Three-Dimensional Topological Twistronics

Fengcheng Wu; Rui-Xing Zhang; Sankar Das Sarma

Three-Dimensional Topological Twistronics

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A generalized Bloch theorem, built on the screw symmetry of a chiral twisted stack, turns a three-dimensional twisted crystal into an effective $k_z$-labeled Bloch Hamiltonian and predicts Weyl nodes, type-I/type-II transitions, and two…

desk verdict A genuinely new framework for 3D twistronics with concrete predictions; the main approximation is a legitimate weak spot but likely shifts numbers rather than invalidating the physics. read the letter →

arxiv 1909.01350 v3 pith:GFKX5TDI submitted 2019-09-03 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el

Fengcheng Wu , Rui-Xing Zhang , Sankar Das Sarma This is my paper · ORCID

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

keywords 3DtwistronicsgeneralizedBlochtheoremnonsymmorphicsymmetrychiraltwistedgraphiteWeylfermionsmagicanglegaugefieldmoirésuperlattice

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

Three-dimensional twistronics asks what happens when every layer of a stack is twisted by the same small angle $\theta$ relative to the previous one, so the whole crystal forms a continuous spiral. This paper claims that such a structure, despite breaking ordinary translational symmetry in every direction, still admits a Bloch-like description: its exact screw symmetry defines an effective out-of-plane momentum $k_z$, and the low-energy Hamiltonian becomes $H \approx \sum_{k_z}\int d^2r\, \psi^\dagger(r)[h(k_\parallel) + e^{ik_z}T(r) + e^{-ik_z}T^\dagger(r)]\psi(r)$. In chiral twisted graphite that Hamiltonian produces Weyl nodes whose character switches between type-I and type-II as $\theta$ is tuned, and two magic angles where the in-plane Fermi velocity vanishes. In a twisted Weyl semimetal the same machinery generates a chiral gauge field with a vortex-antivortex lattice whose core line modes form moiré-scale 3D Weyl fermions. If correct, the twist angle becomes a continuous tuning knob for three-dimensional topological band structure and for enhanced correlations near the magic angles.

What carries the argument

The load-bearing object is the nonsymmorphic screw symmetry: rotating each layer by $\theta$ and translating by one interlayer spacing $d_z$ leaves the Hamiltonian invariant. It generates the generalized Bloch wave $\psi_{k_z}(r) = N^{-1/2}\sum_n e^{-ink_z}\psi_n[\hat{R}(n\theta)r]$, which labels states by $k_z$ even though there is no ordinary out-of-plane lattice translation. The decisive simplification is the small-angle replacement $\hat{R}(\pm\theta)\to I$, which turns interlayer tunneling into the moiré-periodic potential $\Delta(k_z,r) = e^{ik_z}T(r)+e^{-ik_z}T^\dagger(r)$, restoring an in-plane moiré Brillouin zone and making Eq. (5) a genuine Bloch Hamiltonian. All the concrete predictions—Weyl nodes, critical angles, magic angles, and vortex-line modes—follow from diagonalizing this effective Hamiltonian in a plane-wave basis.

What would settle it

Take a finite chiral twisted stack of $N$ graphene layers using the unapproximated rotation in Eq. (4) and diagonalize it directly; if the Weyl node at $k_{1/2}=(0,0,\pi/2)$ does not exist, or if the in-plane velocity $v_\parallel$ does not vanish near $\theta\approx1.09^\circ$ and $1.67^\circ$ once $N$ and the lateral size approach $a_M/\theta$, then the effective Hamiltonian in Eq. (5) is not the right description. Experimentally, angle-resolved photoemission or high-field quantum oscillations on a chiral twisted graphite sample of controlled $\theta$ could look for the predicted vanishing in-plane velocity and the Weyl-node band crossings.

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Extended reading notes

Core claim

The central discovery is that a three-dimensional layered crystal with constant interlayer twist $\theta$, which breaks all ordinary translational symmetries, still has an exact nonsymmorphic symmetry: an in-plane rotation by $\theta$ combined with an out-of-plane translation by one interlayer spacing. This symmetry defines a generalized Bloch wave $\psi_{k_z}(r) = N^{-1/2}\sum_n e^{-ink_z}\psi_n[\hat{R}(n\theta)r]$, so $k_z$ is a good quantum number. After approximating the small-angle rotation $\hat{R}(\pm\theta)$ by the identity, the Hamiltonian takes the effective Bloch form $H \approx \sum_{k_z}\int d^2r\, \psi^\dagger(r)[h(k_\parallel)+e^{ik_z}T(r)+e^{-ik_z}T^\dagger(r)]\psi(r)$, which is valid for system sizes below $a_M/\theta$. In chiral twisted graphite this Hamiltonian yields Weyl nodes protected by $\hat{C}_{3z}$ and $\hat{C}_{2z}\hat{T}$; the node at $k_{1/2}=(0,0,\pi/2)$ undergoes type-I to type-II transitions at $\theta\approx1.22^\circ$ and $1.52^\circ$, and its in-plane velocity vanishes at magic angles $\theta\approx1.09^\circ$ and $1.67^\circ$. In the twisted Weyl semimetal, the twist produces a chiral gauge field with a vortex-antivortex lattice, and the vortex-core line modes combine into moiré-scale 3D Weyl fermions.

Load-bearing premise

The load-bearing step is replacing the rotation matrices $\hat{R}(\pm\theta)$ with the identity in going from Eq. (4) to Eq. (5), which neglects 'moiré of moiré' effects on the scale $a_M/\theta$; the paper states the resulting theory is accurate only for system sizes below $a_M/\theta$, and it also keeps only nearest-layer tunneling. If those neglected effects are not weak, the effective $k_z$ description and the predicted Weyl nodes, critical angles, and magic angles would shift or disappear.

Editorial extensions

If this is right

Chiral twisted graphite should host Weyl nodes along the $k_z$ axis; the node at $\mathbf{k}=(0,0,\pi/2)$ flips between type-I and type-II at $\theta\approx1.22^\circ$ and $1.52^\circ$, and the chirality of the $\gamma$-point node can change sign with $\theta$.
At the two magic angles $\theta\approx1.09^\circ$ and $1.67^\circ$, the in-plane Fermi velocity of that Weyl fermion vanishes, and the low-energy density of states per layer becomes orders of magnitude larger than in monolayer graphene, which should strengthen correlation effects and superconducting instability.
In a twisted Weyl semimetal, the chiral gauge field $\mathcal{A}$ has a vortex-antivortex lattice; the vortex cores bind line modes whose chirality alternates in space, and an out-of-plane electric field should drive real-space pumping of electrons between the two vortex-core sublattices.
Because the framework starts from a generic 2D building block and a generic interlayer tunneling, the same generalized Bloch theory applies to any chiral twisted stack of Dirac or Weyl layers, including photonic and phononic metamaterials.

Reading between the lines

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

If the $\hat{R}(\pm\theta)\approx I$ step is as accurate as claimed, the effective-$k_z$ picture should survive in finite spirals as long as lateral dimensions stay below $a_M/\theta$; a testable consequence is that the predicted magic angles would drift or split in samples approaching that size, which could be checked by finite-size exact diagonalization of the unapproximated Hamiltonian.
The magic-angle condition in chiral twisted graphite is set by the same dimensionless ratio $w/(\hbar v_F|\mathbf{g}_1|)$ that controls twisted bilayer graphene, so other Dirac materials with different Fermi velocities or interlayer tunnelings should show 3D magic-angle Weyl physics at different twist angles.
The vortex-line-mode picture suggests a clean transport signature: a longitudinal electric field along the twist axis should produce a chiral, angle-tunable pumping current between the $R$ and $R_{1/2}$ vortex-core positions, measurable as nonlocal resistance or circular dichroism in a twisted Weyl stack.
Since chiral twisted nanowires have already been synthesized, the screw-symmetry Bloch construction may apply beyond planar van der Waals stacks to twisted nanowire or nanotube arrays, where each 'layer' is a ring of the spiral.

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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, this paper actually opens a new subfield: a generalized Bloch theorem based on the screw symmetry gives a well-defined out-of-plane kz for chiral twisted stacks, and the two applications (chiral twisted graphite, twisted Weyl semimetal) are concrete and testable. Second, the weak point is exactly where the stress-test note puts it: the step from Eq. (4) to Eq. (5) replaces R(±θ) by the identity, restoring in-plane translation symmetry that the real structure does not have. The authors are honest about this in SM S1, but the stated cutoff (system size below a_M/θ) does not cleanly apply to bulk predictions, which are statements about an infinite stack.

The paper does a lot well. The derivation is clear, and the numerical diagonalizations are straightforward and support the qualitative picture. No fitting is done to the target predictions: wAA and wAB are external inputs, the Weyl semimetal parameters are external, and the magic angles and velocities are computed. The first magic angle is reproduced semiquantitatively by the analytic perturbation theory; the second is not (0.46° vs 1.09°), which suggests the truncated momentum-shell approximation, rather than necessarily the whole framework, is starting to break at small angles.

The stress-test concern is fair and worth taking seriously. At the magic angles, the unperturbed in-plane velocity vanishes, so any O(θ) correction to the velocity from the neglected R(−θ) is not small relative to the quantity being predicted. The type-I/type-II transitions are determined by the sign of v1, so a correction to v1 could move or even remove those critical angles. I would therefore read the quantitative angle values with caution. But the existence of Weyl nodes at k=0 and k=π/2 is symmetry-enforced within the effective model and likely robust to the approximation, and the vortex-line-mode picture in the twisted Weyl semimetal is supported by the real-space density plots. The main qualitative claims probably survive.

The citation pattern looks fine. References 19–21 are properly distinguished from the present work. This is a paper for the moiré and topological-semimetal communities, and I would bring it to a reading group.

Recommendation: send to serious peer review. I would ask the referees to scrutinize the R(θ) approximation at the magic angles and to request a quantitative estimate of the neglected terms, perhaps from a finite-stack calculation that keeps the exact rotation. But this is a publishable Letter after revision, not a desk reject.

Referee Report

0 major / 6 minor

Summary. The manuscript develops a generalized Bloch band theory for three-dimensional layered systems in which successive layers are twisted by a constant angle θ about a common axis. The construction uses an exact nonsymmorphic screw symmetry to define an effective out-of-plane momentum kz, and, after replacing the in-plane rotation matrices acting on field operators by the identity (Eq. (4)→(5)), yields a moiré-periodic Hamiltonian with a kz-dependent interlayer coupling. The authors apply the framework to two systems: chiral twisted graphite, predicting Weyl nodes on the kz axis, θ-tuned type-I/type-II transitions, and two magic angles where the in-plane Weyl velocity vanishes; and a twisted Weyl semimetal, predicting a chiral gauge field with a vortex-antivortex lattice, vortex line modes, and moiré-scale Weyl fermions. Supporting material includes an analytic perturbation theory for the first magic angle and plane-wave numerics.

Significance. The paper is original and opens a direction ('3D twistronics') with likely follow-up work. Its strengths are the elegant exact use of the screw symmetry, the use of external input parameters (wAA, wAB, and the Weyl-semimetal parameters) with no fitting of the target predictions, a parameter-free analytic estimate of the first magic angle, and concrete falsifiable predictions. The weakest point is the R(±θ)≈I approximation, which is load-bearing for all subsequent results; the authors are transparent about it in SM S1 (system-size caveat a_M/θ) and provide a perturbation-theory check. I considered the stress-test objection that the approximation becomes uncontrolled at the magic angles because v∥ vanishes there. While the exact twisted structure indeed lacks the artificial C3z and C2zT symmetries of the approximate model, the Weyl nodes are topologically stable and the rotation correction vanishes at k∥=0 where the nodes sit, so I do not regard the objection as a demonstrated failure; a quantitative estimate of the rotation corrections would nevertheless strengthen the paper.

minor comments (6)

[Chiral twisted graphite / SM S3] The main text says the analytic perturbation theory agrees 'quantitatively' with the full band-structure calculation for θ_M,1, but SM S3 reports θ*_M,1≈1.87° versus θ_M,1≈1.67° and describes the agreement as 'semiquantitative'; please align the wording.
[SM S1] The statement 'we assume open boundary condition along z and an infinite number of layers' is confusing, since an infinite stack has no boundary; please clarify that the system is infinite and the screw symmetry is exact for any θ, and that the open-boundary remark is only meant to explain why kz takes continuous values.
[Derivation of Eq. (5)] The small-angle approximation from Eq. (4) to Eq. (5) also implicitly neglects the spinor rotation in the Dirac kinetic term; please state this explicitly in the main text or in the SM so that the scope of the R≈I approximation is fully documented.
[Chiral twisted graphite, magic angles] The second magic angle θ_M,2≈1.09° from the plane-wave calculation differs substantially from the perturbation-theory value θ*_M,2≈0.46°; the main text should acknowledge this discrepancy rather than leaving it only in the SM.
[SM S1 / S3] Given the system-size caveat in SM S1 and the fact that the headline applications concern bulk infinite stacks, please add a short estimate in the SM of how the R(±θ) corrections renormalize v∥ and v1 near the magic and critical angles, so that the robustness of the predicted nodes and velocities is explicitly quantified.
[Fig. 1(b)] The labels θ_C,1 and θ_C,2 in Fig. 1(b) are hard to read at the plotted scale; consider enlarging the relevant angle range or adding a table of the numerical values of θ_M,1, θ_M,2, θ_C,1, and θ_C,2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's predictions are computed from externally specified inputs; the rotation-to-identity step is an approximation, not a definitional reuse of the target results.

full rationale

The derivation chain is self-contained and non-circular. The generalized Bloch transformation (Eq. 3) is an exact symmetry-based change of basis, producing Eq. (4); Eq. (5) then follows from an explicitly stated small-angle approximation in which R(±θ) is replaced by the identity, with the paper acknowledging the neglected O(θ^2) 'moiré of moiré' effects. This is a controlled physical approximation, not an input smuggled in as a prediction. All material parameters entering the two applications are external inputs: for chiral twisted graphite, vF, wAA, and wAB are taken from monolayer graphene and twisted-bilayer graphene literature; the magic angles are obtained by diagonalizing the resulting Hamiltonian and by an independent perturbation theory (SM S3) that gives a closed-form expression for v∥/vF, whose zeros are then computed. The type-I/type-II Weyl transitions follow from the sign of computed velocities v1 and v2, not from any fitted target. For the twisted Weyl semimetal, the model parameters M0, M1, tsp, vW, a0, and Qz are specified externally, and the vortex-line-mode picture and moiré-scale Weyl fermions are verified by direct numerical diagonalization of Eq. (7). The effective gauge field A and pseudo-magnetic field bz are derived rewritings of the interlayer tunneling, used interpretively, not as fitted predictions. No load-bearing self-citation, no uniqueness theorem imported from the authors' prior work, and no renamed known result appear in the argument. The admitted validity limitation of Eq. (5) at the magic angles is a substantive scientific concern about the approximation's accuracy, but it is not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on four modeling assumptions: the infinite-layer screw symmetry, the small-angle replacement of rotations by identity, the continuum Dirac description of graphene layers, and the two-center/local-approximation tunneling model for the Weyl semimetal. No parameter is fitted to the predicted magic angles or velocities.

free parameters (2)

wAA and wAB interlayer tunneling parameters = wAA ≈ 90 meV, wAB ≈ 117 meV
Taken from twisted bilayer graphene literature (Refs 1,17), not fitted to the predictions here; magic angle positions in chiral twisted graphite depend directly on their ratio.
Twisted Weyl semimetal model parameters (M0, M1, tsp, vW, a0, Qz) = M0=869 meV, M1=10.36 eV Å^2, tsp=4 meV, vW=3.74e5 m/s, a0=7.5 Å, Qz=π/2
Hand-chosen to define a minimal Weyl semimetal; the numerical demonstration (e.g., vW*(θ)) is model-dependent, but the qualitative vortex-line-mode picture does not rest on precise values.

assumptions (4)

domain assumption The chiral twisted stack with open boundary conditions and infinitely many layers has an exact nonsymmorphic screw symmetry (θ rotation plus out-of-plane translation).
Introduced in main text around Eq. (3); makes kz a good quantum number. Violated by finite N or periodic boundary unless θ is a rational multiple of 2π.
ad hoc to paper Small-angle approximation: in-plane rotation matrices R(±θ) acting on electron field operators can be replaced by the identity.
Used in the step from Eq. (4) to Eq. (5); error is O(θ|g|) and limits validity to system sizes below a_M/θ as stated in SM S1.
domain assumption Low-energy physics of each twisted graphene layer is captured by a Dirac k·p Hamiltonian at the +K valley with sublattice Pauli matrices σ.
Standard TBG continuum model (Refs 1,17); neglects intervalley scattering and higher-order terms.
ad hoc to paper Interlayer tunneling in twisted Weyl semimetal is derived with a two-center approximation and the local displacement d(r) ≈ θ z×r, retaining only the first shell of moiré reciprocal vectors; intraorbital tunneling spatial modulation is neglected.
SM S4; this model choice generates the vortex-antivortex gauge field and is an approximation, not an exact lattice description.

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Pith. "Pith review of Three-Dimensional Topological Twistronics." pith.science (2026). https://pith.science/paper/GFKX5TDI

@misc{pith2026190901350,
  author       = {Pith},
  title        = {Pith review of: Three-Dimensional Topological Twistronics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFKX5TDI}},
  note         = {Machine review of arXiv:1909.01350}
}

abstract

We introduce a theoretical framework for the new concept of three-dimensional (3D) twistronics by developing a generalized Bloch band theory for 3D layered systems with a constant twist angle $\theta$ between successive layers. Our theory employs a nonsymmorphic symmetry that enables a precise definition of an effective out-of-plane crystal momentum, and also captures the in-plane moir\'e pattern formed between neighboring twisted layers. To demonstrate the novel topological physics that can be achieved through 3D twistronics, we present two examples. In the first example of chiral twisted graphite, Weyl nodes arise because of inversion-symmetry breaking, with $\theta$-tuned transitions between type-I and type-II Weyl fermions, as well as magic angles at which the in-plane velocity vanishes. In the second example of twisted Weyl semimetal, the twist in the lattice structure induces a chiral gauge field $\boldsymbol{\mathcal{A}}$ that has a vortex-antivortex lattice configuration. Line modes bound to the vortex cores of the $\boldsymbol{\mathcal{A}}$ field give rise to 3D Weyl physics in the moir\'e scale. We also discuss possible experimental realizations of 3D twistronics.

Figures

Figures reproduced from arXiv: 1909.01350 by the authors.

**Figure 2.** FIG. 2. Results of chiral twisted graphite. (a) In-plane and (b) out-of-plane band structure. In (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Results of twisted Weyl semimetal. (a) Out-of-plane band structure in the twisted Weyl semimetal. In-plane momentum [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Works this paper leans on

33 extracted references · 15 canonical work pages

[1]

Bistritzer and A

R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci. U.S.A. 108, 12233 (2011)

2011
[2]

S. Carr, D. Massatt, S. Fang, P. Cazeaux, M. Luskin, and E. Kaxiras, Phys. Rev. B 95, 075420 (2017)

2017
[3]

Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018)

2018
[4]

Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- Herrero, Nature 556, 80 (2018)

2018
[5]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan- abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Science 363, 1059 (2019)

2019
[6]

X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, et al. , arXiv:1903.06513

arXiv 1903
[7]

A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Science 365, 605 (2019)

2019
[8]

Serlin, C

M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. Young, arXiv:1907.00261

arXiv 1907

Show all 33 references

[9]

San-Jose, J

P. San-Jose, J. Gonz´ alez, and F. Guinea, Phys. Rev. Lett. 108, 216802 (2012)

2012
[10]

H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Phys. Rev. X 8, 031089 (2018)

2018
[11]

Koshino, N

M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018)

2018
[12]

Kang and O

J. Kang and O. Vafek, Phys. Rev. X 8, 031088 (2018)

2018
[13]

J. Liu, J. Liu, and X. Dai, Phys. Rev. B 99, 155415 (2019)

2019
[14]

Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)

2019
[15]

A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Nature 527, 495 (2015)

2015
[16]

D. I. Pikulin, A. Chen, and M. Franz, Phys. Rev. X 6, 041021 (2016)

2016
[17]

J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007)

2007
[18]

See Supplemental Material at URL for a perturbation theory that estimates the ﬁrst magic angle in chiral twisted graphite, and a derivation of the interlayer tun- neling matrix in twisted Weyl semimetal, which includes Ref. 33
[19]

Khalaf, A

E. Khalaf, A. J. Kruchkov, G. Tarnopolsky, and A. Vish- wanath, Phys. Rev. B 100, 085109 (2019)

2019
[20]

T. Cea, N. R. Walet, and F. Guinea, Nano Lett. 19, 8683 (2019)

2019
[21]

C. Mora, N. Regnault, and B. A. Bernevig, Phys. Rev. Lett. 123, 026402 (2019)

2019
[22]

Sutter, S

P. Sutter, S. Wimer, and E. Sutter, Nature 570, 354 (2019)

2019
[23]

Y. Liu, J. Wang, S. Kim, H. Sun, F. Yang, Z. Fang, N. Tamura, R. Zhang, X. Song, J. Wen, B. Z. Xu, M. Wang, S. Lin, Q. Yu, K. B. Tom, Y. Deng, J. Turner, E. Chan, D. Jin, R. O. Ritchie, A. M. Minor, D. C. Chrzan, M. C. Scott, and J. Yao, Nature 570, 358 (2019)

2019
[24]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Sza- meit, Nature 496, 196 (2013)

2013
[25]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Science349, 622 (2015)

2015
[26]

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, et al. , Science 359, 1013 (2018)

2018
[27]

D. Wang, B. Yang, W. Gao, H. Jia, Q. Yang, X. Chen, M. Wei, C. Liu, M. Navarro-C´ ıa, J. Han, et al. , Nature Physics (2019)

2019
[28]

H. Jia, R. Zhang, W. Gao, Q. Guo, B. Yang, J. Hu, Y. Bi, Y. Xiang, C. Liu, and S. Zhang, Science 363, 148 (2019)

2019
[29]

Xiao, W.-J

M. Xiao, W.-J. Chen, W.-Y. He, and C. T. Chan, Nat. Phys. 11, 920 (2015)

2015
[30]

Yang and B

Z. Yang and B. Zhang, Phys. Rev. Lett. 117, 224301 (2016)

2016
[31]

F. Li, X. Huang, J. Lu, J. Ma, and Z. Liu, Nat. Phys. 14, 30 (2018)

2018
[32]

Dorrell, H

W. Dorrell, H. Pirie, S. M. Gardezi, N. C. Drucker, and J. E. Hoﬀman, arXiv:1909.13854 (2019)

2019 arXiv
[33]

moir´ e of moir´ e

C. Zheng, R. Hoﬀmann, and D. R. Nelson, J. Am. Chem. Soc. 112, 3784 (1990). 6 Supplemental Material This Supplemental Material includes four sections. Sections S1 and S2 provide additional discussion, respectively, on the generalized Bloch theory and on chiral twisted graphite...

1990

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bistritzer and A

R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci. U.S.A. 108, 12233 (2011)

2011

[2] [2]

S. Carr, D. Massatt, S. Fang, P. Cazeaux, M. Luskin, and E. Kaxiras, Phys. Rev. B 95, 075420 (2017)

2017

[3] [3]

Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018)

2018

[4] [4]

Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- Herrero, Nature 556, 80 (2018)

2018

[5] [5]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan- abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Science 363, 1059 (2019)

2019

[6] [6]

X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, et al. , arXiv:1903.06513

arXiv 1903

[7] [7]

A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Science 365, 605 (2019)

2019

[8] [8]

Serlin, C

M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. Young, arXiv:1907.00261

arXiv 1907