims are overstated and the localization-length section has sign errors that need fixing before publication. read the letter →

arxiv 2507.20713 v2 pith:HWF2LHEK submitted 2025-07-28 cond-mat.mes-hall cond-mat.str-el

Z.Z. Alisultanov , A. Kudlis This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.str-el

keywords off-diagonaldisordermultilayertopologicalinsulatorBurkov-BalentsmodelinterlayertunnelingboundstatesdensityofWeylsemimetalanomalousquantumHalleffect

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 asks what random interlayer tunneling - off-diagonal disorder - does to a multilayer topological insulator built from alternating topological and trivial layers. Its central claim is that a single tunneling defect creates an in-gap bound state whose energy crosses the middle of the gap only when the stack is in the trivial phase, never in the topological phase, so the defect acts as a local marker of bulk topology. For many defects, the paper argues, disorder fills the gap with bulk states and can close it: the Weyl semimetal phase survives strong disorder, the anomalous quantum Hall phase does not, and the added bulk states shrink the Hall plateau. It also claims that the disorder statistics control edge-mode localization, with Gaussian disorder able to delocalize the edges and thereby pull the longitudinal conductance away from its quantized value without breaking time-reversal symmetry. A sympathetic reader would care because these effects supply testable signatures - a narrowed Hall plateau and a suppressed edge conductance - for off-diagonal disorder in real van der Waals stacks.

What carries the argument

The load-bearing object is the chiral, off-block-diagonal structure of the Burkov-Balents Hamiltonian in the layer basis, combined with the pole condition for a single defect. The defect perturbation enters as $\delta\hat h = \tau^x\,\delta\Delta_S$, and because the clean Hamiltonian is chiral ($KHK^{-1}=-H$, e.g. $K=\tau^z\otimes\sigma^z$), every eigenstate has a partner at opposite energy. In-gap bound states are poles of the Dyson-resummed Green function, $\det(1-G^0\,\delta\hat h)=0$; inside the gap the local Green operator has eigenvalues $\lambda_\pm = G_x \pm |G_0|$, and the condition $1/\delta\Delta_S = \lambda_\pm(\varepsilon)$ controls whether a bound-state branch crosses zero. For the disordered density of states the paper uses two complementary self-consistent schemes: the locator expansion in localized states (each layer carrying a local propagator) and the $t$-matrix+RPA expansion in Bloch states. At zero energy the projector structure of the locators makes every diagram that returns to its starting layer vanish, so the Green function factorizes into a product of strictly forward locators, giving the inverse localization length $L_c^{-1}=|\langle\ln\Delta_S^i\rangle-\langle\ln\Delta_D^i\rangle|$.

What would settle it

A numerically exact calculation on a finite multilayer with random interlayer tunneling would settle the central claims: if a single defect's in-gap level crosses zero in the topological phase for any finite defect strength, or if the disorder-averaged density of states at moderate disorder does not fill and close the gap, the picture fails. The same calculation can test the Gaussian delocalization by measuring the inverse participation ratio of zero-energy edge states as the fluctuation width grows.

Watch

Extended reading notes

Core claim

The central discovery is that off-diagonal disorder - random interlayer tunneling - acts on the Burkov-Balents multilayer in a phase-selective way. A single Hermitian tunneling defect, a change $\delta\Delta_S$ in one layer's tunneling amplitude, creates an in-gap bound state whose energy crosses zero only in the trivial (normal-insulator) phase, at $\delta\Delta_S = -\Delta_S$; in the topological phase the pole condition $\delta\Delta_S/(\Delta_S+\delta\Delta_S)=1$ has no finite-strength solution, so the state never reaches zero energy, and a non-Hermitian defect splits the level without changing the crossing rule. For a finite density of defects, the paper argues, disorder fills the gap with bulk states and can close it: the Weyl semimetal phase survives strong fluctuations, while the anomalous quantum Hall phase is fragile and its Hall plateau shrinks as disorder grows. In the topological phase at weak disorder the two self-consistent schemes (locator and $t$-matrix+RPA) disagree on whether the gap narrows or widens, a discrepancy the paper reports as not yet understood. At zero energy each Green-function diagram containing a return path vanishes, leaving only forward paths, which yields a disorder-dependent localization length for edge modes; for Gaussian fluctuations the inverse localization length $L_c^{-1}$ can vanish, so opposite edges overlap and the longitudinal conductance drops below $\pi e^2/4h$ even though time-reversal symmetry and chirality are preserved.

Load-bearing premise

The load-bearing premise is that the two approximate calculation schemes used for the disordered density of states are valid in every regime studied; this is strained because at weak disorder in the topological phase one scheme says the gap narrows while the other says it widens, and the paper leaves that disagreement unexplained.

Editorial extensions

If this is right

A single off-diagonal defect can serve as a local, topologically selective probe: its zero-energy crossing occurs only in the trivial phase, not in the topological phase, and adding non-Hermitian loss or gain or asymmetric hopping preserves that rule.
Finite off-diagonal disorder generates bulk in-gap states and can close the gap; in the anomalous quantum Hall regime this shrinks the Hall plateau, which the paper proposes as an explanation for experimental deviations from quantized Hall behavior.
The Weyl semimetal phase is robust against strong off-diagonal disorder, while the anomalous quantum Hall phase is not, so disorder of this type produces a phase-selective stability.
The edge-mode penetration depth is distribution-dependent; Gaussian or Lorentzian disorder enlarges it and in the Gaussian case can make it diverge, enabling inter-edge tunneling and a correction that lowers the longitudinal conductance below $\pi e^2/4h$ without breaking time-reversal symmetry.
In the gapless Dirac or Weyl phase the Green-function series never converges, implying that off-diagonal disorder cannot localize zero-energy states or open a gap there.

Reading between the lines

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

Beyond the paper, the zero-crossing rule could be tested locally with scanning tunneling spectroscopy: tune a single junction between two layers and look for a zero-bias in-gap resonance only when the clean stack is trivial.
Beyond the paper, the unresolved weak-disorder discrepancy in the topological phase - gap narrowing versus widening - is a concrete target for exact diagonalization or transfer-matrix numerics, and whichever approximation survives would change how disorder renormalizes the topological mass.
Beyond the paper, the predicted divergence of the Gaussian localization length suggests a disorder-driven loss of helical protection in the thermodynamic limit; finite-size transport simulations could look for a threshold in fluctuation variance where two-terminal conductance departs from quantization.
Beyond the paper, applying the same forward-path analysis to related layered models, such as antiferromagnetic or superconducting stacks, would test whether the zero-energy diagrammatic cancellation generalizes to other chiral multilayer Hamiltonians.

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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

Colleague,

The one result worth taking away is the single-defect crossing rule: for off-diagonal (interlayer-tunneling) disorder in the Burkov–Balents multilayer, a bound state from a single tunneling defect crosses mid-gap only in the trivial phase, never in the topological phase. That extends Refs. [12,13] from diagonal to off-diagonal disorder and looks right, despite the sloppy signs in Appendix B. The non-Hermitian extension is a reasonable bonus. If I were working on impurity-bound-state diagnostics, I would cite the single-defect part after a cleanup.

What the paper does not deliver is the advertised agreement between the two DOS methods. In Section V.A the authors state explicitly that in the topological phase at weak disorder the locator scheme predicts gap narrowing while the Bloch-state (t-matrix+RPA) scheme predicts gap widening, and that the origin is not understood. That directly contradicts the abstract's "matching densities of states" and the conclusion's "mutually consistent results." The stress-test note is on target here, with one qualification: the discrepancy is specific to the topological phase at weak disorder. Trivial-phase, Weyl, and AQH DOS curves do agree, so the broad claims about Weyl robustness and AQH plateau shrinkage rest on firmer ground than the topological-phase claims. Still, the gap-closure and plateau-narrowing narrative is sold as if the two methods cross-check everything, and that is not true.

There are two concrete technical errors a referee should insist on. Equation (9) states the delocalization condition as A > ln(ΔS/ΔD); from Eq. (7) the threshold is A > ln(ΔD/ΔS). That is a sign error in the central condition for destroying the topological phase. Second, Eq. (62) uses |⟨ln ΔS⟩ − ⟨ln ΔD⟩|, while the localization-length plots show signed values, especially Fig. 15(b) where the inverse length approaches zero “from below.” For Gaussian and Lorentzian distributions the argument ΔS+η has support across zero, so the real logarithm is not defined; the formula needs |ΔS+η|. The Gaussian divergence and the edge-conductance correction therefore need to be re-examined. These are fixable, but they are load-bearing for the localization/transport section.

Who is this for? A theory reader interested in exact single-defect signatures of topology will get value from Sec. IV. The rest is a promising but unfinished treatment of off-diagonal disorder. It deserves a serious referee, because the core defect result is new and likely correct, and the open problems are specific. I would send it to review with the expectation of major revision, not desk-reject it.

Referee Report

4 major / 4 minor

Summary. This manuscript analyzes off-diagonal disorder—random interlayer tunneling—in the Burkov–Balents multilayer topological insulator model. The central single-defect claim is that a Hermitian tunneling defect produces an in-gap bound state whose energy crosses zero only in the trivial phase and never in the topological phase, with the same crossing rule preserved under non-Hermitian perturbations, so that the bound state serves as a local marker of topology. The paper also develops two diagrammatic Green-function schemes (self-consistent locator and t-matrix+RPA in the Bloch basis) and uses them to compute the density of states in the normal, topological, Weyl, and anomalous quantum Hall regimes, concluding that off-diagonal disorder fills the gap with bulk states, closes the gap for strong disorder, leaves the Weyl phase robust, and shrinks the anomalous Hall plateau. Finally, the manuscript studies edge-mode localization lengths for uniform, Gaussian, and Lorentzian disorder and derives a correction to the longitudinal edge conductance from inter-edge tunneling. The single-defect analysis is analytically self-contained and likely independent of the finite-disorder approximations, but several load-bearing statements in the finite-disorder and localization parts are not currently supported.

Significance. If established, the single-defect crossing rule would be a useful and elegant local diagnostic of topology in multilayer topological insulators, extending the Green-function-zeros approach to off-diagonal defects. The manuscript also makes concrete, falsifiable predictions about Hall-plateau narrowing and disorder-dependent edge conductance. Its strengths include exact or analytic treatments of the single-defect problem, analytic localization-length expressions for three disorder distributions, and a self-contained derivation of the diagrammatic expansions without fitting to the target results. However, the finite-disorder conclusions currently rest on two approximations that the paper itself shows to disagree in the topological phase at weak disorder, and the localization-length section contains a sign inconsistency. These issues must be resolved before the finite-disorder claims can be considered reliable.

major comments (4)

[Section II, Eq. (9)] The delocalization condition has the wrong logarithmic argument. From Eq. (7) the transfer factor is (Delta_S/Delta_D) exp(A-1), so the edge mode becomes extended when A-1 > ln(Delta_D/Delta_S), not A-1 > ln(Delta_S/Delta_D). As printed, the inequality is automatically satisfied in the topological phase (Delta_S < Delta_D) even for infinitesimal disorder, which contradicts the text's own discussion and the finite critical line shown in Fig. 2. The derivation and the phase diagram must be corrected.
[Section V.A, Figs. 8-9] The two self-consistent schemes do not give matching densities of states in the topological phase at weak disorder: the locator method predicts gap narrowing while the Bloch-state t-matrix+RPA expansion predicts gap widening, a discrepancy the authors explicitly state is not yet understood. This contradicts the abstract's claim of matching densities of states and leaves the subsequent gap-closure, Weyl-robustness, AQH-fragility, and Hall-plateau conclusions dependent on which approximation is used. Please resolve the discrepancy or substantially qualify the finite-disorder claims.
[Section V.B, Eq. (62) and Fig. 15] Eq. (62) defines L_c^{-1} as the absolute value of the difference of mean logarithms, so it cannot be negative. Yet Fig. 15(b) plots negative values and the text describes L_c^{-1} approaching zero from below and changing sign. The sign convention and the plotted quantity must be stated consistently. The claimed Gaussian divergence of L_c and the resulting inter-edge tunneling correction in Eq. (66) depend on this sign, so the current presentation is internally inconsistent.
[Appendix B, Eqs. (B2)-(B3)] The root-selection statement is incorrect for the in-gap case. For epsilon_perp^2 < (Delta_S - Delta_D)^2, the dimensionless combination x = (epsilon_perp^2 - Delta_S^2 - Delta_D^2)/(2 Delta_S Delta_D) is less than -1, and the root inside the unit circle is z1 = x + sqrt(x^2 - 1), not the 'minus' root as stated. Since Eq. (B3) feeds into the single-defect pole condition Eq. (31) and the eigenvalue analysis Eq. (35), the sign/root convention should be fixed and the crossing rule re-verified.

minor comments (4)

[Throughout] The manuscript contains numerous missing spaces and broken formatting, for example the opening words 'Westudymultilayertopologicalinsulators' and similar throughout; the text should be properly typeset.
[Reference [12]] Reference [12] is incomplete ('R. J. Slager and et. al.'); the full author list and title should be supplied.
[Eq. (66)] The printed form of Eq. (66) appears to omit the normalization: Appendix D, Eq. (D15), defines delta_sigma_xx/(pi e^2/4h), so Eq. (66) should be written as the normalized correction rather than as 'sigma_xx = sigma_xx^Delta pi e^2/4h = ...'.
[Section V, notation] Section numbers are inconsistent: the text refers to 'Sec. 3' and 'Sec. 5', while the paper is organized into Sections III and V; please unify the references.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations are self-contained analytic expansions against the fixed Burkov-Balents Hamiltonian, and the locator-formalism self-citation is not load-bearing.

full rationale

Walking the derivation chain from Eq. (1) through Eq. (66), the clean-limit Hamiltonian and chiral symmetry are fixed inputs. The single-defect pole condition (28)/(31) and the zero-crossing rule are derived from the local Green function G0_ll computed in Appendix B; no fitted parameter is involved. The localization length Eq. (62) follows from the forward-only Green function product at epsilon=0 (Eqs. 56-62), and the density of states is computed from the self-consistent locator equations (17)-(19) and the t-matrix+RPA equation (22), with no parameter fitted to the claimed outputs. The only reuse of the authors' own Ref. [42] is the locator expansion, but Eqs. (14)-(19) are restated in the paper and attributed to external Refs. [46-50], so the self-citation is not load-bearing. The conductance correction (66) contains ad hoc Delta_0 and Gamma, but these parametrize a model estimate, not a fit to the target results. The self-admitted discrepancy in Sec. V.A (locator gap narrowing vs. Bloch-state gap widening in the topological phase at weak disorder) is a genuine limitation that undermines some finite-disorder conclusions, but it is an approximation-validity problem, not a circular reduction of the result to its inputs. The paper is therefore self-contained against its stated model and external benchmarks, with no circular derivation chain.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central results rest on the Burkov-Balents model, the preservation of chiral symmetry under the disorder, the validity of two Green-function approximations that are shown to disagree in one regime, and the ad hoc modeling of sign-changing hopping and inter-edge tunneling conductance. No invented physical entities are introduced.

free parameters (5)

eta_0 = 0.1 to 30 meV in figures
Amplitude of uniform off-diagonal disorder; scanned to produce DOS and localization plots, not fitted to data.
alpha_D = 3, 20, 50 meV^-2 in Fig. 15b
Gaussian width parameter for Delta_D fluctuations.
Gamma_D = 0, 20, 40 meV in Fig. 15c
Lorentzian width parameter for Delta_D fluctuations.
Gamma = 0.1 to 0.8 eV in Fig. 16
Scattering rate in the two-edge conductance model; assumed proportional to the tunneling amplitude Delta_t, not independently determined.
Delta_0 = 1 eV in Fig. 16
Tunneling amplitude for fully overlapping edges in the estimate Delta_t = Delta_0 exp(-L/L_c).

assumptions (6)

domain assumption The Burkov-Balents Hamiltonian with alternating TI and NI layers and tunnelings Delta_S and Delta_D describes the multilayer topological insulator.
Used throughout, Eq. (2).
domain assumption The disordered Hamiltonian retains chiral symmetry K = tau_z tensor sigma_z with KHK^{-1} = -H, so eigenstates come in +/-E pairs.
Section II, used for zero-mode analysis and localization.
ad hoc to paper Off-diagonal disorder is modeled as independent random fluctuations of the interlayer tunneling amplitudes with uniform, Gaussian, or Lorentzian distributions; for Gaussian/Lorentzian the sign of the hopping can change because only ln|Delta| enters the localization length.
Section V.B, Eqs. (60)-(65); sign-changing hopping is not physically motivated for van der Waals tunneling.
domain assumption The self-consistent locator and t-matrix+RPA approximations give the density of states; the t-matrix cancellation trick is valid after disorder averaging.
Section III, Eqs. (17)-(22); the two methods disagree in the topological phase at weak disorder, Section V.A.
domain assumption Ziman's analytic convergence test for the Green-function series determines localization and delocalization in the Dirac phase.
Section V.B, Eqs. (67)-(75).
ad hoc to paper The edge conductance correction can be computed from a two-coupled-Dirac-surfaces model with tunneling Delta_t = Delta_0 exp(-L/L_c) and a phenomenological scattering rate Gamma.
Appendix D, Eq. (66); Gamma proportional to Delta_t is assumed.

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Cite this review

Pith. "Pith review of Theory of off-diagonal disorder in multilayer topological insulator." pith.science (2026). https://pith.science/paper/HWF2LHEK

@misc{pith2026250720713,
  author       = {Pith},
  title        = {Pith review of: Theory of off-diagonal disorder in multilayer topological insulator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWF2LHEK}},
  note         = {Machine review of arXiv:2507.20713}
}

read the original abstract

We study multilayer topological insulators with random interlayer tunneling, known as off-diagonal disorder. Within the Burkov-Balents model a single Hermitian defect creates a bound state whose energy crosses the middle of the gap in the trivial phase but never in the topological phase; a non-Hermitian defect splits this level yet preserves the same crossing rule, so the effect serves as a local marker of topology. However, the key distinction persists: the bound state crosses zero in the trivial phase but not in the topological phase. Two complementary diagrammatic approaches give matching densities of states for the normal, topological, Weyl and anomalous quantum Hall regimes. Off diagonal disorder inserts bulk states into the gap and can close it: the Weyl phase remains robust under strong disorder, whereas the anomalous quantum Hall phase survives only for weak fluctuations, and the added bulk states shrink the Hall plateau, clarifying experimental deviations. Finally, we analyze edge modes. Uniform disorder shortens their localization length slightly, while Gaussian and Lorentzian disorder enlarge it and in the Gaussian case can even delocalize the edges. Although chirality is maintained, the enhanced overlap permits tunneling between opposite edges and pulls the longitudinal conductance away from its quantized value.

Figures

Figures reproduced from arXiv: 2507.20713 by the authors.

**Figure 2.** Phase diagram in the (δ0, ∆D/∆S) plane, where δ0 = η0/∆S measures the amplitude of off–diagonal disorder and ∆D/∆S sets the clean-limit band inversion. The solid curve is the critical line obtained from Eq. (9). Blue region (below the curve): the disorder satisfies the inequality in Eq. (9), the edge state delocalizes, and the topological phase is destroyed. Orange region (above the curve): the inequality is viol… view at source ↗

**Figure 3.** Diagrammatic expansions for the quantities [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 5.** Figure 5: Energy dependence of the two eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 4.** Figure 4: Energy of the defect–induced bound state as a func [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 6.** Figure 6: (a) Van der Waals multilayer composed of alter [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Bound–state energies generated by a single tunnel [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 9.** Figure 9: Density of states ρ(ε) of the multilayer topological insulator in the topological phase for several amplitudes of off–diagonal disorder η0 (values in meV are indicated in the legend). Solid lines: self–consistent locator scheme; dashed lines: Bloch–state (t–matrix+RPA…

**Figure 8.** Figure 8: Density of states ρ(ε) in the trivial–insulator regime for several amplitudes of off–diagonal disorder η0 (numbers in the legend, in meV). Solid lines are obtained within the self–consistent locator scheme, dashed lines from the Bloch–state (t–matrix+RPA) expansion. (a…

**Figure 10.** Figure 10: Density of states ρ(ε) of the multilayer system in the Weyl semimetal phase for several amplitudes of off–diagonal disorder η0 (numbers in meV are given in the legend). Solid lines correspond to the self–consistent locator scheme, dashed lines to the Bloch–state (t–…

**Figure 12.** Figure 12: Hall resistivity ρxy (arbitrary units) in the AQHE phase versus the chemical potential µ for several amplitudes of off–diagonal disorder η0 (values in meV are indicated in the legend). Curves are evaluated from Eq. (52) using the disorder–broadened DOS of [PITH_FULL_…

**Figure 13.** Figure 13: Disorder–induced renormalization of the chemical [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

**Figure 14.** Figure 14: Graphical explanation of why any segment that [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]

**Figure 15.** Figure 15: Inverse localization length L −1 c as a function of the fluctuation amplitude of the intra–layer tunneling, ηS, for three statistics of off–diagonal disorder. (a) Uniform distribution: L −1 c grows monotonically with ηS; larger fluctuations of the inter–layer tunneli…

**Figure 16.** Figure 16: Edge conductivity of a three-dimensional topo [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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∆i S = ∆S + ηi, ∆i D = ∆D,

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It is given by the residue of the Green function: df = Tr Res G ε0 2 . (32) At ε = ε0 2 the residue is Res G ε0 2 = G0 iI ε0 2 δh adj 1 − δh G0 ε0 2 det′ 1 − δh G0 ε0 2 G0 Ij ε0 2 , (33) so that df = X n ⟨n|G0 αβ|I⟩ δhβγ Mγλ ⟨I|G0 λα|n⟩ = ⟨I|G0 λαG0 αβ|I⟩ δhβγ Mγλ = ⟨I| G0 2 λβ|I⟩ δhβγ Mγλ, (34) where Mγλ = adj[1 − δh G0(ε0 2)]γλ/ det′[1 − δh G0(ε0 2)] an...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

∆i S = ∆S + ηi, ∆i D = ∆D,

work page

[2] [2]

It is given by the residue of the Green function: df = Tr Res G ε0 2 . (32) At ε = ε0 2 the residue is Res G ε0 2 = G0 iI ε0 2 δh adj 1 − δh G0 ε0 2 det′ 1 − δh G0 ε0 2 G0 Ij ε0 2 , (33) so that df = X n ⟨n|G0 αβ|I⟩ δhβγ Mγλ ⟨I|G0 λα|n⟩ = ⟨I|G0 λαG0 αβ|I⟩ δhβγ Mγλ = ⟨I| G0 2 λβ|I⟩ δhβγ Mγλ, (34) where Mγλ = adj[1 − δh G0(ε0 2)]γλ/ det′[1 − δh G0(ε0 2)] an...

work page

[3] [3]

Topo- logical

gives ⟨n|b⟩ ∝G0 nI ∼ e−aRnI inside the gap. We now examine the topological character of bound states created by a single off-diagonal defect. For diag- onal disorder of codimension one (point impurity in 1D, line in 2D, etc.), Ref. [12] showed that a bound state always appears in the topological phase, whereas in the trivial phase it depends on details. T...

work page

[4] [4]

Hence SiˆtilSlˆtlj = ∆2 D 4 0 0 0 1 ⊗σ0

= −∆−1 S τ x ⊗σ0. Hence SiˆtilSlˆtlj = ∆2 D 4 0 0 0 1 ⊗σ0. (54) Next consider j = i, which corresponds to a diagram containing one return to site i. Then til = ∆D 2 τ + ⊗ σ0 and tli = ∆D 2 τ − ⊗σ0, giving SiˆtilSlˆtli = 0. (55) Written with explicit indices,S21 i ˆtil 12S21 l ˆtli 11 = 0, as illus- trated schematically in Fig. 14. Thus, at zero energy eve...

work page

[5] [5]

J. Chen, H. Qin, F. Yang, J. Liu, T. Guan, F. Qu, G. Zhang, J. Shi, X. Xie, C. Yang,et al., Gate-voltage control of chemical potential and weak antilocalization in Bi2Se3, Phys. Rev. Lett.105, 176602 (2010)

work page 2010

[6] [6]

∆i S = ∆S + ηi, ∆i D = ∆D + ξi,

work page

[7] [7]

For each case we employ three averaging pro- cedures: (1) a uniform distribution, ⟨

∆i S = ∆S + ηi, ∆i D = ∆D − ηi. For each case we employ three averaging pro- cedures: (1) a uniform distribution, ⟨. . .⟩x = (2x0)−1 R x0 −x0 . . . dx; (2) a Gaussian distribution,⟨. . .⟩x =p α/π R ∞ −∞ . . . e−α(x−x)2 dx; (3) a Lorentzian distribu- tion, ⟨. . .⟩x = π−1 R ∞ −∞ . . .Γ [ (x − x)2 + Γ2]−1dx. All three averages can be evaluated analytically. ...

work page 2023

[8] [8]

K. v. Klitzing, G. Dorda, and M. Pepper, New method forhigh-accuracydeterminationofthefine-structurecon- stantbasedonquantizedHallresistance,Phys.Rev.Lett. 45, 494 (1980)

work page 1980