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REVIEW 3 major objections 5 minor 14 references

Twisted Multilayer Graphene: Superperiodicity and quasicrystals

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Moderate disorder can increase the mean free path of flat-band electrons in magic-angle twisted bilayer graphene.

desk verdict The MATBLG disorder-induced delocalization is likely real but the ℓ numbers rest on clean-system v_F and a 600-fs plateau; the thesis is honest, useful, and worth refereeing. read the letter →

arxiv 2607.25411 v1 pith:XBUM3KJI submitted 2026-07-28 cond-mat.mes-hall physics.comp-ph

Pedro Alcázar Guerrero This is my paper · ORCID
classification cond-mat.mes-hallphysics.comp-ph
keywords twistedbilayergraphenemagicangleflatbandsrandomon-sitedisordermeanfreepathquantummetricquasicrystalspinlifetime
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 thesis uses large-scale real-space quantum-transport simulations to argue that geometry itself—twist angles, moiré superlattices, quasicrystalline order, and atomic corrugation—acts as an effective potential shaping electronic and spin transport in graphene stacks. Its central claim is that in magic-angle twisted bilayer graphene, within a bounded window of random on-site disorder where the flat bands remain visible, increasing disorder broadens those bands and reduces the scattering rate, so the mean free path grows instead of shrinking; only above that window do the flat bands dissolve and normal localization scaling resume. The same calculations give a first estimate of the sub-ballistic diffusion exponent (about 0.84) for dodecagonal graphene quasicrystals and show that this quasicrystalline order and its self-similar localization patterns are fragile against disorder and an added third layer. A separate study of suspended corrugated graphene shows that curvature-induced fluctuating spin-orbit fields can bring spin lifetimes down to nanoseconds even while charge transport stays nearly ballistic.

What carries the argument

The carrying mechanism is the time-dependent diffusion coefficient D(E,t) = ½ d⟨ΔX²⟩/dt, computed by a Chebyshev-polynomial expansion of the real-space tight-binding Hamiltonian on million-atom samples. In a disordered system D(t) is expected to plateau; the plateau value yields the mean free path through the semiclassical relation ℓ = v_F τ_p with τ_p = 2D/v_F². The paper's main step is to identify that at the magic-angle flat band this plateau rises as the random disorder W is increased from 3γ0/4 to 3γ0/2, and to explain it as disorder-induced band broadening that reduces the scattering rate. The quantum metric is obtained from an optical-conductivity sum rule, G_xx ∝ ∫ dω Re σ_xx(ω)/ω, w

What would settle it

Extend the disordered flat-band simulations to longer evolution times (beyond 600 fs) and larger systems at W ≈ γ0/4 and γ0/2; if the diffusion coefficient does not plateau or the effective Fermi velocity is renormalized by disorder, the reported 76 nm and 306 nm mean free paths would be extraction artifacts. On the experimental side, a transport measurement that tunes disorder in a magic-angle device and observes a non-monotonic flat-band mobility would confirm the mechanism.

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Extended reading notes

Core claim

This thesis establishes, through large-scale real-space quantum-transport simulations of atomistically relaxed structures, that in magic-angle twisted bilayer graphene the flat bands are not only localization-prone but also respond counterintuitively to disorder. For random on-site disorder strengths in a finite window—roughly W ≈ 3γ0/4 to 3γ0/2, with flat-band features washed out by W ≈ 2γ0—the mean free path at charge neutrality increases with disorder rather than decreasing. The proposed mechanism is that disorder broadens the very narrow flat bands, lowering the density of available final states and hence the scattering rate, while delocalizing the real-space wave functions away from the

Load-bearing premise

The mean free paths are extracted by assuming that the diffusion coefficient reaches a true plateau within the 600 fs simulation window and that the clean-system Fermi velocity remains the correct conversion factor; for the weakest disorders that plateau is not actually reached, so the largest reported mean free paths rest on a perturbation-theory extrapolation rather than a directly observed diffusive regime.

Editorial extensions

If this is right

  • If the disorder-induced delocalization is real, the flat-band mean free path in magic-angle twisted bilayer graphene is non-monotonic: it rises with disorder up to about W ≈ 3γ0/2, then falls, implying a noise-tolerance window for flat-band transport.
  • The accompanying rise in the quantum metric extracted from optical conductivity makes the same single-particle mechanism measurable through the optical sum, not just through dc transport.
  • The graphene quasicrystal's sub-ballistic exponent α ≈ 0.84 predicts anomalous, non-Drude optical and temperature scaling, useful as an experimental fingerprint of quasicrystalline order.
  • Quasicrystalline resonances are fragile: they are destroyed by moderate disorder and by proximity to a third layer, so observing them requires very clean, isolated 30-degree interfaces.
  • Curvature-induced Rashba fields set a nanosecond-scale upper bound on spin lifetimes in suspended graphene, even where charge mean free paths remain long.

Reading between the lines

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

  • Inference: The disorder window studied is comparable to charge inhomogeneity from common substrates, so sample-to-sample variations in correlated-phase transport may be partly a disorder-window effect rather than intrinsic physics.
  • Inference: The same band-broadening mechanism might be sought in other flat-band moiré systems, such as transition-metal dichalcogenide heterobilayers, where an analogous disorder window could produce disorder-enhanced mobility testable in gated devices.
  • Inference: The quantum-metric–mean-free-path correlation suggests that disorder could act as a dial for superfluid weight in moiré superconductors; optical-conductivity measurements at controlled disorder would provide a direct test.
  • Inference: Because the quasicrystal sub-ballistic exponent is extracted from a very short time window, extending the calculation to larger approximants or to true 30-degree samples with absorbing boundaries would test whether the power law persists or is a short-time transient.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript (a PhD thesis deposited on arXiv) uses large-scale real-space Kubo/Kernel Polynomial Method simulations to study transport in superperiodic, quasicrystalline, and disordered graphene stacks. Its central claim, developed in Sec. 5.2, is that in magic-angle twisted bilayer graphene a finite window of Anderson disorder (W ≈ 3γ0/4 to 3γ0/2) broadens the flat bands and increases the diffusion coefficient and mean free path at charge neutrality — a disorder-induced delocalization that reverses at W = 2γ0. Supporting results include a sub-ballistic exponent α ≈ 0.84 for dodecagonal graphene quasicrystal approximants (Sec. 5.3.3), fragility of quasicrystalline states to disorder and to an added third layer (Secs. 5.3.4, 5.4), and nanosecond spin lifetimes in corrugated monolayer graphene from curvature-induced Rashba fields (Ch. 7). The manuscript is transparent about its limitations, including an explicit statement that the magnetism chapter (Ch. 6) is a work in progress and that the diffusive regime is not accessed at the weakest disorders.

Significance. If the central claim survives scrutiny, it is a conceptually important counterexample to the usual disorder-localization scaling: moderate disorder can delocalize flat-band states by broadening the flat bands and reducing the effective scattering rate. The connection to the quantum metric via the SWM sum rule is a valuable geometric perspective, and the quasicrystal exponent and spin-lifetime results address open questions. The manuscript's strengths are its large-scale atomistic KPM calculations, explicit structural relaxation, a code-availability appendix, and unusually honest caveats about unaccessed diffusive regimes and unreproduced literature results. However, the main claim's mean-free-path extraction pipeline contains a load-bearing fragility: the reported ℓ values use the clean-system Fermi velocity and a 600 fs plateau assumption, and the largest ℓ values are Fermi-golden-rule extrapolations anchored to a single converged point. Fixing this requires new analysis rather than editing.

major comments (3)
  1. [Sec. 5.3.3 / Fig. 5.6 and Fig. 5.14] The disorder-induced delocalization claim is supported by ℓ(E,W) in Fig. 5.5, obtained from the diffusion coefficient through ℓ = 2D/v_F using the clean-system Fermi velocity v_F(E) (inset of Fig. 5.3). Because Anderson disorder broadens the flat bands and changes the spectral composition at E=0, the effective ⟨v²⟩ at the flat band is not guaranteed to equal the clean-system v_F². A rise in D(0) from W=3γ0/4 to W=3γ0/2 can therefore be produced by a rise in ⟨v²⟩ even if the true scattering time τ_p is unchanged or reduced. Please provide a disorder-resolved v_F(E,W) — for example from the short-time ballistic slope of D(t) or from a direct evaluation of the velocity operator spectral function — and show that τ_p = 2D/v_F² also increases. In addition, the 600 fs window is not evidently a converged diffusive plateau for the W=3γ0/4 case: the clean system required 6000 fs to reach its asymp
  2. [Chapter 6] The first estimate of the sub-ballistic exponent α ≈ 0.84 for dodecagonal graphene quasicrystals is fitted from D(t) in the interval t < 13 fs in a 29.8° periodic approximant. The manuscript itself states that this window is before the electronic spreading reaches the approximant's superperiodicity; the shadowed region in Fig. 5.14 marks this limitation. A power-law fit in such a short, pre-periodicity window can reflect a transient inherited from the initial condition or from the periodic approximant rather than the asymptotic quasicrystalline exponent. Please provide a stability analysis: fits over several time windows, comparison with the 31° approximant and/or the Koshino-Moon continuum model, system-size dependence, and an estimate of the contamination from the initial state. Without this, the 'first estimate' is a promising but unsecured transient.
  3. Chapter 6 is presented as a thesis chapter on magnetism in twisted bilayer graphene, but the text explicitly states (p. x) that it 'represents a work in progress' and that it was 'not possible with the available computational resources to replicate previous results in the literature with our formalism.' The numerical results shown in Figs. 6.1–6.3 are convergence failures or magnetizations that remain at the tolerance threshold. Including an unreplicated chapter in the manuscript undermines the 'unified picture' promised in the conclusions. The authors should either remove this chapter or reframe it as an explicit, self-contained negative result with a technical analysis of why convergence is not achieved (for example, KPM broadening versus the relevant gap scale, initial conditions, or mean-field instability). This is a self-acknowledged missing result, not a presentation issue.
minor comments (5)
  1. [Sec. 5.2.5 / Fig. 5.5 (bottom)] The sentence 'The increase in QM at lower disorder is expected for weakly disordered cases, since the cleaner the system, the longer the corresponding mean free path and localization length' appears to conflict with the delocalization interpretation and should be clarified. As written, it is difficult to see why a longer mean free path in a cleaner system implies an increased quantum metric at lower disorder.
  2. [Sec. 4.1] The simulation length is stated as L = 2084; since the text describes periodic boundary conditions and powers of two are standard in such calculations, please confirm whether this is a typo for L = 2048.
  3. [Table 4.1] The caption says 'Choice of parameters for our twisted bilayer graphene simulator,' but the section models a gate-defined Bernal bilayer with an artificial superlattice, not a twisted bilayer. Please correct the caption to avoid confusion.
  4. [Multiple sections] Notation is inconsistent for the diffusion coefficient: D(t) is sometimes the time-dependent quantity ½ dΔX²/dt and sometimes the asymptotic diffusion constant. Please define explicitly whether the plotted values are D(t = 600 fs) or the saturated plateau value, and use separate symbols for the two.
  5. [App. C] The code availability appendix would be more useful with a URL or repository identifier and a list of key dependencies/versions, rather than a general statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MATBLG mean-free-path trend, quasicrystal exponent, and spin lifetimes are direct simulation outputs; acknowledged extrapolation and plateau limitations are methodological, not definitional.

full rationale

I find no circular step that meets the evidentiary bar. The MATBLG mean free path is obtained by applying Eqs. (3.16)-(3.17) to the simulated Kubo diffusion coefficient D(E,t) and a clean-system Fermi velocity read from the ballistic slope (Sec. 5.2.3, Fig. 5.3 inset). The disorder comparison W=3g0/4 vs W=3g0/2 is a direct simulation output, not a quantity fitted to the claimed disorder-induced delocalization. The quantum metric is independently computed from the optical conductivity via the SWM sum rule (Eq. 5.4) and is not an input to the transport extraction. The quasicrystal sub-ballistic exponent alpha~0.84 is a power-law fit to the simulated D(t) (Sec. 5.3.3), and the spin lifetimes in Ch. 7 are outputs of time-evolution simulations. Model parameters and methods are attributed to external works (Nguyen et al., Trambly de Laissardiere, Kolmogorov-Crespi, Fan et al.), so the derivation is not carried by a self-citation chain. The paper explicitly discloses the methodological weak points: Sec. 5.2.5 admits the diffusive regime is not accessed for the weakest disorders and labels the ell~76 nm and 306 nm values as a Fermi-golden-rule rough estimate anchored to one converged point; Sec. 5.3.3 restricts the sub-ballistic fit to t<13 fs; Ch. 6 states it could not replicate prior mean-field results with available resources. These are reliability/interpretation concerns about finite-time extraction and extrapolation, not cases where a fitted parameter is renamed as a prediction or where a result is equivalent to its input by construction. Self-citations (Guerrero et al. 2025a,b; Cummings et al. 2025) are provenance for published versions of the same in-thesis simulations and are not load-bearing circular justifications. Honest non-finding is therefore the appropriate outcome.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The claim stack rests on standard single-particle tight-binding + KPM transport (well-established), with domain assumptions that are mostly disclosed: Anderson disorder as a model of realistic MATBLG disorder; relaxed p_z Slater-Koster TB with parameters tuned to reproduce the flat bands; diffusive-plateau extraction of ℓ within 600 fs (partially satisfied — the paper admits the weakest-disorder regime is not accessible); the SWM quantum metric as a flat-band localization measure (an interpretive step, since the metric is the occupied-manifold integral); periodic approximants for the 30° quasicrystal at t<13 fs; and corrugation+SOC modeling for spin lifetimes. Free parameters: TB parameters adjusted to the flat bands, fitted exponents (α's, T^-0.7), the Fermi-golden-rule anchor for ℓ extrapolation, KPM broadening vs the flat-band width, and scanned disorder strengths whose window is selected post hoc. Invented entities: none — no new particles, forces, or dimensions are postulated; the 'fluctuating Rashba fields' of Ch. 7 are emergent effective fields derived from curvature-induced SOC terms, not new postulates, and the quantum metric and quasicrystalline states are pre-existing concepts from the cited literature.

free parameters (7)
  • MATBLG Slater-Koster parameter set = V^0_ppσ = 367.5 meV, qπ/a0 = qσ/d0 = 22.18 nm^-1, rc = 0.614 nm, λc = 0.0265 nm
    Sec. 5.1.2 states 'the TB parameters are adjusted' to reproduce the flat bands of relaxed MATBLG at ~1.1°; the flat-band physics being probed is therefore a tuned output of the model.
  • KPM broadening relative to the flat-band width = 66 meV for σ(ω)/G_xx; Ch. 5 DoS broadening unstated
    Sec. 5.2.5 uses a 66 meV broadening for the quantum metric; the MATBLG flat bands are only a few meV wide, so the 'flat-band features remain robust' statement is assessed at a resolution coarser than the band, making the DoS peak shape near E=0 partly a kernel artifact.
  • Sub-ballistic exponents α = 0.46 (Fibonacci chain), 0.05-0.2 (2D QC potential), ≈0.84 (QC peaks)
    Power-law fits D∝t^α in Secs. 4.1.2, 4.2.4, 5.3.3, without error bars; α≈0.84 is fitted from a t<13 fs window in a 29.8° approximant; the 1D exponent is stated to be tunable by the potential strength, so part of the anomaly is controlled by construction.
  • Fermi-golden-rule anchor for ℓ(W) = ℓ = 33.8 nm at W = 3γ0/4 → ℓ ≈ 76 nm (W=1/2), 306 nm (W=1/4)
    Sec. 5.2.5: weak-disorder mean free paths are extrapolated with ℓ∝1/W² anchored to the single converged simulation point, presented as an estimate, not a simulation.
  • Spin-lifetime temperature exponent = τ_s ∝ T^-0.7
    Sec. 7.4 (Fig. 7.6a inset): exponent obtained by fitting simulation data and used to compare with analytical fluctuating-SOC models; no uncertainty reported.
  • Anderson disorder strengths and the disorder window = W = 3γ0/4, 3γ0/2, 2γ0
    Scan control parameters (Sec. 5.2); the 'finite disorder window' where the delocalization holds is defined after observing where the flat-band DoS peak survives, i.e., a post-hoc window.
  • Superperiodic/quasiperiodic potential parameters (A_n, g_n, θ_n) = values not fully stated in visible text; N_per = 1, 3, 6 geometries
    Ch. 4 model inputs; the paper states the potential strength can tune α across (-1,1), so the anomalous exponents reported for these models are partly tunable by construction.
assumptions (8)
  • domain assumption The p_z Slater-Koster tight-binding Hamiltonian (Eqs. 5.2-5.3) with the chosen parameters captures the low-energy flat-band physics of relaxed MATBLG.
    Invoked in Sec. 5.1.2; all of Ch. 5 rests on this model, and the parameters are themselves adjusted to reproduce the flat bands, so the model is tuned to the effect under study.
  • domain assumption Anderson on-site disorder (uniform [-W,W]) adequately represents the dominant disorder in experimental MATBLG and quasicrystal samples.
    Secs. 5.1-5.3 use Anderson disorder exclusively; the central delocalization claim is specific to this disorder model, and the connection to experimental disorder (electron-hole puddles) is asserted once for hBN substrates.
  • standard math The real-space Kubo/Chester-Thellung formalism (Sec. 3.1) yields correct quantum diffusion and mean free paths at zero temperature.
    Established formalism (Chester-Thellung 1959; Fan et al. 2021), but the ℓ extraction additionally requires a well-defined diffusive plateau (see next axiom).
  • domain assumption The diffusive regime is reached within the 600 fs simulation window for the disordered flat-band states, so D(t) plateau values define ℓ = D/v_F.
    Sec. 5.2.4 relies on D(t) saturation. The paper explicitly states the diffusive regime is not accessed for the lowest disorder strengths (Sec. 5.2.5), where the largest ℓ values are extrapolated; for the flat band τ_p = ℓ/v_F grows with ℓ, so the plateau onset moves beyond the window precisely where the effect is largest.
  • domain assumption The SWM-sum-rule quantum metric G_xx (Eq. 5.4) is a valid flat-band localization measure whose disorder dependence reports the Wannier spread of the flat-band states.
    Sec. 5.2.5; G_xx integrates σ(ω)/ω over the whole occupied manifold, so attributing its disorder increase to the flat-band states assumes the other bands' geometric contribution is disorder-independent, which is not demonstrated.
  • domain assumption Periodic approximants at 29.8° and 31° represent the 30° quasicrystal for the transport window studied (t < 13 fs).
    Secs. 5.3.1, 5.3.4; the paper itself flags that larger approximants are needed to fully discard disorder-induced delocalization in the QC, so the approximant correspondence is only partially validated.
  • domain assumption Curvature-induced SOC terms (Eqs. 7.12-7.14) plus the corrugation model of Sec. 7.3 capture the dominant spin-relaxation mechanism in suspended graphene.
    Ch. 7; parameters taken from Cummings et al. 2025 (Table 7.1). The claim that this reconciles the theory-experiment gap in spin lifetimes rests on the corrugation height-correlation statistics being representative of real suspended graphene at 300 K.
  • domain assumption The clean-system Fermi velocity (from the ballistic slope D = v_F² t) remains the correct velocity scale for converting the disordered-system D into ℓ.
    Secs. 3.1.4, 5.2.3-5.2.4; the text does not state which v_F is used under disorder. If disorder renormalizes v_F in the broadened flat band, ℓ = v_F τ_p is miscalibrated and part of the reported ℓ enhancement could be a velocity effect.

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Pith. "Pith review of Twisted Multilayer Graphene: Superperiodicity and quasicrystals." pith.science (2026). https://pith.science/paper/XBUM3KJI

@misc{pith2026260725411,
  author       = {Pith},
  title        = {Pith review of: Twisted Multilayer Graphene: Superperiodicity and quasicrystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBUM3KJI}},
  note         = {Machine review of arXiv:2607.25411}
}
read the original abstract

This thesis investigates how superperiodicity, quasiperiodicity, and disorder shape electronic and spin transport in graphene-based systems, with an emphasis on experimentally relevant length scales and realistic atomistic modeling. Using large-scale real-space quantum-transport methods, it first establishes controlled transport fingerprints that distinguish conventional Bloch propagation in periodic structures from the anomalous dynamics induced by quasiperiodic modulations. Building on this framework, the thesis analyzes magic-angle twisted bilayer graphene and shows that, within a finite disorder window where flat-band features remain robust, moderate Anderson disorder can counterintuitively enhance the mean free path. This disorder-induced delocalization is further linked to changes in the quantum metric extracted from optical conductivity, revealing a direct connection between transport, electronic geometry, and the real-space extent of the underlying states. The study then turns to graphene quasicrystal approximants and hybrid multilayer stacks, identifying sub-ballistic transport and self-similar localization patterns as signatures of quasicrystalline order, while also demonstrating their strong fragility against disorder and interlayer proximity effects. Finally, the thesis addresses spin transport in suspended monolayer graphene, showing that atomic-scale corrugations generate short-range fluctuating Rashba fields that can limit spin lifetimes to the nanosecond range even when charge transport remains close to ballistic. Taken together, these results provide a unified picture of how geometry, disorder, and structural complexity govern transport phenomena in twisted and corrugated graphene systems.

Figures

Figures reproduced from arXiv: 2607.25411 by the authors.

Figure 2.1
Figure 2.1. a. Crystalline structure of graphene where the unit cell is shad￾owed and the colors correspond to the A (blue) and B (orange) sublat￾tices. b. First Brillouin zone of graphene where the equivalent K points are marked as triangles pointing up and the K’ with triangles pointing down. a1 =  √ 3 2 acc, 3 2 acc a2 =  − √ 3 2 acc, 3 2 acc . Here acc is the distance between two nearest neighbors. Thus, we can find the… view at source ↗
Figure 2.2
Figure 2.2. Decay of the different overlap functions used for this integral [PITH_FULL_IMAGE:figures/full_fig_p043_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Band structure computed from a nearest-neighbor approxima [PITH_FULL_IMAGE:figures/full_fig_p044_2_3.png] view at source ↗
Figures from the paper (53 more)
Figure 2.4
Figure 2.4. Figure 2.4: Different possible stackings of graphene and their correspond [PITH_FULL_IMAGE:figures/full_fig_p048_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Two cases of moire patterns. In panel a: two identical hexag- ´ onal (graphene) structures are rotated 12◦ . In panel B the unit cells are parallel but there is a mismatch between the lattice constant in red and blue unit cells. lattice vectors A1, A2, so that if we …
Figure 2.6
Figure 2.6. Figure 2.6: Mismatch between the first Brillouin zone of the top (red) and [PITH_FULL_IMAGE:figures/full_fig_p054_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: First Brillouin zone of the bottom (blue) and top (red) graphene [PITH_FULL_IMAGE:figures/full_fig_p058_2_7.png]
Figure 2.8
Figure 2.8. Figure 2.8: Band structure of two different angles for the twisted [PITH_FULL_IMAGE:figures/full_fig_p060_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: Band structure of the quasicrystal twisted bilayer graphene [PITH_FULL_IMAGE:figures/full_fig_p064_2_9.png]
Figure 2.10
Figure 2.10. Figure 2.10: (A) A low-energy electron diffraction pattern of graphene [PITH_FULL_IMAGE:figures/full_fig_p067_2_10.png]
Figure 3.1
Figure 3.1. Figure 3.1: Transport of a Gaussian pulse in a linear chain defined as in [PITH_FULL_IMAGE:figures/full_fig_p077_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Scheme of the workflow for the mean-field Hartree-Fock cal [PITH_FULL_IMAGE:figures/full_fig_p091_3_2.png]
Figure 4.1
Figure 4.1. Figure 4.1: Transport in a superperiodic linear chain. a: Top panel repre [PITH_FULL_IMAGE:figures/full_fig_p097_4_1.png]
Figure 4
Figure 4. Figure 4: a that the velocity first increases before reaching [PITH_FULL_IMAGE:figures/full_fig_p099_4.png]
Figure 4.2
Figure 4.2. Figure 4.2: Evolution of a Gaussian wave packet as a function of time for [PITH_FULL_IMAGE:figures/full_fig_p101_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: Shape of possible potentials emerging from equation (4.12) on [PITH_FULL_IMAGE:figures/full_fig_p105_4_3.png]
Figure 4.4
Figure 4.4. Figure 4.4: Density of states (left), diffusion coefficient (right) and elec [PITH_FULL_IMAGE:figures/full_fig_p107_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Density of states (left), diffusion coefficient in X (right) and [PITH_FULL_IMAGE:figures/full_fig_p109_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: Local densities of states in the valence and conduction band [PITH_FULL_IMAGE:figures/full_fig_p110_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: Figure similar to Fig. 4.4 for the quasicrystalline case. In the [PITH_FULL_IMAGE:figures/full_fig_p111_4_7.png]
Figure 4
Figure 4. Figure 4: summarizes our numerical results for this quasicrystalline [PITH_FULL_IMAGE:figures/full_fig_p112_4.png]
Figure 4.8
Figure 4.8. Figure 4.8: Projected density of states computed from (4.13) for top (blue) [PITH_FULL_IMAGE:figures/full_fig_p116_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Description of a device to model the electrostatic potential [PITH_FULL_IMAGE:figures/full_fig_p117_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: z profile for a Poisson in a gap with parameters extracted from [PITH_FULL_IMAGE:figures/full_fig_p121_4_10.png]
Figure 4.11
Figure 4.11. Figure 4.11: Top left panel shows the region of the unit cell where bands [PITH_FULL_IMAGE:figures/full_fig_p122_4_11.png]
Figure 5.1
Figure 5.1. Figure 5.1: Total density of states of MATBLG for disorder strengths of [PITH_FULL_IMAGE:figures/full_fig_p131_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Electronic spreading as a function of time for the clean system [PITH_FULL_IMAGE:figures/full_fig_p133_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Diffusion coefficient as a function of time for clean MATBLG [PITH_FULL_IMAGE:figures/full_fig_p134_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Time dependent diffusion coefficient at energies of [PITH_FULL_IMAGE:figures/full_fig_p135_5_4.png]
Figure 5
Figure 5. Figure 5: (bottom panel) shows [PITH_FULL_IMAGE:figures/full_fig_p137_5.png]
Figure 5.5
Figure 5.5. Figure 5.5: Top panel: mean free path for Anderson strengths of [PITH_FULL_IMAGE:figures/full_fig_p138_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: Top left: Density of states of 29.8 ◦ and 31◦ approximants, we have marked the quasicrystalline peaks α (purple), β (green) and γ (or￾ange). Bottom left: Diffusion coefficient of the 29.8 ◦ approximant as a function of time, we observe a sub-ballistic behavior at the…
Figure 5.7
Figure 5.7. Figure 5.7: Left: Time evolution of the diffusion coefficient of the [PITH_FULL_IMAGE:figures/full_fig_p142_5_7.png]
Figure 5.8
Figure 5.8. Figure 5.8: Middle layer of the QC+MATBLG bilayers for the [PITH_FULL_IMAGE:figures/full_fig_p145_5_8.png]
Figure 5.9
Figure 5.9. Figure 5.9: Density of states (top panel) and Fermi velocities (bottom [PITH_FULL_IMAGE:figures/full_fig_p146_5_9.png]
Figure 5
Figure 5. Figure 5: compares the DoS of MATBLG (reproduced from Fig. 5.1) in pur [PITH_FULL_IMAGE:figures/full_fig_p147_5.png]
Figure 5.10
Figure 5.10. Figure 5.10: Layer-projected DoS (bottom left panel) and LDoS of each [PITH_FULL_IMAGE:figures/full_fig_p148_5_10.png]
Figure 5.11
Figure 5.11. Figure 5.11: Time-dependent diffusion coefficient in the flat band for dif [PITH_FULL_IMAGE:figures/full_fig_p149_5_11.png]
Figure 5
Figure 5. Figure 5: ) presents the anomalous trend previously reported and discussed [PITH_FULL_IMAGE:figures/full_fig_p150_5.png]
Figure 5.12
Figure 5.12. Figure 5.12: DoS (top panel) and Fermi velocity (bottom panel) in [PITH_FULL_IMAGE:figures/full_fig_p151_5_12.png]
Figure 5.13
Figure 5.13. Figure 5.13: Local density of states of 29.8 ◦ -twisted bilayer graphene (left) and the two bottom layers of the 29.8 ◦+1.1 ◦ trilayer at the α peak (vertical dotted line in [PITH_FULL_IMAGE:figures/full_fig_p152_5_13.png]
Figure 5
Figure 5. Figure 5: ), reflecting the emergence of a dodecagonal order that is incom [PITH_FULL_IMAGE:figures/full_fig_p152_5.png]
Figure 5.14
Figure 5.14. Figure 5.14: Time-dependent diffusion coefficient of the quasicrystalline [PITH_FULL_IMAGE:figures/full_fig_p153_5_14.png]
Figure 6
Figure 6. Figure 6: , we observe a typical convergence behavior of the problem. De [PITH_FULL_IMAGE:figures/full_fig_p165_6.png]
Figure 6.1
Figure 6.1. Figure 6.1: Convergence behavior for θ ≈ 1.47◦ and U/V 0 ppπ = 2. Left: Evolution of the maximum magnetization with the number of iterations. We see the magnetization reach values comparable with the convergence threshold. Inset of the left panel shows the evolution of the conve…
Figure 6.2
Figure 6.2. Figure 6.2: Figure analogous to that of Fig. 6.1 for the ferromagnetic case. [PITH_FULL_IMAGE:figures/full_fig_p167_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: Top left: Evolution of magnetizations for the nanoribbon [PITH_FULL_IMAGE:figures/full_fig_p168_6_3.png]
Figure 7.1
Figure 7.1. Figure 7.1: Effects of different spin-orbit couplings in the [PITH_FULL_IMAGE:figures/full_fig_p171_7_1.png]
Figure 7.2
Figure 7.2. Figure 7.2: Schematic illustrations for Dyakonov-Perel (a) and Elliott [PITH_FULL_IMAGE:figures/full_fig_p175_7_2.png]
Figure 7.3
Figure 7.3. Figure 7.3: Example of the curvature modeled for Sec. 7.3. [PITH_FULL_IMAGE:figures/full_fig_p178_7_3.png]
Figure 7.4
Figure 7.4. Figure 7.4: Height (a) and local curvature (b) real-space maps for a ther [PITH_FULL_IMAGE:figures/full_fig_p182_7_4.png]
Figure 7.5
Figure 7.5. Figure 7.5: a: Diffusion coefficient (main) and mean-squared displacement [PITH_FULL_IMAGE:figures/full_fig_p184_7_5.png]
Figure 7
Figure 7. Figure 7: b shows the spin-lifetime anisotropy [PITH_FULL_IMAGE:figures/full_fig_p186_7.png]
Figure 7.6
Figure 7.6. Figure 7.6: a: Spin relaxation time in x of corrugated graphene as a func￾tion of chemical potential at different values for the thermalization temper￾ature. Dashed lines correspond to predictions from fluctuating SOC fields, see (7.20). The inset shows the scaling of τs with te…
Figure 7.7
Figure 7.7. Figure 7.7: Temperature scaling of the root-mean-squared [PITH_FULL_IMAGE:figures/full_fig_p189_7_7.png]
Figure 7
Figure 7. Figure 7: displays the temperature dependence of [PITH_FULL_IMAGE:figures/full_fig_p189_7.png]

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Works this paper leans on

14 extracted references · 1 linked inside Pith

  1. [1]

    J., Moon, P., Kim, T.-H., Kim, H.-W., Shin, H.-C., Kim, E

    [Ahn et al., 2018] Ahn, S. J., Moon, P., Kim, T.-H., Kim, H.-W., Shin, H.-C., Kim, E. H., Cha, H. W., Kahng, S.-J., Kim, P., Koshino, M., Son, Y .-W., Yang, C.-W., and Ahn, J. R. (2018). Dirac electrons in a dodecagonal graphene quasicrystal.Science, 361(6404):782–786. [Akzyanov et al., 2014] Akzyanov, R. S., Sboychakov, A. O., Rozhkov, A. V ., Rakhmanov,...

    2018

  2. [2]

    manuscript

    [Sichau et al., 2019] Sichau, J., Prada, M., Anlauf, T., Lyon, T. J., Bosn- jak, B., Tiemann, L., and Blick, R. H. (2019). Resonance microwave measurements of an intrinsic spin-orbit coupling gap in graphene: A possible indication of a topological state.Phys. Rev. Lett., 122:046403. [Sierra et al., 2021] Sierra, J. F., Fabian, J., Kawakami, R. K., Roche, ...

    2019

  3. [12]

    manuscript

    [Tong et al., 2022] Tong, L.-H., Tong, Q., Yang, L.-Z., Zhou, Y .-Y ., Wu, Q., Tian, Y ., Zhang, L., Zhang, L., Qin, Z., and Yin, L.-J. (2022). Spec- troscopic visualization of flat bands in magic-angle twisted monolayer- bilayer graphene: Coexistence of localization and delocalization.Phys. Rev. Lett., 128:126401. [T¨orm¨a et al., 2022] T ¨orm¨a, P., Peo...

    2022

  4. [35]

    manuscript

    Academic Press. [Haydock et al., 1972] Haydock, R., Heine, V ., and Kelly, M. J. (1972). Electronic structure based on the local atomic environment for tight- binding bands.Journal of Physics C: Solid State Physics, 5(20):2845. [Haydock et al., 1975] Haydock, R., Heine, V ., and Kelly, M. J. (1975). Electronic structure based on the local atomic environme...

    arXiv 1972

  5. [50]

    manuscript

    Springer Nature Singapore, Singapore. [Ames, 1997] Ames, W. F. (1997).Numerical methods for partial differ- ential equations.London: Academic Press. [Anderson, 1958] Anderson, P. W. (1958). Absence of diffusion in certain random lattices.Phys. Rev., 109:1492–1505. [Ashcroft and Mermin, 1976] Ashcroft, N. W. and Mermin, N. D. (1976). Solid state physics. H...

    1997

  6. [64]

    manuscript

    [Maci´a-Barber, 2020] Maci ´a-Barber, E. (2020).Quasicrystals: Funda- mentals and Applications. Taylor & Francis CRC Press. 186 “manuscript” — 2026/7/29 — 0:40 — page 187 — #219 [Mandelshtam and Taylor, 1995] Mandelshtam, V . A. and Taylor, H. S. (1995). A simple recursion polynomial expansion of the green’s func- tion with absorbing boundary conditions. ...

    2020

  7. [230]

    C., Yin, Y ., Tadich, A., Haldon, M., Solari, S., Riley, J., Huwald, E., Daniels, K

    [Grubisic-Cabo et al., 2024] Grubisic-Cabo, A., Kotsakidis, J. C., Yin, Y ., Tadich, A., Haldon, M., Solari, S., Riley, J., Huwald, E., Daniels, K. M., Myers-Ward, R. L., Edmonds, M. T., Medhekar, N. V ., Gaskill, D. K., and Fuhrer, M. S. (2024). Quasi-freestanding aa-stacked bilayer graphene induced by calcium intercalation of the graphene-silicon car- b...

    2024

  8. [306]

    manuscript

    [Wu et al., 2018] Wu, F., Lovorn, T., Tutuc, E., and MacDonald, A. (2018). Hubbard model physics in transition metal dichalcogenide moir´e bands.Physical Review Letters, 121(2). [Xie et al., 2021] Xie, Y ., Pierce, A. T., Park, J. M., Parker, D. E., Khalaf, E., Ledwith, P., Cao, Y ., Lee, S. H., Chen, S., Forrester, P. R., Watan- abe, K., Taniguchi, T., V...

    2018

Show all 14 references
  1. [584]

    manuscript

    [Park et al., 2008] Park, C.-H., Yang, L., Son, Y .-W., Cohen, M. L., and Louie, S. G. (2008). New generation of massless dirac fermions in graphene under external periodic potentials.Phys. Rev. Lett., 101:126804. [Park et al., 2021] Park, J. M., Cao, Y ., Watanabe, K., Tanigu...

    2008

  2. [1215]

    manuscript

    [Quarteroni et al., 2000] Quarteroni, A., Sacco, R., and Saleri, F. (2000). Numerical Mathematics. Springer. [Raes et al., 2017] Raes, B., Cummings, A. W., Bonell, F., Costache, M. V ., Sierra, J. F., Roche, S., and Valenzuela, S. O. (2017). Spin pre- cession in anisotropic me...

    2000

  3. [1524]

    manuscript

    [Song et al., 2018] Song, K., Soriano, D., Cummings, A. W., Rob- les, R., Ordej ´on, P., and Roche, S. (2018). Spin proximity ef- fects in graphene/topological insulator heterostructures.Nano Letters, 18(3):2033–2039. [Souza et al., 2000] Souza, I., Wilkens, T., and Martin, R....

    2018

  4. [2023]

    manuscript

    [Guerrero et al., 2025a] Guerrero, P. A., Nguyen, V .-H., Cummings, A. W., Charlier, J.-C., and Roche, S. (2025a). Proximity effects between a graphene quasicrystal and magic-angle twisted bilayer graphene. Phys. Rev. B, 112:214202. [Guerrero et al., 2025b] Guerrero, P. A., Ng...

    2026

  5. [2024]

    [Vidarte et al., 2025] Vidarte, K. J. U., Riffo, F. P., Morell, E. S., and Lewenkopf, C. (2025). Stoner ferromagnetism in low-angle twisted bi- layer graphene at three-quarters filling.Physical Review B, 111(4). [Weiße et al., 2006] Weiße, A., Wellein, G., Alvermann, A., and F...

    2025

  6. [3971]

    [Tilak et al., 2021] Tilak, N., Lai, X., Wu, S., Zhang, Z., Xu, M., Ribeiro, R., Canfield, P., and Andrei, E

    [The Elk Code, ] The Elk Code.http://elk.sourceforge.net/. [Tilak et al., 2021] Tilak, N., Lai, X., Wu, S., Zhang, Z., Xu, M., Ribeiro, R., Canfield, P., and Andrei, E. (2021). Flat band carrier confinement in magic-angle twisted bilayer graphene.Nature Communications,

    2021

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