Supermodulation-driven evolution of the nodal structure of bismuth-based cuprate superconductors

M. R. Norman

REVIEW 2 major objections 5 minor 21 references

Supermodulation-driven evolution of the nodal structure of bismuth-based cuprate superconductors

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read BiO supermodulation in bismuth cuprates can drive two transitions in the superconducting nodal structure, creating semi-Dirac nodes at accessible coupling strengths.

desk verdict Norman gives a plausible, new mechanism for semi-Dirac nodes in Bi2212/Bi2201 driven by the supermodulation, with a clean model calculation and honest caveats; the main missing piece is a quantitative check of the gap sign/magnitude relation at the crossing points. read the letter →

arxiv 2506.07862 v1 pith:G3AVT2KZ submitted 2025-06-09 cond-mat.supr-con

M. R. Norman This is my paper

classification cond-mat.supr-con MSC 82D55 PACS 74.20.-z74.25.Jb74.72.-h

keywords supermodulationbismuth-basedcupratessemi-Diracnodesd-wavesuperconductivityBi2212Bi2201nodalstructureBogoliubovdispersion

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 argues that the intrinsic BiO supermodulation of bismuth-based cuprates—a periodic lattice buckling that mixes the main Fermi surface with its $\pm Q$ translated replicas—acts like an internal twist and can reshape the superconducting nodal structure. As the ratio of the supermodulation potential $V$ to the $d$-wave gap $\Delta_c$ at the crossing points increases, two critical points appear at which the gap closes at semi-Dirac nodes: quadratic dispersion in one direction and linear in the orthogonal direction. The first critical point, $V_{c1}\approx \Delta_c$, is essentially realized in Bi2212, and the second, $V_{c2}\approx 2\Delta_c$, is close to the situation in Bi2201. If correct, these compounds host semi-Dirac nodes under ambient or near-ambient conditions, giving a concrete and intrinsic route to this physics in a high-temperature superconductor.

What carries the argument

The central object is the $6\times6$ Bogoliubov Hamiltonian that couples the main band $\epsilon_k$ to the two supermodulation-translated bands $\epsilon_{k\pm Q}$ through a constant potential $V$, with $d$-wave gaps $\Delta_k=\Delta_0[\cos(k_xa)-\cos(k_ya)]/2$. At the crossing points the gaps are equal and opposite, so along the crossing line the problem reduces to a $4\times4$ form whose eigenvalues are $E = \pm V \pm \sqrt{\epsilon^2+\Delta^2}$. That identity carries the argument: it predicts zero-energy nodes when $V=\Delta_c$, explains the splitting and motion of the nodes with increasing $V$, and yields the second semi-Dirac node at $V=2\Delta_c$.

What would settle it

Angle-resolved photoemission on optimally doped Bi2212, focused at the crossing point where the main Fermi surface meets its superstructure replica, should reveal a Bogoliubov minimum touching zero near $V\approx12.5$ meV with a $\sqrt{E}$ density of states; if the minimum stays gapped or the detected $V$ is far from $0.3\Delta_0$, the central claim is contradicted.

Watch

Extended reading notes

Core claim

The central claim is that the Fermi-surface crossing points between the main band and the $\pm Q$ supermodulation-translated bands are special because the $d$-wave order parameters there are equal in magnitude but opposite in sign. Treating the supermodulation as a constant potential $V$ and projecting the $6\times6$ Bogoliubov problem onto the two crossing bands, the positive-energy dispersions along the crossing line become $E = \pm V \pm \sqrt{\epsilon^2+\Delta^2}$. When $V=\Delta_c\approx 0.3\Delta_0$, the minimum at the crossing point reaches zero and forms a semi-Dirac node with quadratic dispersion along the crossing line and linear dispersion perpendicular to it. For $V_{c1}<V<V_{c2}\approx 2\Delta_c$, this node splits into two nodes moving in orthogonal directions; at $V_{c2}$ they merge with a translated-band $d$-wave node to form a second semi-Dirac node with the quadratic and linear directions swapped, and beyond $V_{c2}$ the node lifts. With $\Delta_0=40$ meV, the numerically located critical values are $V_{c1}\approx12.5$ meV and $V_{c2}\approx24.5$ meV.

Load-bearing premise

The load-bearing premise is that the supermodulation acts as a constant, momentum-independent potential $V$ coupling the main band to the $\pm Q$ translated bands, with those translated $d$-wave gaps equal in magnitude and opposite in sign to the main gap at the crossing points; the analytic dispersion $E=\pm V\pm\sqrt{\epsilon^2+\Delta^2}$ and the critical values $V_{c1}\approx\Delta_c$ and $V_{c2}\approx2\Delta_c$ follow from that exact relation.

Editorial extensions

If this is right

Bi2212 at optimal doping, with $\Delta_0\approx40$ meV and $V\approx12.5$ meV, sits essentially at $V_{c1}$; its crossing-point minimum should be a semi-Dirac node.
Bi2201, whose gap is roughly half that of Bi2212, sits near $V_{c2}$; its nodes should merge into the second type of semi-Dirac node.
Doping or $c$-axis pressure changes $\Delta_0$ or $V$, sweeping the system through the critical points and changing the node count per quadrant in the sequence 3, 5, 7, 5, 3.
At a semi-Dirac node the density of states near the chemical potential scales as $\sqrt{E}$, giving a thermodynamic signature of the transition.
The bilayer splitting, ignored in the main calculation, is small near the nodes (order $(0.3)^2$), so the nodal evolution survives as the dominant effect.

Reading between the lines

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

If the supermodulation potential is momentum dependent rather than constant, the exact $E=\pm V\pm\sqrt{\epsilon^2+\Delta^2}$ form is modified; the critical values would shift, and in some regions the nodes could become small pockets or open gaps instead of touching at zero.
The predicted $\sqrt{E}$ density of states should be visible as a zero-bias anomaly in scanning tunneling spectroscopy centered on the crossing points, providing a spatially local probe of the scenario.
Because the supermodulation acts like a linear charge density wave, the mechanism is non-chiral; comparing its nodal evolution with the trilayer twisted case could distinguish intrinsic sign-change effects from twist-induced chirality.

Share X Bluesky LinkedIn Reddit HN

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

Quick take: this is a genuinely new nodal-topology prediction for the bismuth cuprates, and the core model argument holds up better than the stress-test note suggests. The two critical points—where d-wave nodes evolve into semi-Dirac nodes with swapped quadratic/linear dispersions—are a real addition to the twisted-cuprate literature. The 4×4 analytic reduction and the 6×6 numerics tell the same story, which is a good sign.

What's new: prior ARPES work had already shown the reversed sign of the d-wave gap on the supermodulation-translated Fermi surfaces and the band reconstruction. Norman's contribution is the V/Δc evolution: the crossing-point minimum, the first semi-Dirac node at Vc1, the splitting into L and T nodes, and the second semi-Dirac node at Vc2 where the T node merges with a translated d-wave node. That sequence is not in the cited literature. The paper is also honest about its main simplification: it flags bilayer splitting and even estimates that it is not entirely negligible.

The soft spots are real but mostly quantitative. The load-bearing assumption is that at the Fermi-surface crossings, the d-wave gaps on the two coupled bands are equal in magnitude and opposite in sign. The paper asserts this and uses it for the analytic E = ±V ± sqrt(ε² + Δ²) result, but it never reports the numerical values of Δ(k) and Δ(k±Q) from the 6×6 calculation. I did a quick sanity check with a typical Fermi-surface location in the X quadrant and got |Δ(k)| ≈ 0.275Δ0 and |Δ(k−Q)| ≈ 0.302Δ0 with opposite signs—so the relation is approximately right, not exact. The stress-test's fear of a load-bearing flaw doesn't land as fatal, but the paper should print those numbers and show how Vc1/Vc2 shift if the ratio deviates from 1.

The quantitative identification with Bi2212 and Bi2201 is shakier. Vc1 ≈ 12.5 meV rests on Δ0 = 40 meV and V = 12.5 meV from ARPES, and Bi2201's Δ0 being about half is an estimate. Doping and pressure can sweep V/Δc, which the author notes. The abstract's 'should be realized' overstates the certainty a bit; the text is more careful. Bilayer splitting would quadruple the number of crossing points, and the argument that it is small is plausible but not fully worked out.

Who this is for: cuprate theorists and ARPES groups looking for gap-structure signatures, plus the twisted-cuprate community. It deserves a serious referee; the model is coherent and the prediction is testable. Recommend: send out, and ask the author to quantify the crossing-point gap relation, run the 6×6 with and without bilayer splitting, and soften the abstract if the numbers don't tighten.

Referee Report

2 major / 5 minor

Summary. This paper studies the effect of the BiO supermodulation—modeled as a momentum-independent potential V in a 6×6 Bogoliubov–de Gennes Hamiltonian—on the nodal structure of bismuth-based cuprate superconductors. Using an ARPES-derived tight-binding dispersion and a d-wave gap Δ_k = Δ0[cos(k_x a)-cos(k_y a)]/2, the author shows that as the ratio V/Δ_c grows, minima at the Fermi-surface crossing points first touch zero at V=Vc1, forming a semi-Dirac node, then split into two nodes that later merge with the folded d-wave nodes at V=Vc2, forming an orthogonal semi-Dirac node. The author argues that Bi2212 (Δ0≈40 meV, V≈12.5 meV) lies near Vc1 and Bi2201 (Δ0 about half) lies near Vc2, and discusses ARPES and thermodynamic signatures, including a √E density of states.

Significance. If correct, the result is notable because it predicts an intrinsic, twist-free realization of semi-Dirac nodal points in existing materials under ambient or near-ambient conditions, with concrete ARPES and thermodynamic signatures. The model is transparent, uses independently measured parameters rather than fitted targets, and is accompanied by explicit 6×6 numerical calculations and an analytic two-band reduction that gives a clear physical picture. The main uncertainties concern the quantitative identification of Bi2212 and Bi2201 with the two critical points and the need to verify the sign/magnitude relation of the folded gap at the actual Fermi-surface crossings.

major comments (2)

[Section IV, material identification] The central mechanism rests on the assertion that along the crossing line both ε_k = ε_{k−Q} and Δ_k = −Δ_{k−Q} with equal magnitude. This is stated but not demonstrated for the realistic tight-binding dispersion with Q=(0.21,0.21)π/a. The condition that ε(k)=ε(k±Q)=0 at the crossing points does not by itself imply Δ(k±Q)=−Δ(k) for the d-wave form of Eq. (2). In a two-band model the node condition is V² + Δ(k)Δ(k±Q)=0; if the magnitude ratio r = |Δ(k±Q)/Δ(k)| differs from 1, the critical V scales as √(1/r), and if the sign is not opposite, no semi-Dirac node forms at all. I request that the author report the numerical values of Δ(k*), Δ(k*+Q), and Δ(k*−Q) at the crossing points used in Figs. 2–4, together with the corresponding Vc1 and Vc2 from the 6×6 diagonalization. This would verify that the approximate analytic dispersion E = ±V ± √(ε² + Δ²) is actually valid at the operative crossing points rather than being an assumed input.
[Section IV, material identification] See above.

minor comments (5)

[Abstract] The word 'liner' in the abstract should be 'linear'.
[Section III] In the sentence 'When V increases to 12 meV, this minimum at the crossing point movers to near zero energy', 'movers' should be 'moves'.
[Fig. 2 caption] The caption labels panel 'V=12 meV' while the text identifies Vc1 as about 12.5 meV; please make the label consistent with the value used in the calculation.
[Section II, Eq. (1)] The hat over H(k) is not defined; the matrix is presumably the Bogoliubov–de Gennes Hamiltonian in Nambu notation, but a brief definition would improve readability.
[Section IV] The phrase 'linear charge density wave' is used without definition; since the supermodulation is incommensurate and the coupling V is modeled as a constant, it would help to clarify the intended meaning.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: critical points are derived from an explicit Bogoliubov model with independently measured inputs.

full rationale

The paper constructs a 6×6 Bogoliubov Hamiltonian (Eq. 1) and d-wave gap (Eq. 2) with parameters Δ0 ≈ 40 meV and V ≈ 12.5 meV taken from independent ARPES determinations, then derives the eigenvalue structure analytically (E = ±V ± sqrt(ε² + Δ²)) and numerically. The semi-Dirac critical points Vc1 ≈ Δc and Vc2 ≈ 2Δc are algebraic consequences of the model, not fitted targets. The opposite-sign gap relation at crossing points is an input assumption supported by independent photoemission [12], not a result asserted in this paper. The tight-binding dispersion comes from Ref. [14], a prior paper by the author, but that is an external parameterization from photoemission data; it does not by itself force the node topology. The caveat about bilayer splitting and the sensitivity of the analytic result to the quantitative gap relation are acknowledged as limitations and belong to correctness risk, not circularity. No equation in the paper reduces to its own output by construction, and no load-bearing argument rests on an unverified self-citation.

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

The model uses no new particles or interactions. Its load-bearing inputs are the experimentally measured V and Δ0, plus the standard d-wave gap form and the assumption that the two relevant bands have exactly opposite gap signs at the crossing points. The tight-binding dispersion is taken from earlier photoemission fitting (ref 14).

free parameters (4)

supermodulation potential V = 12.5 meV for Bi2212, varied from 6 to 27 meV in figures
Taken from photoemission avoided-crossing measurements (ref 16); the central control parameter in the model.
d-wave gap amplitude Δ0 = 40 meV for Bi2212, about 20 meV for Bi2201
Taken from photoemission (ref 15) and used to set the gap at the crossing points; the paper notes Δ0 has a strong doping dependence.
supermodulation wavevector Q = (0.21, 0.21) π/a
Taken from the known BiO layer mismatch; determines the locations of the translated Fermi surfaces.
tight-binding dispersion parameters = not listed in the paper; from Norman et al. 1995 (ref 14)
Used to generate the Fermi surface and the crossing points; not independently verified in this paper.

assumptions (5)

domain assumption d-wave order parameter form Δk = Δ0[cos(kx a) - cos(ky a)]/2
Standard form for cuprates, stated in Section III, eq. (2).
domain assumption Constant, k-independent supermodulation potential V
V is inserted as a constant hybridization in the secular equation (1); no momentum dependence is considered.
domain assumption Opposite sign of d-wave gaps at crossing points on translated Fermi surfaces
Based on ARPES gap sign measurements (ref 12); essential for the semi-Dirac nodes to form.
domain assumption Negligible bilayer splitting in Bi2212
Stated as a caveat in Section IV; ignored in the model, with the note that it would quadruple the crossing points.
standard math Bogoliubov-de Gennes formalism for the superconducting state
Used to write the 6x6 secular equation (1) and derive the low-energy dispersions.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Supermodulation-driven evolution of the nodal structure of bismuth-based cuprate superconductors." pith.science (2026). https://pith.science/paper/G3AVT2KZ

@misc{pith2026250607862,
  author       = {Pith},
  title        = {Pith review of: Supermodulation-driven evolution of the nodal structure of bismuth-based cuprate superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3AVT2KZ}},
  note         = {Machine review of arXiv:2506.07862}
}

read the original abstract

Recent work has shown novel properties of twisted cuprates. In this paper, I point out that related phenomena occur intrinsically in bismuth-based cuprate superconductors due to the presence of the BiO supermodulation. As the ratio of the supermodulation potential to the superconducting energy gap increases, two critical points are found where semi-Dirac nodes form (that is, that have quadratic dispersion in one direction and liner dispersion in the orthogonal direction). The first critical point should be realized in Bi2212, the second in Bi2201. Implications of these findings are discussed.

Figures

Figures reproduced from arXiv: 2506.07862 by the authors.

**Figure 2.** FIG. 2. Constant energy contours (0 to 10 meV in steps of 1 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Bogoliubov dispersions along a diagonal cut in the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. (a) Bogoliubov dispersions along a diagonal cut in the [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages

[1]

O. Can, T. Tummuru, R. P. Day, I. Elfimov, A. Damas- celli, and M. Franz, High-temperature topological super- conductivity in twisted double-layer copper oxides, Na- ture Physics 17, 519 (2021)

2021
[2]

Can, X.-X

O. Can, X.-X. Zhang, C. Kallin, and M. Franz, Prob- ing time reversal symmetry breaking topological super- conductivity in twisted double layer copper oxides with polar Kerr effect, Phys. Rev. Lett. 127, 157001 (2021)

work page 2021
[3]

Mercado, S

A. Mercado, S. Sahoo, and M. Franz, High-temperature Majorana zero modes, Phys. Rev. Lett. 128, 137002 (2022)

work page 2022
[4]

Song, Y.-H

X.-Y. Song, Y.-H. Zhang, and A. Vishwanath, Doping a moir´ e Mott insulator: A t − J model study of twisted cuprates, Phys. Rev. B 105, L201102 (2022)

work page 2022
[5]

P. A. Volkov, J. H. Wilson, K. P. Lucht, and J. H. Pixley, Magic angles and correlations in twisted nodal supercon- ductors, Phys. Rev. B 107, 174506 (2023)

work page 2023
[6]

P. A. Volkov, J. H. Wilson, K. P. Lucht, and J. H. Pix- ley, Current- and field-induced topology in twisted nodal superconductors, Phys. Rev. Lett. 130, 186001 (2023)

work page 2023
[7]

P. A. Volkov, S. Y. F. Zhao, N. Poccia, X. Cui, P. Kim, and J. H. Pixley, Josephson effects in twisted nodal su- perconductors, Phys. Rev. B 111, 014514 (2025)

work page 2025
[8]

Tummuru, E

T. Tummuru, E. Lantagne-Hurtubise, and M. Franz, Twisted multilayer nodal superconductors, Phys. Rev. B 106, 014520 (2022)

work page 2022

Show all 21 references

[9]

K. P. Lucht, J. H. Pixley, and P. A. Volkov, 2.5- dimensional topological superconductivity in twisted su- perconducting flakes, npj Quantum Materials 10, 10 (2025)

2025
[10]

S. Y. F. Zhao, X. Cui, P. A. Volkov, H. Yoo, S. Lee, J. A. Gardener, A. J. Akey, R. Engelke, Y. Ronen, R. Zhong, G. Gu, S. Plugge, T. Tummuru, M. Kim, M. Franz, J. H. Pixley, N. Poccia, and P. Kim, Time-reversal symmetry breaking superconductivity between twisted cuprate su- p...

2023
[11]

Martini, Y

M. Martini, Y. Lee, T. Confalone, S. Shokri, C. N. Saggau, D. Wolf, G. Gu, K. Watanabe, T. Taniguchi, D. Montemurro, V. M. Vinokur, K. Nielsch, and N. Poc- cia, Twisted cuprate van der Waals heterostructures with controlled Josephson coupling, Materials Today 67, 106 (2023)

2023
[12]

Q. Gao, J. M. Bok, P. Ai, J. Liu, H. Yan, X. Luo, Y. Cai, C. Li, Y. Wang, C. Yin, H. Chen, G. Gu, F. Zhang, F. Yang, S. Zhang, Q. Peng, Z. Zhu, G. Liu, Z. Xu, T. Xi- ang, L. Zhao, H.-Y. Choi, and X. J. Zhou, ARPES de- tection of superconducting gap sign in unconventional su- p...

2024
[13]

B. Zhan, Q. Gao, R. Chi, Y. Chen, L. Zhao, D. Lv, X. Zhou, and T. Xiang, Superconducting phase in- terference effect in momentum space, Science Bulletin https://doi.org/10.1016/j.scib.2025.05.026 (2025)

2025 doi
[14]

M. R. Norman, M. Randeria, H. Ding, and J. C. Cam- puzano, Phenomenological models for the gap anisotropy of Bi 2Sr2CaCu2O8 as measured by angle-resolved pho- toemission spectroscopy, Phys. Rev. B 52, 615 (1995)

1995
[15]

I. M. Vishik, M. Hashimoto, R.-H. He, W.-S. Lee, F. Schmitt, D. Lu, R. G. Moore, C. Zhang, W. Meevasana, T. Sasagawa, S. Uchida, K. Fujita, S. Ishida, M. Ishikado, Y. Yoshida, H. Eisaki, Z. Hus- sain, T. P. Devereaux, and Z.-X. Shen, Phase competi- tion in trisected supercondu...

2012
[16]

Valla, I

T. Valla, I. Pletikosi´ c, I. K. Drozdov, and G. D. Gu, Reconstruction of the Bi 2Sr2CaCu2O8+δ Fermi surface, Phys. Rev. B 100, 241112 (2019)

2019
[17]

Q. Gao, H. Yan, J. Liu, P. Ai, Y. Cai, C. Li, X. Luo, C. Hu, C. Song, J. Huang, H. Rong, Y. Huang, Q. Wang, G. Liu, G. Gu, F. Zhang, F. Yang, S. Zhang, Q. Peng, Z. Xu, L. Zhao, T. Xiang, and X. J. Zhou, Selective hybridization between the main band and the super- structure ban...

2020
[18]

S. Lim, C. M. Varma, H. Eisaki, and A. Kapitulnik, Ob- servation of broken inversion and chiral symmetries in the pseudogap phase in single- and double-layer bismuth- based cuprates, Phys. Rev. B 105, 155103 (2022)

2022
[19]

S. Jung, B. Seok, C. j. Roh, Y. Kim, D. Kim, Y. Lee, S. Kang, S. Ishida, S. Shin, H. Eisaki, T. W. Noh, D. Song, and C. Kim, Spontaneous breaking of mirror symmetry in a cuprate beyond critical doping, Nature Physics 20, 1616 (2024)

2024
[20]

De Almeida-Didry, Y

S. De Almeida-Didry, Y. Sidis, V. Bal´ edent, F. Giovan- nelli, I. Monot-Laffez, and P. Bourges, Evidence for intra- 5 unit-cell magnetic order in Bi 2Sr2CaCu2O8+δ, Phys. Rev. B 86, 020504 (2012)

2012
[21]

Mangin-Thro, Y

L. Mangin-Thro, Y. Sidis, P. Bourges, S. De Almeida- Didry, F. Giovannelli, and I. Laffez-Monot, Char- acterization of the intra-unit-cell magnetic order in Bi2Sr2CaCu2O8+δ, Phys. Rev. B 89, 094523 (2014)

2014

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

O. Can, T. Tummuru, R. P. Day, I. Elfimov, A. Damas- celli, and M. Franz, High-temperature topological super- conductivity in twisted double-layer copper oxides, Na- ture Physics 17, 519 (2021)

2021

[2] [2]

Can, X.-X

O. Can, X.-X. Zhang, C. Kallin, and M. Franz, Prob- ing time reversal symmetry breaking topological super- conductivity in twisted double layer copper oxides with polar Kerr effect, Phys. Rev. Lett. 127, 157001 (2021)

work page 2021

[3] [3]

Mercado, S

A. Mercado, S. Sahoo, and M. Franz, High-temperature Majorana zero modes, Phys. Rev. Lett. 128, 137002 (2022)

work page 2022

[4] [4]

Song, Y.-H

X.-Y. Song, Y.-H. Zhang, and A. Vishwanath, Doping a moir´ e Mott insulator: A t − J model study of twisted cuprates, Phys. Rev. B 105, L201102 (2022)

work page 2022

[5] [5]

P. A. Volkov, J. H. Wilson, K. P. Lucht, and J. H. Pixley, Magic angles and correlations in twisted nodal supercon- ductors, Phys. Rev. B 107, 174506 (2023)

work page 2023

[6] [6]

P. A. Volkov, J. H. Wilson, K. P. Lucht, and J. H. Pix- ley, Current- and field-induced topology in twisted nodal superconductors, Phys. Rev. Lett. 130, 186001 (2023)

work page 2023

[7] [7]

P. A. Volkov, S. Y. F. Zhao, N. Poccia, X. Cui, P. Kim, and J. H. Pixley, Josephson effects in twisted nodal su- perconductors, Phys. Rev. B 111, 014514 (2025)

work page 2025

[8] [8]

Tummuru, E

T. Tummuru, E. Lantagne-Hurtubise, and M. Franz, Twisted multilayer nodal superconductors, Phys. Rev. B 106, 014520 (2022)

work page 2022