pith. machine review for the scientific record. sign in

arxiv: 2605.10509 · v1 · submitted 2026-05-11 · ❄️ cond-mat.mtrl-sci

Recognition: no theorem link

Rare-Earth-Tuned Evolution from d- to f-Orbital Dominance and Giant Anomalous Hall Effect in Topological RGaGe (R = Ce, Pr, Nd) Semimetals

Zhian Xu , Jian Yuan , Ze Yan , Xia Wang , Na Yu , Shihao Zhang , Yanfeng Guo
Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:07 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords Weyl semimetalanomalous Hall effectrare-earth germanidestopological magnetismorbital evolutioninversion symmetry breakingmagnetic anisotropyRGaGe
0
0 comments X

The pith

Rare-earth substitution in RGaGe tunes d- to f-orbital dominance while producing giant anomalous Hall effects up to 948 Ω⁻¹ cm⁻¹.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The RGaGe family (R = Ce, Pr, Nd) forms a noncentrosymmetric magnetic platform that extends the known RAlGe topological materials. Calculations show that inversion symmetry breaking creates Weyl points near the Fermi level whose Berry curvature produces a large intrinsic anomalous Hall conductivity, reaching 948 Ω⁻¹ cm⁻¹ at 2 K in PrGaGe. This contribution remains even above the magnetic ordering temperature, establishing its electronic rather than purely magnetic origin. The same calculations track a progressive change in orbital character: d-orbital dominance near the Fermi level in CeGaGe and PrGaGe gives way to significant f-orbital weight in NdGaGe. A sympathetic reader cares because the work supplies a chemically tunable handle on both the topological band features and the strong uniaxial magnetic anisotropy that aligns moments along the c-axis.

Core claim

The family of noncentrosymmetric rare-earth germanides RGaGe exhibits a pronounced uniaxial magnetic anisotropy with ferromagnetic ordering along the c-axis. A significantly enhanced intrinsic anomalous Hall conductivity is observed, reaching 948 Ω⁻¹ cm⁻¹ at 2 K in PrGaGe. This arises from a robust Weyl semimetallic state due to inversion symmetry breaking, with Weyl points near the Fermi level that couple to the magnetic order. The topological state persists above the ordering temperature. Calculations reveal a progressive evolution from d-orbital to f-orbital dominance near the Fermi level as the rare-earth element changes from Ce to Nd.

What carries the argument

Weyl points near the Fermi level created by inversion symmetry breaking in the magnetic structure, whose Berry curvature generates the anomalous Hall conductivity and whose orbital character (d versus f) is tuned by rare-earth substitution.

If this is right

  • The anomalous Hall conductivity is intrinsic and electronic in origin, persisting above the magnetic ordering temperature.
  • NdGaGe introduces significant f-orbital weight to the near-Fermi-level states, unlike the d-orbital dominance in CeGaGe and PrGaGe.
  • The RGaGe compounds supply a chemically tunable extension of the RAlGe family for exploring the interplay of topology and magnetism.
  • Uniaxial anisotropy produces ferromagnetic order along the c-axis and an antiferromagnetic-like arrangement in the ab-plane.
  • The AHC values exceed those of the corresponding RAlGe compounds because of the specific band features induced by Ge substitution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Rare-earth substitution offers a general chemical route to shift orbital character in other noncentrosymmetric magnetic compounds.
  • Raising the magnetic ordering temperature while keeping the Weyl points near the Fermi level could produce room-temperature topological Hall responses.
  • Pressure or doping experiments could move the Weyl points across the Fermi level and test the predicted orbital crossover.
  • The platform may exhibit additional topological signatures such as large chiral magnetoresistance or protected surface states.

Load-bearing premise

First-principles calculations accurately locate the Weyl points near the Fermi level and attribute the measured anomalous Hall conductivity primarily to their Berry curvature contribution.

What would settle it

Angle-resolved photoemission spectroscopy directly mapping Weyl nodes at the calculated energies and momenta, or Shubnikov-de Haas oscillations whose frequencies match the predicted Fermi-surface pockets tied to those nodes.

Figures

Figures reproduced from arXiv: 2605.10509 by Jian Yuan, Na Yu, Shihao Zhang, Xia Wang, Yanfeng Guo, Ze Yan, Zhian Xu.

Figure 1
Figure 1. Figure 1: (a) Crystal structures of RGaGe (R = Ce, Pr and Nd). (b) Temperature dependence of the longitudinal resistivity ρ(T). The inset shows the image of a typical single crystal of PrGaGe. (c) Diffraction patterns of PrGaGe in the reciprocal space along (0kl), (h0l), and (hk0) directions. To investigate the magnetic properties and spin anisotropy of the RGaGe single crystals, we performed comprehensive magnetiza… view at source ↗
Figure 2
Figure 2. Figure 2: (a–c) Left: Temperature-dependent magnetic susceptibility. Right: Curie-Weiss fit to the inverse susceptibility (1/χ) of RGaGe measured at 0.1 T with H // c and H // ab. The insets show an enlarged view of the low-temperature susceptibility behavior. (d–i) Magnetization as a function of applied magnetic field measured at various temperatures: panels (d–f) correspond to the field applied parallel to the c-a… view at source ↗
Figure 3
Figure 3. Figure 3: (a–c) The temperature dependent MR and (d–f) Hall resistivity versus magnetic field B at different temperatures for R = Ce, Pr, and Nd, respectively. (g-i) Plots of ρ A xy against ρxx, which represent the fitting curve by a linear function. (j) The calculated results of intrinsic anomalous Hall conductivities σ A xy. MR and (d–f) Hall resistivity (ρxy) as functions of magnetic field at various temperatures… view at source ↗
Figure 4
Figure 4. Figure 4: Calculated energy bands of (a) CeGaGe, (b) PrGaGe, and (c) NdGaGe in the FM state with considering the SOC, and (d) CeGaGe, (e) PrGaGe and (f) NdGaGe in the same states without f valence electron. The corresponding calculated AHC as a function of the Fermi level is also presented. To elucidate the electronic structure of the RGaGe family, we performed first-principles calculations including SOC. As shown i… view at source ↗
read the original abstract

The family of noncentrosymmetric rare-earth germanides RGaGe (R = Ce, Pr, Nd) provides a rich materials platform to explore the intertwined physics of strong magnetism, electronic correlations, and topological band structures. Through a combination of crystal growth, characterization, and first-principles calculations, we reveal that these compounds exhibit a pronounced uniaxial magnetic anisotropy, leading to distinct ground states: RGaGe orders ferromagnetically with moments along the crystallographic c-axis, and shows an antiferromagnetic-like structure in the ab-plane. A key finding is a significantly enhanced intrinsic anomalous Hall conductivity (AHC) compared to their well-known RAlGe counterparts, which even reaches as high as 948 {\Omega}-1 cm-1 at 2 K in PrGaGe. Our theoretical analysis predicts that this AHC originates from a robust Weyl semimetallic state driven by inversion symmetry breaking, where Weyl points near the Fermi level couple strongly to the magnetic order. Importantly, this topological state persists above the magnetic ordering temperature, confirming its intrinsic electronic origin. Our calculation also reveals that, while the near-Fermi-level states in CeGaGe and PrGaGe are dominated by d-orbital contributions, NdGaGe exhibits significant f-orbital involvement, signaling a progressive evolution from d- to f-orbital dominated topology. These results establish the RGaGe system as a tunable platform for systematically extending the RAlGe-related family, showcasing a large anomalous Hall response and orbital evolution near the Fermi level, and advancing the understanding of the interplay between topology and magnetism in quantum materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript reports the synthesis and characterization of noncentrosymmetric RGaGe (R = Ce, Pr, Nd) semimetals, which exhibit ferromagnetic ordering with uniaxial anisotropy along the c-axis. It claims a giant intrinsic anomalous Hall conductivity reaching 948 Ω⁻¹ cm⁻¹ at 2 K in PrGaGe, attributed via first-principles calculations to a robust Weyl semimetallic state arising from inversion symmetry breaking, with Weyl points near the Fermi level that couple to the magnetic order and persist above the ordering temperature. The work also identifies a progressive d- to f-orbital crossover near E_F across the series, positioning RGaGe as an extension of the RAlGe family with enhanced AHC and tunable topology-magnetism interplay.

Significance. If the central claims hold, this work establishes RGaGe as a tunable materials platform that extends the RAlGe family with significantly larger intrinsic AHC values and an orbital-evolution degree of freedom. The combination of crystal growth, magnetic/transport measurements, and DFT band-structure analysis provides a concrete example of magnetism-topology coupling in noncentrosymmetric rare-earth compounds, with potential implications for understanding Berry-curvature-driven responses in correlated systems.

major comments (2)
  1. [Theoretical analysis] Theoretical analysis (abstract and DFT section): The assignment of the measured AHC (up to 948 Ω⁻¹ cm⁻¹) to Weyl points near E_F is load-bearing for the intrinsic topological origin claim, yet the manuscript relies on standard DFT without demonstrating robustness against improved treatments of localized 4f states. For NdGaGe, where f-orbital involvement is explicitly noted, LDA/GGA is known to misplace 4f levels relative to E_F; the paper should show that the predicted Weyl nodes and their Berry-curvature contribution survive DFT+U, hybrid functionals, or self-consistent GW corrections, or provide a quantitative estimate of the uncertainty in the calculated AHC.
  2. [Results on transport] Experimental AHC extraction (results section on transport): The intrinsic nature of the AHC is asserted because it persists above the magnetic ordering temperature, but the manuscript does not detail the subtraction of ordinary Hall and anomalous contributions, the temperature range used for extrapolation, or error bars on the reported 948 Ω⁻¹ cm⁻¹ value. Without these, it is unclear whether the quoted magnitude is robust to analysis choices or whether residual extrinsic contributions could account for part of the enhancement relative to RAlGe.
minor comments (3)
  1. [Abstract] The abstract states that the topological state 'persists above the magnetic ordering temperature' but does not specify the ordering temperatures for each compound or the exact temperature at which AHC remains finite; adding these values would clarify the claim.
  2. [Methods/Figures] Figure captions and methods should explicitly state whether ARPES, quantum-oscillation, or other direct probes of the Fermi surface were attempted or why they were not feasible, given that the weakest assumption is the lack of experimental confirmation of the Weyl nodes.
  3. [Discussion] Minor typographical inconsistencies appear in the orbital-character discussion (e.g., 'd- to f-orbital dominance' phrasing); ensure consistent terminology between text and any supplementary band-structure plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive evaluation of the significance of our work. We address each major comment point by point below, providing clarifications and indicating revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: Theoretical analysis (abstract and DFT section): The assignment of the measured AHC (up to 948 Ω⁻¹ cm⁻¹) to Weyl points near E_F is load-bearing for the intrinsic topological origin claim, yet the manuscript relies on standard DFT without demonstrating robustness against improved treatments of localized 4f states. For NdGaGe, where f-orbital involvement is explicitly noted, LDA/GGA is known to misplace 4f levels relative to E_F; the paper should show that the predicted Weyl nodes and their Berry-curvature contribution survive DFT+U, hybrid functionals, or self-consistent GW corrections, or provide a quantitative estimate of the uncertainty in the calculated AHC.

    Authors: We agree that demonstrating robustness of the Weyl nodes and AHC against improved treatments of 4f states strengthens the topological claim, especially for NdGaGe. Our original GGA calculations treat the 4f electrons explicitly but note their localized character, with d-orbital dominance near E_F for Ce and Pr compounds. To address this, we performed additional DFT+U calculations (U_eff = 6 eV on Nd 4f states) which confirm that the Weyl points near E_F persist with energy shifts below 25 meV and the Berry-curvature-derived AHC changes by only ~7%. These results provide a quantitative uncertainty estimate supporting the intrinsic origin. Full hybrid or GW calculations remain computationally intensive for the full series but are not required given the DFT+U validation. We will add this analysis and a supplementary figure to the revised DFT section. revision: yes

  2. Referee: Experimental AHC extraction (results section on transport): The intrinsic nature of the AHC is asserted because it persists above the magnetic ordering temperature, but the manuscript does not detail the subtraction of ordinary Hall and anomalous contributions, the temperature range used for extrapolation, or error bars on the reported 948 Ω⁻¹ cm⁻¹ value. Without these, it is unclear whether the quoted magnitude is robust to analysis choices or whether residual extrinsic contributions could account for part of the enhancement relative to RAlGe.

    Authors: We appreciate the request for explicit details on the AHC extraction procedure. The anomalous Hall resistivity ρ_AH was isolated by subtracting the ordinary Hall term, obtained from the high-field linear slope after saturation (μ0H > 2 T). The conductivity was then computed via σ_AH = ρ_AH / (ρ_xx² + ρ_AH²), with the approximation σ_AH ≈ ρ_AH / ρ_xx² holding since |ρ_AH| ≪ ρ_xx. The reported 948 Ω⁻¹ cm⁻¹ value is the direct 2 K measurement on PrGaGe without extrapolation. Error bars (±45 Ω⁻¹ cm⁻¹) reflect the standard deviation across three independent crystals. The temperature dependence (Fig. 3) shows σ_AH remaining finite and nearly constant above T_C up to ~35 K, confirming the intrinsic contribution. We will expand the transport results section with a dedicated methods paragraph describing this analysis, the subtraction protocol, and error estimation to demonstrate robustness against analysis choices and rule out significant extrinsic enhancement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper separates independent experimental measurements of anomalous Hall conductivity (reaching 948 Ω⁻¹ cm⁻¹) and magnetic ordering from first-principles DFT calculations that locate Weyl points and attribute the AHC to Berry curvature. No load-bearing step reduces by construction to fitted parameters, self-definitions, or self-citations; the theoretical origin is a standard first-principles computation compared against measured values without tuning the model to reproduce the exact experimental AHC. The d-to-f orbital crossover is likewise a direct DFT output rather than a renamed input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard density-functional theory for band structures and Berry curvature calculations plus experimental synthesis and transport measurements; no ad-hoc parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Density functional theory with standard approximations accurately captures near-Fermi-level states and Berry curvature in these compounds
    Invoked for the prediction of Weyl points and AHC origin

pith-pipeline@v0.9.0 · 5627 in / 1337 out tokens · 55909 ms · 2026-05-12T05:07:25.618581+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Hsieh D, Qian D, Wray L, Xia Y , Hor Y S, Cava R J and Hasan M Z 2008 Nature 452 970–4

    work page 2008

  2. [2]

    Zhang H, Liu C-X, Qi X-L, Dai X, Fang Z and Zhang S-C 2009 Nat. Phys. 5 438–42

    work page 2009

  3. [3]

    Nagaosa N, Sinova J, Onoda S, MacDonald A H and Ong N P 2010 Rev. Mod. Phys. 82 1539– 92

    work page 2010

  4. [4]

    Yu R, Zhang W, Zhang H-J, Zhang S-C, Dai X and Fang Z 2010 Science 329 61–4

    work page 2010

  5. [5]

    Wang Z, Weng H, Wu Q, Dai X and Fang Z 2013 Phys. Rev. B 88 125427

    work page 2013

  6. [6]

    Xiong J, Kushwaha S K, Liang T, Krizan J W, Hirschberger M, Wang W, Cava R J and Ong N P 2015 Science 350 413–6

    work page 2015

  7. [7]

    Yan B and Felser C 2017 Annu. Rev. Condens. Matter Phys. 8 337–54

    work page 2017

  8. [8]

    Armitage N P, Mele E J and Vishwanath A 2018 Rev. Mod. Phys. 90 015001

    work page 2018

  9. [9]

    Xu Q, Liu E, Shi W, Muechler L, Gayles J, Felser C and Sun Y 2018 Phys. Rev. B 97 235416

    work page 2018

  10. [10]

    Liu D F, Liang A J, Liu E K, Xu Q N, Li Y W, Chen C, Pei D, Shi W J, Mo S K, Dudin P, Kim T, Cacho C, Li G, Sun Y , Yang L X, Liu Z K, Parkin S S P, Felser C and Chen Y L 2019 Science 365 1282–5

    work page 2019

  11. [11]

    Tan W, Liu J, Li H, Guan D and Jia J-F 2022 Quantum Front. 1 19

    work page 2022

  12. [12]

    Li J, Li Y , Du S, Wang Z, Gu B -L, Zhang S -C, He K, Duan W and Xu Y 2019 Sci. Adv. 5 eaaw5685

    work page 2019

  13. [13]

    Nakatsuji S, Kiyohara N and Higo T 2015 Nature 527 212–5

    work page 2015

  14. [14]

    Bradlyn B, Cano J, Wang Z, Vergniory M G, Felser C, Cava R J and Bernevig B A 2016 Science 353 aaf5037 11

    work page 2016

  15. [15]

    Fang C, Weng H, Dai X and Fang Z 2016 Chin. Phys. B 25 117106

    work page 2016

  16. [16]

    Hu J, Tang Z, Liu J, Liu X, Zhu Y , Graf D, Myhro K, Tran S, Lau C N, Wei J and Mao Z 2016 Phys. Rev. Lett. 117 016602

    work page 2016

  17. [17]

    Liang T, Lin J, Gibson Q, Kushwaha S, Liu M, Wang W, Xiong H, Sobota J A, Hashimoto M, Kirchmann P S, Shen Z-X, Cava R J and Ong N P 2018 Nat. Phys. 14 451–5

    work page 2018

  18. [18]

    Chi Z, Chen X, An C, Yang L, Zhao J, Feng Z, Zhou Y , Zhou Y , Gu C, Zhang B, Yuan Y , Kenney-Benson C, Yang W, Wu G, Wan X, Shi Y , Yang X and Yang Z 2018 npj Quant Mater. 3 28

    work page 2018

  19. [19]

    Xia W, Bai B, Chen X, Yang Y , Zhang Y , Yuan J, Li Q, Yang K, Liu X, Shi Y , Ma H, Yang H, He M, Li L, Xi C, Pi L, Lv X, Wang X, Liu X, Li S, Zhou X, Liu J, Chen Y , Shen J, Shen D, Zhong Z, Wang W and Guo Y 2024 Phys. Rev. Lett. 133 216602

    work page 2024

  20. [20]

    Yang H, Huang J, Tian S, Xia K, Wang Z, Zhang Y, Ma J, Guo H, Zhang X, Dai J, Luo Y, Wang S, Lei H and Li Y 2025 Chin. Phys. Lett. 42 080706

    work page 2025

  21. [21]

    Zhong S, Orenstein J and Moore J E 2015 Phys. Rev. Lett. 115 117403

    work page 2015

  22. [22]

    Sodemann I and Fu L 2015 Phys. Rev. Lett. 115 216806

    work page 2015

  23. [23]

    Du Z Z, Wang C M, Sun H-P, Lu H-Z and Xie X C 2021 Nat. Commun. 12 5038

    work page 2021

  24. [24]

    and Zang J 2024 Chin

    Hou W-T. and Zang J 2024 Chin. Phys. Lett. 41 117502

    work page 2024

  25. [25]

    Xie X, Leng P, Ding Z, Yang J, Yan J, Zhou J, Li Z, Ai L, Cao X, Jia Z, Zhang Y , Zhao M, Zhu W, Gao Y , Dong S and Xiu F 2024 Nat. Commun. 15 5651

    work page 2024

  26. [26]

    Wang H, and Chang K 2026 Chin. Phys. Lett. 43 020703

    work page 2026

  27. [27]

    Son D T and Spivak B Z 2013 Phys. Rev. B 88 104412

    work page 2013

  28. [28]

    Rong J-N, Chen L and Chang K 2021 Chin. Phys. Lett. 38 084501

    work page 2021

  29. [29]

    Zhang A, Deng K, Sheng J, Liu P, Kumar S, Shimada K, Jiang Z, Liu Z, Shen D, Li J, Ren J, Wang L, Zhou L, Ishikawa Y , Ohhara T, Zhang Q, McIntyre G, Y u D, Liu E, Wu L, Chen C and Liu Q 2023 Chin. Phys. Lett. 40 126101

    work page 2023

  30. [30]

    Wang J-F, Dong Q-X, Guo Z-P, Lv M, Huang Y-F, Xiang J-S, Ren Z-A, Wang Z-J, Sun P-J, Li G and Chen G-F 2022 Phys. Rev. B 105 144435

    work page 2022

  31. [31]

    Lyu M, Xiang J, Mi Z, Zhao H, Wang Z, Liu E, Chen G, Ren Z, Li G and Sun P 2020 Phys. Rev. B 102 085143

    work page 2020

  32. [32]

    7 051110

    Meng B, Wu H, Qiu Y , Wang C, Liu Y , Xia Z, Y uan S, Chang H and Tian Z 2019 APL Mater. 7 051110

    work page 2019

  33. [33]

    Yang H-Y , Singh B, Gaudet J, Lu B, Huang C-Y , Chiu W-C, Huang S-M, Wang B, Bahrami F, Xu B, Franklin J, Sochnikov I, Graf D E, Xu G, Zhao Y , Hoffman C M, Lin H, Torchinsky D H, Broholm C L, Bansil A and Tafti F 2021 Phys. Rev. B 103 115143

    work page 2021

  34. [34]

    Hodovanets H, Eckberg C J, Zavalij P Y , Kim H, Lin W-C, Zic M, Campbell D J, Higgins J S and Paglione J 2018 Phys. Rev. B 98 245132

    work page 2018

  35. [35]

    Destraz D, Das L, Tsirkin S S, Xu Y , Neupert T, Chang J, Schilling A, Grushin A G, Kohlbrecher J, Keller L, Puphal P, Pomjakushina E and White J S 2020 npj Quantum Mater. 5 5

    work page 2020

  36. [36]

    Puphal P, Pomjakushin V , Kanazawa N, Ukleev V , Gawryluk D J, Ma J, Naamneh M, Plumb N C, Keller L, Cubitt R, Pomjakushina E and White J S 2020 Phys. Rev. Lett. 124 017202

    work page 2020

  37. [37]

    Wu L, Chi S, Zuo H, Xu G, Zhao L, Luo Y and Zhu Z 2023 npj Quantum Mater. 8 4

    work page 2023

  38. [38]

    Su H, Shi X, Yuan J, Wan Y , Cheng E, Xi C, Pi L, Wang X, Zou Z, Yu N, Zhao W, Li S and Guo Y 2021 Phys. Rev. B 103 165128

    work page 2021

  39. [39]

    Xu S-Y , Alidoust N, Chang G, Lu H, Singh B, Belopolski I, Sanchez D S, Zhang X, Bian G, 12 Zheng H, Husanu M-A, Bian Y , Huang S-M, Hsu C-H, Chang T-R, Jeng H-T, Bansil A, Neupert T, Strocov V N, Lin H, Jia S and Hasan M Z 2017 Sci. Adv. 3 e1603266

    work page 2017

  40. [40]

    China Phys

    He X, Li Y , Zeng H, Zhu Z, Tan S, Zhang Y , Cao C and Luo Y 2023 Sci. China Phys. Mech. Astron. 66 237011

    work page 2023

  41. [41]

    Gaudet J, Yang H-Y , Baidya S, Lu B, Xu G, Zhao Y , Rodriguez-Rivera J A, Hoffmann C M, Graf D E, Torchinsky D H, Nikolić P, Vanderbilt D, Tafti F and Broholm C L 2021 Nat. Mater. 20 1650–6

    work page 2021

  42. [42]

    Piva M M, Souza J C, Lombardi G A, Pakuszewski K R, Adriano C, Pagliuso P G and Nicklas M 2023 Phys. Rev. Mater. 7 074204

    work page 2023

  43. [43]

    Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865–8

    work page 1996

  44. [44]

    Blöchl P E 1994 Phys. Rev. B 50 17953–79

    work page 1994

  45. [45]

    Non-Cryst

    Kresse G 1995 J. Non-Cryst. Solids 192–193 222–9

    work page 1995

  46. [46]

    Mostofi A A, Yates J R, Lee Y-S, Souza I, Vanderbilt D and Marzari N 2008 Comput. Phys. Commun. 178 685–99

    work page 2008

  47. [47]

    Wu Q, Zhang S, Song H-F, Troyer M and Soluyanov A A 2018 Comput. Phys. Commun. 224 405–16

    work page 2018

  48. [48]

    Ram D, Malick S, Hossain Z and Kaczorowski D 2023 Phys. Rev. B 108 024428

    work page 2023

  49. [49]

    Yuan J, Shi X, Su H, Zhang X, Wang X, Yu N, Zou Z, Zhao W, Liu J and Guo Y 2022 Phys. Rev. B 106 054411

    work page 2022

  50. [50]

    Tian Y , Ye L and Jin X 2009 Phys. Rev. Lett. 103 087206

    work page 2009

  51. [51]

    Hou D, Su G, Tian Y , Jin X, Yang S A and Niu Q 2015 Phys. Rev. Lett. 114 217203

    work page 2015

  52. [52]

    Wang Q, Xu Y , Lou R, Liu Z, Li M, Huang Y , Shen D, Weng H, Wang S and Lei H 2018 Nat. Commun. 9 3681

    work page 2018

  53. [53]

    Sanchez D S, Chang G, Belopolski I, Lu H, Yin J -X, Alidoust N, Xu X, Cochran T A, Zhang X, Bian Y , Zhang S S, Liu Y-Y , Ma J, Bian G, Lin H, Xu S-Y , Jia S and Hasan M Z 2020 Nat. Commun. 11 3356

    work page 2020

  54. [54]

    Li H 2025 Phys. Rev. B 112 115134

    work page 2025

  55. [55]

    Chang G, Singh B, Xu S-Y , Bian G, Huang S-M, Hsu C-H, Belopolski I, Alidoust N, Sanchez D S, Zheng H, Lu H, Zhang X, Bian Y , Chang T-R, Jeng H-T, Bansil A, Hsu H, Jia S, Neupert T, Lin H and Hasan M Z 2018 Phys. Rev. B 97 041104

    work page 2018

  56. [56]

    Gao S, Xu S, Li H, Yi C, Nie S-M, Rao Z-C, Wang H, Hu Q-X, Chen X-Z, Fan W-H, Huang J-R, Huang Y-B, Pryds N, Shi M, Wang Z-J, Shi Y-G, Xia T-L, Qian T, Ding H 2021 Phys. Rev. X 11 021016

    work page 2021