Transport of Topological Semimetals

In the classical limit, negative longitudinal magnetoresistance in type II Weyl semimetals becomes isotropic, similar to type I semimetals.

“Further studies also found that, in the classical limit characterized by ω c τ 1 (as opposed to the quantum limit or semiclassical limit, where ω c τ 1, where ω c is the cyclotron frequency and τ is the transport relaxation time), negative longitudinal MR in type II WSMs becomes isotropic, similar to that in type I semimetals (303, 308).”

• cites [303] 303. Reis RDd, Ajeesh MO, Kumar N, Arnold F, Shekhar C, et al. 2016. On the search for the chiral anomaly in Weyl semimetals: the negative longitudinal magnetoresistance. New J. Phys. 18:085006 306.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [308] 308. Nandy S, Sharma G, Taraphder A, Tewari S. 2017. Chiral anomaly as the origin of the planar Hall effect in Weyl semimetals. Phys. Rev. Lett. 119:176804 311. sourceDisputed

  The paper finds negative longitudinal magnetoresistance in type-II Weyl semimetals for magnetic fields in any direction, but it does not find it isotropic. Section III (p. 4) reports a direction-dependent response in the low-field quasiclassical regime: magnetoconductivity is approximately linear in the field along the tilt axis and quadratic perpendicular to it, for both type-I and type-II semimetals. (LLM, conf=0.88)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured

Low-energy electrons in topological semimetals are highly robust against crystalline disorder and imperfections due to linear dispersion and spin/pseudospin momentum locking, leading to very high electron mobilities.

“Because of linear dispersion and spin (pseudospin) momentum locking, low-energy electrons in topological semimetals are highly robust against crystalline disorder and imperfections, leading to very high electron mobilities (45,46).”

• cites [45] 45. Liang T, Gibson Q, Ali MN, Liu M, Cava RJ, Ong NP. 2015. Ultrahigh mobility and giant magnetore- sistance in the Dirac semimetal Cd 3 As 2 . Nat. Mater. 14:280-84 sourcePartly supported

  The main text (p. 1) reports that suppressed backscattering in Cd3As2 produces ultrahigh mobility, and the supplement (S5, p. 11) says lattice disorder has little effect on forward drift. But the authors call the protection mechanism unknown; they do not establish linear dispersion or spin/pseudospin momentum locking as its cause, or show the result generally for topological semimetals. (LLM, conf=0.90)

  Experimental — accuracy not yet measured
• cites [46] 46. Shekhar C, Nayak AK, Sun Y, Schmidt M, Nicklas M, et al. 2015. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nat. Phys. 11:645-49 sourcePartly supported

  The Abstract says Weyl and Dirac semimetals usually have very high mobilities and suggests a link to their electronic dispersion. The transport discussion (section labelled “14.5 T; refs”; page unavailable) attributes NbP’s ultrahigh mobility to large Fermi velocity and low effective mass. Our copy does not attribute that mobility to spin or pseudospin momentum locking or establish robustness against crystalline disorder and imperfections. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

Fermi arcs lead to unusual quantum cyclotron orbits (Weyl orbits) as observed in quantum oscillation measurements.

“The Fermi arcs also lead to unusual quantum cyclotron orbits (the Weyl orbits) as observed in quantum oscillation measurements (43,44).”

• cites [43] 43. Potter AC, Kimchi I, Vishwanath A. 2014. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals. Nat. Commun. 5:5161 sourcePartly supported

  The Abstract describes unusual closed magnetic orbits involving Fermi arcs, and the Quantum oscillations section predicts that they produce periodic oscillations. The paper presents semiclassical analysis and numerical simulation, not an observation in quantum oscillation measurements; the Discussion calls the signatures experimentally testable. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites [44] 44. Moll PJW, Nair NL, Helm T, Potter AC, Kimchi I, et al. 2016. Transport evidence for Fermi-arc- mediated chirality transfer in the Dirac semimetal Cd 3 As 2 . Nature 535:266-70 sourcePartly supported

  The paper reports Shubnikov–de Haas oscillations in Cd3As2 consistent with Weyl orbits involving Fermi-arc surface states (p. 1), and says the measured oscillation frequency agrees with the Weyl-orbit prediction (p. 2). But it does not establish an unqualified observation of Weyl orbits: on p. 4, it says the experiments cannot rule out another origin for the oscillations. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

Application of parallel electric and magnetic fields can break the apparent conservation of chiral charge.

“The application of parallel electric and magnetic fields can break the apparent conservation of the chiral charge (10,11,52,53).”

• cites [10] 10. Armitage NP, Mele EJ, Vishwanath A. 2018. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90:015001 sourceSupported
• cites [11] 11. Bernevig A, Weng H, Fang Z, Dai X. 2018. Recent progress in the study of topological semimetals. J. Phys. Soc. Jpn. 87:041001 sourceNot checked — no full text of the source
• cites [52] 52. Jho Y-S, Kim K-S. 2013. Interplay between interaction and chiral anomaly: anisotropy in the electrical resistivity of interacting Weyl metals. Phys. Rev. B 87:205133 sourceSupported
• cites [53] 53. Son DT, Spivak BZ. 2013. Chiral anomaly and classical negative magnetoresistance of Weyl metals. Phys. Rev. B 88:104412 sourceSupported

Solid-state crystals can support band crossings beyond the Dirac/Weyl paradigm, including three-, four-, six-, and eightfold degeneracies.

“In contrast to high-energy physics, solid-state crystals can support band crossings beyond the Dirac/Weyl paradigm (31)(32)(33)(34)(35)(36).These band crossings, broadly referred as unconventional fermions, include three-, four-, six-, and eightfold degeneracies (31).”

• cites (31)(32)(33)(34)(35)(36)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [31] 31. Bradlyn B, Cano J, Wang Z, Vergniory MG, Felser C, et al. 2016. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353:aaf5037 sourceSupported

Transverse MR in WSMs/DSMs such as Cd3As2, PtBi2, WTe2, and NbP can reach up to 0.1-1 million percent at low temperatures (0.5-5 K) and 9 T, without saturation up to 30-100 T.

“The transverse MR, measured with the field perpendicular to the current direction, can reach up to 0.1-1 million percent at low temperatures (0.5-5 K) and a field of 9 T (see Table 1), without any sign of saturation up to 30-100 T in WSMs/DSMs such as Cd 3 As 2 , PtBi 2 , WTe 2 , and NbP (46,48,171,172).”

• cites [46] 46. Shekhar C, Nayak AK, Sun Y, Schmidt M, Nicklas M, et al. 2015. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nat. Phys. 11:645-49 sourceSupported
• cites [48] 48. Ali MN, Xiong J, Flynn S, Tao J, Gibson QD, et al. 2014. Large, non-saturating magnetoresistance in WTe 2 . Nature 514:205-8 sourcePartly supported

  The WTe2 results (p. 3) support a very large transverse magnetoresistance at low temperature, still increasing without saturation at 45 T. They report 452,700% at 4.5 K and 14.7 T, however, not the claimed magnitude at 9 T. This paper does not provide evidence for the other named materials. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites [171] 171. Wang K, Graf D, Lei H, Tozer SW, Petrovic C. 2011. Quantum transport of two-dimensional Dirac fermions in SrMnBi 2 . Phys. Rev. B 84:220401 174. sourceCould not verify

  The paper’s Conclusion (p. 13) discusses density response and Friedel oscillations in two-dimensional Dirac-fermion models, not transverse magnetoresistance. Searches for magnetoresistance, the named materials, and the quoted field strengths and percentages found no passage addressing the claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [172] 172. Novak M, Sasaki S, Segawa K, Ando Y. 2015. Large linear magnetoresistance in the Dirac semimetal TlBiSSe. Phys. Rev. B 91:041203 175. sourcePartly supported

  The paper reports transverse, nonsaturating magnetoresistance in the Dirac semimetal TlBiSSe (pp. 2, 4) and mentions gigantic magnetoresistance in Cd3As2 (p. 4). It does not support the claimed 0.1–1 million percent at 9 T, nonsaturation up to 30–100 T, or the claims about PtBi2, WTe2, and NbP. For TlBiSSe, it reports almost 10,000% at 14 T and 1.8 K (p. 4). (LLM, conf=0.89)

  Experimental — accuracy not yet measured

Carrier compensation is not achieved in some topological semimetals.

“However, reports also indicate that carrier compensation is not achieved in some topological semimetals (188,189).”

• cites [188] 188. Chamber RG. 1990. Electrons in Metals and Semiconductors. New York: Chapman and Hall 191.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [189] 189. Luo Y, Ghimire NJ, Wartenbe M, Choi H, Neupane M, et al. 2015. Electron-hole compensation effect between topologically trivial electrons and nontrivial holes in NbAs. Phys. Rev. B 92:205134 192. sourceCould not verify

  The Introduction (p. 1) and Conclusion (p. 7) describe electron–hole compensation in NbAs. I searched the paper’s discussion of other topological semimetals and carrier densities, but found no passage reporting a topological semimetal in which compensation is not achieved. The NbAs finding does not rule out the claim about other materials. (LLM, conf=0.72)

  Experimental — accuracy not yet measured

Type II WSMs have strongly tilted Weyl cones.

“By contrast, type II WSMs have strongly tilted Weyl cones (Figure 1b) (28).”

• cites [28] 28. Soluyanov AA, Gresch D, Wang Z, Wu Q, Troyer M, et al. 2015. Type-II Weyl semimetals. Nature 527:495-98 sourceSupported

DSMs host Dirac fermions as low-energy quasiparticle excitations.

“DSMs host Dirac fermions as low-energy quasiparticle excitations (12-14, 38, 126-131).”

• cites [12] 12. Young SM, Zaheer S, Teo JCY, Kane CL, Mele EJ, Rappe AM. 2012. Dirac semimetal in three dimen- sions. Phys. Rev. Lett. 108:140405 sourceSupported
• cites [13] 13. Wang Z, Sun Y, Chen X-Q, Franchini C, Xu G, et al. 2012. Dirac semimetal and topological phase transitions in A 3 Bi (A = Na, K, Rb). Phys. Rev. B 85:195320Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [14] 14. Wang Z, Weng H, Wu Q, Dai X, Fang Z. 2013. Three-dimensional Dirac semimetal and quantum transport in Cd 3 As 2 . Phys. Rev. B 88:125427Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [38] 38. Xu S-Y, Liu C, Kushwaha SK, Sankar R, Krizan JW, et al. 2015. Observation of Fermi arc surface states in a topological metal. Science 347:294-98 sourceNot checked — no full text of the source
• cites [126] 126. Liu ZK, Jiang J, Zhou B, Wang ZJ, Zhang Y, et al. 2014. A stable three-dimensional topological Dirac semimetal Cd 3 As 2 . Nat. Mater 13:677-81 129. sourceSupported
• cites [127] 127. Borisenko S, Gibson Q, Evtushinsky D, Zabolotnyy V, Büchner B, Cava RJ. 2014. Experimental real- ization of a three-dimensional Dirac semimetal. Phys. Rev. Lett. 113:027603 130. sourceSupported
• cites [128] 128. Yi H, Wang Z, Chen C, Shi Y, Feng Y, et al. 2014. Evidence of topological surface state in three- dimensional Dirac semimetal Cd 3 As 2 . Sci. Rep. 4:6106 131. sourceNot checked — no full text of the source
• cites [129] 129. Yang B-J, Nagaosa N. 2014. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nat. Commun. 5:4898 132. sourceSupported
• cites [130] 130. Steinberg JA, Young SM, Zaheer S, Kane CL, Mele EJ, Rappe AM. 2014. Bulk Dirac points in distorted spinels. Phys. Rev. Lett. 112:036403 133. sourceSupported
• cites [131] 131. Watanabe H, Po HC, Vishwanath A, Zaletel M. 2015. Filling constraints for spin-orbit coupled insula- tors in symmorphic and nonsymmorphic crystals. PNAS 112:14551-56 134. sourceSupported

Sublinear MR may be attributed to weak antilocalization caused by strong spin-orbit coupling.

“while sublinear MR may be attributed to the weak antilocalization caused by strong SOC (196).”

• cites [196] 196. Beidenkopf H, Roushan P, Seo J, Gorman L, Drozdov I, et al. 2011. Spatial fluctuations of helical Dirac fermions on the surface of topological insulators. Nat. Phys. 7:939-43 199. sourceCould not verify

  The paper’s discussion of backscattering and localization in its main text does not connect weak antilocalization to sublinear magnetoresistance or strong spin-orbit coupling. Searches for those terms and for magnetoresistance found no passage in our copy making the claimed attribution; weak antilocalization appears in a reference title, not as the paper’s explanation of sublinear MR. (LLM, conf=0.73)

  Experimental — accuracy not yet measured

Quantum relaxation time and transport relaxation time are essentially different even though both arise from scattering by static impurities and defects.

“While both arise from the scattering by static impurities and defects, these two quantities are essentially different (217, 218):”

• cites [217] 217. Xiong J, Luo Y, Khoo Y, Jia S, Cava RJ, Ong NP. 2012. High-field Shubnikov-de Haas oscillations in the topological insulator Bi 2 Te 2 Se. Phys. Rev. B 86:045314 220. sourceCould not verify

  Section V (p. 5) defines a lifetime through the Dingle temperature and uses it to estimate surface mobility, but does not distinguish quantum from transport relaxation time. Section III (p. 3) discusses defects and carrier density, not whether both times arise from scattering by static impurities and defects. Searches for the two relaxation times and scattering did not locate the attributed comparison. (LLM, conf=0.72)

  Experimental — accuracy not yet measured
• cites [218] 218. Pariari A, Dutta P, Mandal P. 2015. Probing the Fermi surface of three-dimensional Dirac semimetal Cd 3 As 2 through the de Haas-van Alphen technique. Phys. Rev. B 91:155139 221. sourceCould not verify

  The Fermi-surface discussion (p. 4) reports a Dingle temperature from oscillation damping, but does not compare quantum and transport relaxation times. Searches for relaxation time, scattering, and the claimed distinction found no passage in our copy that addresses it. (LLM, conf=0.74)

  Experimental — accuracy not yet measured

The zero energy of the zeroth Landau level can be understood through the Berry phase arising from cyclotron motion of carriers in momentum space.

“Such a zero energy can be understood in terms of the Berry phase arising from the cyclotron motion of carriers in momentum space (206).”

• cites [206] 206. Xiao D, Chang M-C, Niu Q. 2010. Berry phase effects on electronic properties. Rev. Mod. Phys. 82:1959- 2007 209. sourceSupported

Klein tunneling has been demonstrated in graphene with a potential barrier created by a local gate.

“Klein tunneling has been demonstrated in graphene, with a potential barrier created by a local gate (257, 258).”

• cites [257] 257. O'Brien TE, Diez M, Beenakker CWJ. 2016. Magnetic breakdown and Klein tunneling in a type-II Weyl semimetal. Phys. Rev. Lett. 116:236401 260. sourceNot checked — no full text of the source
• cites [258] 258. Zhang Y, Bulmash D, Hosur P, Potter AC, Vishwanath A. 2016. Quantum oscillations from generic surface Fermi arcs and bulk chiral modes in Weyl semimetals. Sci. Rep. 6:23741 261. sourceCould not verify

  The Methods and Abstract discuss quantum oscillations in Weyl semimetals, not a graphene tunneling experiment. Searches for Klein tunneling and for graphene with a locally gated potential barrier found no passage addressing the claim; page numbers are unavailable in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Additional frequencies attributed to magnetic breakdown have been observed in high-field quantum oscillation studies on several topological semimetals.

“The additional frequencies ascribed to magnetic breakdown have been observed in high-field quantum oscillation studies on several topological semimetals (171,226,252).”

• cites [171] 171. Wang K, Graf D, Lei H, Tozer SW, Petrovic C. 2011. Quantum transport of two-dimensional Dirac fermions in SrMnBi 2 . Phys. Rev. B 84:220401 174. sourceCould not verify

  The paper’s opening summary (p. 1) describes theoretical calculations of excitonic effects, and its Conclusion (p. 13) discusses density response and Friedel oscillations. Searches for magnetic breakdown, high-field quantum oscillations, and topological semimetals found no passage reporting the claimed experimental observations. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [226] 226. Hu J, Zhu Y, Gui X, Graf D, Tang Z, et al. 2018. Quantum oscillation evidence of a topological semimetal phase in ZrSnTe. Phys. Rev. B 97:155101 229. sourcePartly supported

  The ZrSnTe results (p. 5) report multiple frequencies in high-field quantum oscillations, supporting the observation of additional frequencies in a topological semimetal. The paper attributes the measured frequencies to topological bands, not magnetic breakdown; a search for “magnetic breakdown” found no passage making that attribution. This source alone also does not establish observations in several semimetals. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [252] 252. Ru-Keng S, Siu GG, Xiu C. 1993. Barrier penetration and Klein paradox. J. Phys. A Math. Gen. 26:1001 255.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Under magnetic fields, electrons can transport on a cyclotron orbit that connects one surface Fermi arc to the opposite Fermi arc by coupling to bulk states.

“It has been predicted that under magnetic fields, electrons can transport on a cyclotron orbit that connects one surface Fermi arc to the opposite Fermi arc by coupling to bulk states (Figure 4d) (43,260).”

• cites [43] 43. Potter AC, Kimchi I, Vishwanath A. 2014. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals. Nat. Commun. 5:5161 sourceSupported
• cites [260] 260. Li P, Wen Y, He X, Zhang Q, Xia C, et al. 2017. Evidence for topological type-II Weyl semimetal WTe 2 . Nat. Commun. 8:2150 263. sourceSupported

Multiple oscillation frequencies are often observed in most topological semimetals, such as the TaAs family and WHM materials with PbFCl-type structure.

“However, multiple oscillation frequencies are often observed in most topological semimetals, such as those of the TaAs family (179, 180, 182, 183, 191-193, 227, 243) and WHM materials with PbFCl-type structure (W = Zr or Hf; H = Si, Ge, or Sn; M = S, Se, or Te) (156,221,222,226,228,232,233,241,242,244,245).”

• cites [179] 179. Zhang C, Guo C, Lu H, Zhang X, Yuan Z, et al. 2015. Large magnetoresistance over an extended tem- perature regime in monophosphides of tantalum and niobium. Phys. Rev. B 92:041203(R) 182. sourcePartly supported

  The paper reports Shubnikov–de Haas oscillations in TaP and NbP (p. 4) and FFT-derived frequencies for those TaAs-family materials (p. 6). But the Fig. 4 caption (p. 11) assigns F1 and F2 to different samples; the copy does not establish that multiple frequencies occur in each material or that this is common across most topological semimetals. The WHM example has a separate citation group and is not attributed to this source. (LLM, conf=0.79)

  Experimental — accuracy not yet measured
• cites [180] 180. Zhang C-L, Yuan Z, Jiang Q-D, Tong B, Zhang C, et al. 2017. Electron scattering in tantalum monoarsenide. Phys. Rev. B 95:085202 183. sourceCould not verify

  The NbAs results (pp. 2–4) report Shubnikov–de Haas oscillations and identify a principal frequency of 25 T, but do not establish multiple frequencies. The paper also does not address how often multiple frequencies occur across the TaAs family or WHM materials (searched: multiple oscillation frequencies, 25 T, WHM and PbFCl). (LLM, conf=0.72)

  Experimental — accuracy not yet measured
• cites [182] 182. Wang YL, Thoutam LR, Xiao ZL, Hu J, Das S, et al. 2015. Origin of the turn-on temperature behavior in WTe 2 . Phys. Rev. B 92:180402 185. sourceCould not verify

  The discussion on p. 4 mentions quantum-oscillation measurements only to compare electron and hole carrier densities in WTe₂. The introduction (p. 1) mentions TaAs but does not discuss its oscillation frequencies. Searches for multiple frequencies and PbFCl-type WHM materials found no passage addressing the attributed claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [183] 183. Zhao Y, Liu H, Yan J, An W, Liu J, et al. 2015. Anisotropic magnetotransport and exotic longitudinal linear magnetoresistance in WTe 2 crystals. Phys. Rev. B 92:041104 186. sourceSupported
• cites [191] 191. Du J, Wang H, Mao Q, Khan R, Xu B, et al. 2016. Large unsaturated positive and negative magnetore- sistance in Weyl semimetal TaP. Sci. China Phys. Mech. Astron. 59:657406 194.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [192] 192. Ghimire NJ, Yongkang L, Neupane M, Williams DJ, Bauer ED, Ronning F. 2015. Magnetotransport of single crystalline NbAs. J. Phys. Condens. Matter 27:152201 195.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [193] 193. Abrikosov AA. 1998. Quantum magnetoresistance. Phys. Rev. B 58:2788-94 196. sourceNot checked — no full text of the source
• cites [227] 227. Zheng W, Schönemann R, Aryal N, Zhou Q, Rhodes D, et al. 2018. Detailed study of the Fermi surfaces of the type-II Dirac semimetallic candidates XTe 2 (X = Pd, Pt). Phys. Rev. B 97:235154 230. sourcePartly supported

  The Results report multiple quantum-oscillation frequencies in PdTe₂ and PtTe₂, supporting an example of the phenomenon. The paper does not establish that this is often observed in most topological semimetals; its Introduction mentions the TaAs family in connection with Weyl nodes, not oscillation frequencies, and the cited passages do not address PbFCl-type WHM materials. Page numbers are unavailable in our copy. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites [243] 243. Pezzini S, van Delft MR, Schoop LM, Lotsch BV, Carrington A, et al. 2018. Unconventional mass enhancement around the Dirac nodal loop in ZrSiS. Nat. Phys. 14:178-83 246. sourceSupported
• cites [156] 156. Chen C, Xu X, Jiang J, Wu SC, Qi YP, et al. 2017. Dirac line nodes and effect of spin-orbit coupling in the nonsymmorphic critical semimetals MSiS (M = Hf, Zr). Phys. Rev. B 95:125126 159. sourcePartly supported

  The paper identifies HfSiS as having the PbFCl structure type (p. 3) and discusses de Haas–van Alphen quantum oscillations in this material family (p. 6). It does not say that multiple oscillation frequencies were observed, so it does not support that part of the claim. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

In type II WSMs, magnetic breakdown has been predicted to be associated with the Klein paradox.

“In type II WSMs, the magnetic breakdown has been predicted to be associated with the Klein paradox, which states that the tunneling barrier is nearly”

• cites [253] 253. Calogeracos A, Dombey N. 1999. History and physics of the Klein paradox. Contemp. Phys. 40:313-21 256. sourceCould not verify

  The Introduction (p. 3) explains the Klein paradox as transmission through strong repulsive potentials without the usual exponential damping. Our copy does not address type II Weyl semimetals or predict an association between magnetic breakdown and the Klein paradox; searches for those terms found no relevant passage. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

For the nodal-line semimetal ZrSiS, the effective mass obtained from an FFT amplitude fit increases greatly when a narrower field range is used.

“For example, for the NLSM ZrSiS, the effective mass obtained from the fit of the FFT amplitude is greatly increased when a narrower field range is used for the FFT analysis.”

• cites [221] 221. Jeon S, Zhou BB, Gyenis A, Feldman BE, Kimchi I, et al. 2014. Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd 3 As 2 . Nat. Mater 13:851-56 224. sourceCould not verify

  The paper describes STM measurements of Cd3As2 (p. 1) and Fourier transforms used for quasiparticle-interference maps (p. 8). Searches for ZrSiS and for effective-mass fitting over narrower FFT field ranges found no passage addressing the claimed result. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Because the zeroth LL is pinned at EF, the quantum limit can be reached in relatively low fields in YbMnBi2.

“Because the zeroth LL is pinned at E F (269), the quantum limit can be reached in relatively low fields in this material (247).”

• cites [269] 269. Stormer HL, Tsui DC, Gossard AC. 1999. The fractional quantum Hall effect. Rev. Mod. Phys. 71:S298- 305 272. sourceNot checked — no full text of the source
• cites [247] 247. Wang Q, Guo P-J, Sun S, Li C, Liu K, et al. 2018. Extremely large magnetoresistance and high-density Dirac-like fermions in ZrB 2 . Phys. Rev. B 97:205105 250. sourceCould not verify

  The paper discusses magnetotransport and quantum oscillations in ZrB₂ (pp. 1, 2, 4), but the passages do not address a low-field quantum limit in YbMnBi₂. Searches for YbMnBi₂, the quantum limit, and a zeroth Landau level pinned at the Fermi energy found no supporting passage in our copy. (LLM, conf=0.77)

  Experimental — accuracy not yet measured

The magnetic torque anomaly observed in NbAs can be quantitatively described by the topological character of the electronic dispersion.

“Indeed, the magnetic torque anomaly, which has been observed in NbAs, can be quantitatively described by the topological character of the electronic dispersion ( 224).”

• cites [224] 224. Hu J, Zhu YL, Graf D, Tang ZJ, Liu JY, Mao ZQ. 2017. Quantum oscillation studies of topological semimetal candidate ZrGeM (M = S, Se, Te). Phys. Rev. B 95:205134 227. sourceCould not verify

  The quantum-oscillation section (pp. 4–5) discusses magnetic-torque measurements and topological bands in ZrSnTe, not a torque anomaly in NbAs. A search for NbAs and a quantitative explanation of its torque anomaly found only a NbAs magnetotransport citation in the references (p. 11). (LLM, conf=0.76)

  Experimental — accuracy not yet measured

In YbMnBi2, the zeroth LL was probed via interlayer transport.

“In another structurally similar compound, YbMnBi 2 , the zeroth LL was probed via interlayer transport (247).”

• cites [247] 247. Wang Q, Guo P-J, Sun S, Li C, Liu K, et al. 2018. Extremely large magnetoresistance and high-density Dirac-like fermions in ZrB 2 . Phys. Rev. B 97:205105 250. sourceCould not verify

  The Introduction (p. 1) and Results and Discussion (pp. 2–5) discuss magnetotransport and electronic structure in TmSb, not YbMnBi2. Searches for YbMnBi2, the zeroth Landau level, and interlayer transport found no passage addressing the claimed probe. (LLM, conf=0.77)

  Experimental — accuracy not yet measured

This has been demonstrated in nanostructures of Cd3As2 and WTe2.

“This has been demonstrated in nanostructures of Cd 3 As 2 (Figure 4e,f ) (44,261) and WTe 2 (262).”

• cites [44] 44. Moll PJW, Nair NL, Helm T, Potter AC, Kimchi I, et al. 2016. Transport evidence for Fermi-arc- mediated chirality transfer in the Dirac semimetal Cd 3 As 2 . Nature 535:266-70 sourceSupported
• cites [261] 261. Zhang Y, Tan Y-W, Stormer HL, Kim P. 2005. Experimental observation of the quantum Hall effect and Berry's phase in graphene. Nature 438:201-4 264. sourceCould not verify

  The experimental report (p. 1) describes quantum Hall measurements in graphene, not Cd3As2 or WTe2 nanostructures. Searches for both materials in our copy found no passage addressing the attributed demonstrations; the preceding sentence needed to identify what “this” refers to was not provided. (LLM, conf=0.78)

  Experimental — accuracy not yet measured
• cites [262] 262. Xu Y, Miotkowski I, Liu C, Tian J, Nam H, et al. 2014. Observation of topological surface state quantum Hall effect in an intrinsic three-dimensional topological insulator. Nat. Phys. 10:956-63 265. sourceCould not verify

  The Introduction (p. 2) reports a quantum Hall effect in BiSbTeSe2, not in Cd3As2 or WTe2. Searches for Cd3As2, WTe2, and nanostructures found no passage in our copy addressing the claimed demonstrations in those materials. (LLM, conf=0.77)

  Experimental — accuracy not yet measured

Another phenomenon directly associated with electron condensation to the zeroth LL in topological semimetals is anomalous magnetization.

“Another phenomenon directly associated with electron condensation to the zeroth LL in topological semimetals is anomalous magnetization (224).”

• cites [224] 224. Hu J, Zhu YL, Graf D, Tang ZJ, Liu JY, Mao ZQ. 2017. Quantum oscillation studies of topological semimetal candidate ZrGeM (M = S, Se, Te). Phys. Rev. B 95:205134 227. sourceCould not verify

  The discussion on p. 2 mentions interlayer quantum tunneling from the zeroth Landau level, while p. 4 reports magnetic-torque oscillations in ZrSnTe. Neither passage links electron condensation into the zeroth Landau level to anomalous magnetization. Searches for that link and for magnetization at the quantum limit found no supporting passage in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Another example of unusual transport in the quantum limit due to degeneracy enhancement is the quantum tunneling of relativistic fermions in YbMnBi2.

“Another example of unusual transport in the quantum limit due to degeneracy enhancement is the aforementioned quantum tunneling of relativistic fermions in YbMnBi 2 (247).”

• cites [247] 247. Wang Q, Guo P-J, Sun S, Li C, Liu K, et al. 2018. Extremely large magnetoresistance and high-density Dirac-like fermions in ZrB 2 . Phys. Rev. B 97:205105 250. sourceCould not verify

  The Introduction (p. 1) and Results and Discussion (p. 5) discuss magnetoresistance in TmSb, attributing it to electron–hole compensation and high carrier mobility. Searches for YbMnBi2, relativistic-fermion tunneling, and degeneracy enhancement in the quantum limit found no passage addressing the cited claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Linear MR has been widely observed in many recently reported topological semimetals.

“Linear MR has been widely observed in many of the recently reported topological semimetals (45,172,173,175,234,235,(281)(282)(283).”

• cites [45] 45. Liang T, Gibson Q, Ali MN, Liu M, Cava RJ, Ong NP. 2015. Ultrahigh mobility and giant magnetore- sistance in the Dirac semimetal Cd 3 As 2 . Nat. Mater. 14:280-84 sourceSupported
• cites [172] 172. Novak M, Sasaki S, Segawa K, Ando Y. 2015. Large linear magnetoresistance in the Dirac semimetal TlBiSSe. Phys. Rev. B 91:041203 175. sourceSupported
• cites [173] 173. Yi-Yan W, Qiao-He Y, Tian-Long X. 2016. Large linear magnetoresistance in a new Dirac material BaMnBi 2 . Chin. Phys. B 25:107503 176.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Masuda et al. reported a half-integer QHE in a bulk DSM EuMnBi2 with a layered structure.

“Masuda et al. (177) reported a half-integer QHE in a bulk DSM EuMnBi 2 with a layered structure (Figure 5b).”

• cites [177] 177. Wang Z, Zheng Y, Shen Z, Lu Y, Fang H, et al. 2016. Helicity-protected ultrahigh mobility Weyl fermions in NbP. Phys. Rev. B 93:121112 180. sourceCould not verify

  The paper describes Weyl fermions and quantum oscillations in NbP (pp. 1, 3–4). Searches for Masuda, EuMnBi2, and a half-integer quantum Hall effect found no passage addressing the attributed report. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

A mass enhancement in the quantum limit has been observed for ZrTe5.

“Indeed, a mass enhancement in the quantum limit has been observed for ZrTe 5 (272).”

• cites [272] 272. Wang H, Liu H, Li Y, Liu Y, Wang J, et al. 2018. Discovery of log-periodic oscillations in ultra-quantum topological materials. Sci. Adv. 4:eaau5096 275. sourceCould not verify

  The Materials and Methods section (“Calculation of QL”; page unavailable) estimates the quantum-limit field for the authors’ ZrTe5 crystals and says their observed oscillations occur beyond it. It does not report an observed mass enhancement. Searches for “mass enhancement” and enhanced effective mass in the quantum limit found no supporting passage in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

In real materials, an integer factor may be applied for G0 due to degeneracy, such as graphene with a factor of 4 from spin and valley degeneracies.

“In real materials, an integer factor may be applied for G 0 due to degeneracy, such as graphene with a factor of 4 that originates from spin and valley degeneracies (216,263).”

• cites [216] 216. Hwang EH, Das Sarma S. 2008. Single-particle relaxation time versus transport scattering time in a two-dimensional graphene layer. Phys. Rev. B 77:195412 219. sourcePartly supported

  Section II (p. 4) defines g as the spin and valley degeneracy factor and says that, in general, g = 4 for graphene. The paper applies g to Fermi-wave-vector and screening expressions, but the passages do not say that a factor of 4 is applied to the conductance quantum G₀. (LLM, conf=0.85)

  Experimental — accuracy not yet measured
• cites [263] 263. Brüne C, Liu CX, Novik EG, Hankiewicz EM, Buhmann H, et al. 2011. Quantum Hall effect from the topological surface states of strained bulk HgTe. Phys. Rev. Lett. 106:126803 266. sourcePartly supported

  Appendix C (p. 5) says that a twofold Landau-level degeneracy in HgTe affects which Hall plateaus appear, supporting the general point that degeneracy matters. The paper mentions graphene in the main text (p. 2), but does not state that graphene has a factor of four arising from spin and valley degeneracies; that specific example is unsupported by this source. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The chiral magnetic effect was first observed in Dirac systems such as Bi0.97Sb0.03, Na3Bi, Cd3As2, and ZrTe5 before the experimental discovery of WSMs.

“The chiral magnetic effect was first observed in Dirac systems such as Bi 0.97 Sb 0.03 (301), Na 3 Bi (Figure 8c) (105), Cd 3 As 2 (45, 106), and ZrTe 5 (Figure 8d) (107) before the experimental discovery of WSMs.”

• cites [301] 301. Lv Y-Y, Li X, Zhang B-B, Deng WY, Yao S-H, et al. 2017. Experimental observation of anisotropic Adler-Bell-Jackiw anomaly in type-II Weyl semimetal WTe 1.98 crystals at the quasiclassical regime. Phys. Rev. Lett. 118:096603 304.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [105] 105. Li C-Z, Wang L-X, Liu H, Wang J, Liao Z-M, Yu D-P. 2015. Giant negative magnetoresistance induced by the chiral anomaly in individual Cd 3 As 2 nanowires. Nat. Commun. 6:10137 107. sourcePartly supported

  The Abstract and Discussion report negative magnetoresistance in Cd3As2 nanowires as evidence of the chiral anomaly, supporting Cd3As2 as an example of the phenomenon. The paper does not establish that this observation preceded the experimental discovery of Weyl semimetals, and it does not explicitly call its Cd3As2 result an observation of the chiral magnetic effect. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [45] 45. Liang T, Gibson Q, Ali MN, Liu M, Cava RJ, Ong NP. 2015. Ultrahigh mobility and giant magnetore- sistance in the Dirac semimetal Cd 3 As 2 . Nat. Mater. 14:280-84 sourcePartly supported

  For the Cd3As2 part of the sentence, the paper reports a weak but distinct negative magnetoresistance term (p. 6). It says further experiments are needed to compare that term meaningfully with the predicted chiral term, so it does not establish an observation of the chiral magnetic effect. The paper also does not establish the sentence’s ‘first’ or pre-WSM chronology. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

A TRS-breaking Weyl state has been predicted or established in many magnetic compounds.

“A TRS-breaking Weyl state has also been predicted or established in many magnetic compounds.”

• cites [95] 95. Chang G, Xu S-Y, Zheng H, Singh B, Hsu C-H, et al. 2016. Room-temperature magnetic topological Weyl fermion and nodal line semimetal states in half-metallic Heusler Co 2 TiX (X = Si, Ge, or Sn). Sci. Rep. 6:38839 sourcePartly supported

  The discussion on p. 3 says Co₂TiSn and several other Heusler compounds had been proposed as candidates for a magnetic topological phase; the references on p. 12 identify predictions of magnetic Weyl states. But the paper does not establish a TRS-breaking Weyl state in many magnetic compounds: it reports line nodes in Co₂MnGa (p. 4) and says few topological magnets had been demonstrated (p. 8). (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites [96] 96. Wang Z, Vergniory MG, Kushwaha S, Hirschberger M, Chulkov EV, et al. 2016. Time-reversal- breaking Weyl fermions in magnetic Heusler alloys. Phys. Rev. Lett. 117:236401 sourceSupported
• cites [97] 97. Ernst B, Sahoo R, Sun Y, Nayak J, Muechler L, et al. 2017. Manifestation of the Berry curvature in Co 2 TiSn Heusler films. arXiv:1710.04393 [cond-mat.mtrl-sci] sourceSupported

Recent observations of the chiral anomaly together with band structure calculations suggest a magnetic field-driven Weyl state in AFM RPtBi.

“The recent observations of the chiral anomaly-a unique feature of Weyl fermions-together with band structure calculations suggest a magnetic field-driven Weyl state in AFM RPtBi (109,110).”

• cites [109] 109. Shekhar C, Nayak AK, Singh S, Kumar N, Wu S-C, et al. 2016. Observation of chiral magneto-transport in RPtBi topological Heusler compounds. arXiv:1604.01641 [cond-mat.mtrl-sci] 111. sourceSupported
• cites [110] 110. Wu Y, Mou D, Jo NH, Sun K, Huang L, et al. 2016. Observation of Fermi arcs in type-II Weyl semimetal candidate WTe 2 . Phys. Rev. B 94:121113(R) 112.Not checked — no full text of the source

It has been theoretically predicted that the locations of the Weyl points of these compounds in momentum space can be tuned by the magnetization direction.

“It has been theoretically predicted that the locations of the Weyl points of these compounds in momentum space can be tuned by the magnetization direction (96,97).”

• cites [96] 96. Wang Z, Vergniory MG, Kushwaha S, Hirschberger M, Chulkov EV, et al. 2016. Time-reversal- breaking Weyl fermions in magnetic Heusler alloys. Phys. Rev. Lett. 117:236401 sourceSupported
• cites [97] 97. Ernst B, Sahoo R, Sun Y, Nayak J, Muechler L, et al. 2017. Manifestation of the Berry curvature in Co 2 TiSn Heusler films. arXiv:1710.04393 [cond-mat.mtrl-sci] sourcePartly supported

  The Berry curvature calculation section reports that, for Co₂TiSn with magnetization along z, two Weyl points emerge along k_z. It explains how that magnetization direction changes the crystal symmetries, but does not show that changing the direction tunes the Weyl points’ momentum-space locations; its calculations use the (001) direction. (LLM, conf=0.81)

  Experimental — accuracy not yet measured

These properties, together with the predicted giant anomalous Hall conductivity, make these materials potentially useful for spintronic and thermoelectric applications.

“These properties, together with the predicted giant anomalous Hall conductivity (98,293), make these materials potentially useful for spintronic and thermoelectric applications.”

• cites [98] 98. Kübler J, Felser C. 2016. Weyl points in the ferromagnetic Heusler compound Co 2 MnAl. Europhys. Lett. 114:47005 100. sourcePartly supported

  The Summary reports a calculated anomalous Hall conductivity of 100 S/cm for Co₂TiSn and identifies its potential for spintronics applications. It does not describe the predicted conductivity as giant or discuss thermoelectric applications. (LLM, conf=0.87)

  Experimental — accuracy not yet measured
• cites [293] 293. Chadov S, Qi X, Kübler J, Fecher GH, Felser C, Zhang SC. 2010. Tunable multifunctional topological insulators in ternary Heusler compounds. Nat. Mater 9:541-45 296.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Other magnetic systems such as FM kagomé metal Fe2Sn3, FM spinel CuCr2Se4-xBrx, and magnetic semiconductors have also been reported to display the intrinsic AHE.

“Other magnetic systems such as FM kagomé metal Fe 2 Sn 3 (299), FM spinel CuCr 2 Se 4-x Br x (289), and magnetic semiconductors (288, 291) have also been reported to display the intrinsic AHE.”

• cites [299] 299. Kim H-J, Kim K-S, Wang JF, Sasaki M, Satoh N, et al. 2013. Dirac versus Weyl fermions in topological insulators: Adler-Bell-Jackiw anomaly in transport phenomena. Phys. Rev. Lett. 111:246603 302. sourceCould not verify

  The Conclusion (page unavailable) discusses magnetoconductivity in Bi1-xSbx and the chiral anomaly, not intrinsic anomalous Hall effect in magnetic semiconductors. Searches for intrinsic AHE and the named magnetic materials found no passage in our copy reporting the claimed effect. (LLM, conf=0.73)

  Experimental — accuracy not yet measured
• cites [289] 289. Manyala N, Sidis Y, DiTusa JF, Aeppli G, Young DP, Fisk Z. 2004. Large anomalous Hall effect in a silicon-based magnetic semiconductor. Nat. Mater 3:255-62 292. sourceSupported
• cites [288] 288. Husmann A, Singh LJ. 2006. Temperature dependence of the anomalous Hall conductivity in the Heusler alloy Co 2 CrAl. Phys. Rev. B 73:172417 291. sourceNot checked — no full text of the source

Remarkable anomalous behavior has also been observed in Mn3Sn's Nernst effect.

“Furthermore, remarkable anomalous behavior has also been observed in this material's Nernst effect (57).”

• cites [57] 57. Ikhlas M, Tomita T, Koretsune T, Suzuki M-T, Nishio-Hamane D, et al. 2017. Large anomalous Nernst effect at room temperature in a chiral antiferromagnet. Nat. Phys. 13:1085-90 sourceSupported

An abnormal Hall signal under an in-plane magnetic field was first reported in ZrTe5.

“In experimental studies on DSMs and WSMs, an abnormal Hall signal under in-plane magnetic field was first reported in ZrTe 5 (317).”

• cites [317] 317. Fu L, Kane CL, Mele EJ. 2007. Topological insulators in three dimensions. Phys. Rev. Lett. 98:106803 320. sourceCould not verify

  The paper discusses theoretical topological-insulator phases and mentions bismuth as a possible material (pp. 1, 4). Searches for ZrTe5 and an abnormal Hall signal under an in-plane magnetic field found no passage addressing the claimed experimental report or its priority. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

Since the experimental discoveries of the WSM state in materials such as TaAs class (type I) and (W/Mo)Te2 (type II), many research groups have reported observation of negative longitudinal MR in those materials and have attributed it to the chiral magnetic effect.

“Since the experimental discoveries of the WSM state in materials such as TaAs class (type I) materials (22,23,25,27,39,(86)(87)(88)(89)(90)(91)(92) and (W/Mo)Te 2 (type II) materials (28,(111)(112)(113)(114)(115)(116)(117)(118)(119)(120)(121)(122), many research groups have reported observation of negative longitudinal MR in those materials and have attributed it to the chiral magnetic effect (109”

• cites [109] 109. Shekhar C, Nayak AK, Singh S, Kumar N, Wu S-C, et al. 2016. Observation of chiral magneto-transport in RPtBi topological Heusler compounds. arXiv:1604.01641 [cond-mat.mtrl-sci] 111. sourcePartly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [110] 110. Wu Y, Mou D, Jo NH, Sun K, Huang L, et al. 2016. Observation of Fermi arcs in type-II Weyl semimetal candidate WTe 2 . Phys. Rev. B 94:121113(R) 112.Not checked — no full text of the source
• cites [179] 179. Zhang C, Guo C, Lu H, Zhang X, Yuan Z, et al. 2015. Large magnetoresistance over an extended tem- perature regime in monophosphides of tantalum and niobium. Phys. Rev. B 92:041203(R) 182. sourcePartly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

An in-plane Hall voltage can be generated when electric and magnetic fields are coplanar, due to the chiral anomaly, producing the planar Hall effect.

“However, in-plane Hall voltage can be generated in the presence of coplanar electric and magnetic fields (Figure 9a) due to the chiral anomaly, leading to the so-called PHE (309-315).”

• cites [309] 309. Li P, Zhang CH, Zhang JW, Wen Y, Zhang XX. 2018. Giant planar Hall effect in the Dirac semimetal ZrTe 5-δ . Phys. Rev. B 98:121108(R) 312. sourceSupported
• cites [310] 310. Li H, Wang H-W, He H, Wang J, Shen S-Q. 2018. Giant anisotropic magnetoresistance and planar Hall effect in the Dirac semimetal Cd 3 As 2 . Phys. Rev. B 97:201110 313.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [311] 311. Kumar N, Guin SN, Felser C, Shekhar C. 2018. Planar Hall effect in the Weyl semimetal GdPtBi. Phys. Rev. B 98:041103 314. sourceSupported
• cites [312] 312. Singha R, Roy S, Pariari A, Satpati B, Mandal P. 2018. Planar Hall effect in the type II Dirac semimetal VAl 3 . Phys. Rev. B 98:081103(R) 315.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [313] 313. Wang YJ, Gong JX, Liang DD, Ge M, Wang JR, et al. 2018. Planar Hall effect in type-II Weyl semimetal WTe 2 . arXiv:1801.05929 [cond-mat.mtrl-sci] 316. sourceSupported
• cites [314] 314. West FG. 1963. Rotating-field technique for galvanomagnetic measurements. J. Appl. Phys. 34:1171-73 317. sourceNot checked — no full text of the source
• cites [315] 315. Liang T, Lin J, Gibson Q, Kushwaha S, Liu M, et al. 2018. Anomalous Hall effect in ZrTe 5 . Nat. Phys. 14:451-55 318. sourceSupported

A strict sin 2φ dependence of the planar Hall effect was later observed in ZrTe5, Cd3As2, GdPtBi, WTe2, and VAl3.

“A strict sin 2ϕ dependence was later observed in a number of materials, including ZrTe 5 , Cd3As 2 , GdPtBi, WTe 2 , and VAl 3 (311)(312)(313)(314)(315).”

• cites [311] 311. Kumar N, Guin SN, Felser C, Shekhar C. 2018. Planar Hall effect in the Weyl semimetal GdPtBi. Phys. Rev. B 98:041103 314. sourceSupported
• cites [312] 312. Singha R, Roy S, Pariari A, Satpati B, Mandal P. 2018. Planar Hall effect in the type II Dirac semimetal VAl 3 . Phys. Rev. B 98:081103(R) 315.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [313] 313. Wang YJ, Gong JX, Liang DD, Ge M, Wang JR, et al. 2018. Planar Hall effect in type-II Weyl semimetal WTe 2 . arXiv:1801.05929 [cond-mat.mtrl-sci] 316. sourceSupported
  Experimental — accuracy not yet measured

The planar Hall effect does not satisfy antisymmetry; that is, ρxy = -ρyx.

“Unlike the conventional Hall effect, the PHE does not satisfy antisymmetry; i.e., ρ xy = -ρ yx .”

• cites [309] 309. Li P, Zhang CH, Zhang JW, Wen Y, Zhang XX. 2018. Giant planar Hall effect in the Dirac semimetal ZrTe 5-δ . Phys. Rev. B 98:121108(R) 312. sourceSupported
• cites [310] 310. Li H, Wang H-W, He H, Wang J, Shen S-Q. 2018. Giant anisotropic magnetoresistance and planar Hall effect in the Dirac semimetal Cd 3 As 2 . Phys. Rev. B 97:201110 313.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

At small angle, the angular dependence of magnetoconductivity has a Lorentzian profile with an angular width given by φ ≈ εF vF / lB^2 τ τc.

“At a small angle, the angular dependence of magnetoconductivity has a Lorentzian profile with angular width (309): ϕ ≈ ε F v F /l B 2 τ τ c ,19.”

• cites [309] 309. Li P, Zhang CH, Zhang JW, Wen Y, Zhang XX. 2018. Giant planar Hall effect in the Dirac semimetal ZrTe 5-δ . Phys. Rev. B 98:121108(R) 312. sourcePartly supported

  On p. 4, the paper explicitly says the small-angle magnetoconductivity has a Lorentzian form with angular width Δθ ∼ Lₐ/L꜀. The equation that would express this width in terms of εF, vF, ℓB, τ, and τc is not decoded in our copy, so the specific formula attributed to it cannot be verified. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The planar Hall effect does not originate from the Lorentz force.

“This is because the PHE does not originate from the Lorentz force (309, 310).”

• cites [309] 309. Li P, Zhang CH, Zhang JW, Wen Y, Zhang XX. 2018. Giant planar Hall effect in the Dirac semimetal ZrTe 5-δ . Phys. Rev. B 98:121108(R) 312. sourceSupported
• cites [310] 310. Li H, Wang H-W, He H, Wang J, Shen S-Q. 2018. Giant anisotropic magnetoresistance and planar Hall effect in the Dirac semimetal Cd 3 As 2 . Phys. Rev. B 97:201110 313.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The planar Hall effect has twofold anisotropy under rotating in-plane and out-of-plane fields, unlike the onefold symmetry of the conventional Hall effect.

“With rotating inplane field (Figure 9a) and out-of-plane field (Figure 9b), the twofold anisotropy of the PHE (Figure 9c) clearly differs from the onefold symmetry seen for the conventional Hall effect (Figure 9d) (313).”

• cites [313] 313. Wang YJ, Gong JX, Liang DD, Ge M, Wang JR, et al. 2018. Planar Hall effect in type-II Weyl semimetal WTe 2 . arXiv:1801.05929 [cond-mat.mtrl-sci] 316. sourceSupported

Negative longitudinal magnetoresistance in type II Weyl semimetals is strongly anisotropic.

“Therefore, their negative longitudinal MR is strongly anisotropic (28,306,307); this has been observed in WTe 2 (302,303).”

• cites [28] 28. Soluyanov AA, Gresch D, Wang Z, Wu Q, Troyer M, et al. 2015. Type-II Weyl semimetals. Nature 527:495-98 sourcePartly supported

  In “Novel chiral anomaly in type-II Weyl semimetals” (p. 5), the paper says the chiral anomaly depends on magnetic-field direction and has immediate consequences for transport. It does not specifically state that negative longitudinal magnetoresistance is strongly anisotropic; that is an inference from its result, not a reported finding. (LLM, conf=0.83)

  Experimental — accuracy not yet measured
• cites [306] 306. Sharma G, Goswami P, Tewari S. 2017. Chiral anomaly and longitudinal magnetotransport in type-II Weyl semimetals. Phys. Rev. B 96:045112 309. sourcePartly supported

  The lattice-model results (p. 4) show direction-dependent longitudinal magnetoconductivity, supporting anisotropy in the corresponding negative magnetoresistance. But the paper describes *strong* anisotropy with a vanishing perpendicular response as an earlier suggestion that its own results challenge: it finds a nonzero response at all angles (Discussion and Conclusions, p. 5). (LLM, conf=0.83)

  Experimental — accuracy not yet measured
• cites [307] 307. Burkov AA. 2017. Giant planar Hall effect in topological metals. Phys. Rev. B 96:041110 310.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Therefore, with increasing magnetic field, the voltage drop between voltage contacts may even decrease for asymmetric point-like electrical contacts and irregular sample shape, leading to negative longitudinal MR.

“Therefore, with increasing magnetic field, the voltage drop between voltage contacts may even decrease for asymmetric point-like electrical contacts and irregular sample shape, leading to negative longitudinal MR (10,304,305).”

• cites [10] 10. Armitage NP, Mele EJ, Vishwanath A. 2018. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90:015001 sourceSupported
• cites [304] 304. Udagawa M, Bergholtz EJ. 2016. Field-selective anomaly and chiral mode reversal in type-II Weyl ma- terials. Phys. Rev. Lett. 117:086401 307. sourcePartly supported

  Section II (page unavailable) describes small current contacts that accentuate field-induced current jetting. The passage on case 1 says the edge-contact voltage decreases as the magnetic field increases, yielding a spurious negative longitudinal magnetoresistance. The paper does not establish the claim’s specific conditions of asymmetric contacts and irregular sample shape; it describes platelike samples with contacts placed along the spine and edge. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [305] 305. Yu Z-M, Yao Y, Yang SA. 2016. Predicted unusual magnetoresponse in type-II Weyl semimetals. Phys. Rev. Lett. 117:077202 308. sourceCould not verify

  The paper’s Optical Conductivity discussion (p. 3) addresses magnetic-field effects on optical absorption, not voltage measured between contacts. Searches for asymmetric contacts, irregular sample shape, voltage drop, and negative longitudinal magnetoresistance found no passage addressing the claimed mechanism. (LLM, conf=0.75)

  Experimental — accuracy not yet measured

The narrowing of the AMR peak at high magnetic field can be derived from the longitudinal resistivity expression.

“Another unusual property that can be derived from Equation 18is the narrowing of the AMR peak at high magnetic field (309).”

• cites [309] 309. Li P, Zhang CH, Zhang JW, Wen Y, Zhang XX. 2018. Giant planar Hall effect in the Dirac semimetal ZrTe 5-δ . Phys. Rev. B 98:121108(R) 312. sourcePartly supported

  On p. 4, the paper derives angular narrowing of the longitudinal magnetoconductivity peak using Eq. (15), not Eq. (18). It reports that the angular width decreases roughly as 1/B at low fields but appears to saturate at higher fields. Thus the narrowing is supported, but its attribution to Eq. (18) and its unqualified description as a high-field effect are not. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

Current jetting is also expected to be weak in materials with small transverse MR [e.g., GdPtBi] due to reduced-conductance anisotropy under magnetic fields.

“Current jetting is also expected to be weak in materials with small transverse MR [e.g., GdPtBi (304)] due to reduced-conductance anisotropy under magnetic fields.”

• cites [304] 304. Udagawa M, Bergholtz EJ. 2016. Field-selective anomaly and chiral mode reversal in type-II Weyl ma- terials. Phys. Rev. Lett. 117:086401 307. sourceCould not verify

  The paper discusses field-dependent chiral modes (p. 2) and optical conductivity (pp. 3–4), not current jetting. Searches for GdPtBi, small transverse magnetoresistance, and conductance anisotropy found no passage addressing the cited claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

For example, a quadratic field dependence of negative MR anticipated for a non-quantum limit regime has been observed for most of the above materials.

“For example, a quadratic field dependence of negative MR anticipated for a non-quantum limit regime has been observed for most of the above materials (107, 110, 301).”

• cites [107] 107. Nakajima Y, Hu R, Kirshenbaum K, Hughes A, Syers P, et al. 2015. Topological RPdBi half-Heusler semimetals: a new family of noncentrosymmetric magnetic superconductors. Sci. Adv. 1:e1500242 109. sourceCould not verify

  The passages on charge transport and electrical resistivity (section labeled Abstract; page unavailable) discuss temperature dependence and Hall carrier density, not negative magnetoresistance. Searches for quadratic negative MR and its field dependence in the non-quantum limit found no passage addressing the claim. (LLM, conf=0.74)

  Experimental — accuracy not yet measured
• cites [110] 110. Wu Y, Mou D, Jo NH, Sun K, Huang L, et al. 2016. Observation of Fermi arcs in type-II Weyl semimetal candidate WTe 2 . Phys. Rev. B 94:121113(R) 112.Not checked — no full text of the source
• cites [301] 301. Lv Y-Y, Li X, Zhang B-B, Deng WY, Yao S-H, et al. 2017. Experimental observation of anisotropic Adler-Bell-Jackiw anomaly in type-II Weyl semimetal WTe 1.98 crystals at the quasiclassical regime. Phys. Rev. Lett. 118:096603 304.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Half-Heusler RPtBi is another group of materials that exhibits the magnetic field-induced chiral magnetic effect.

“Half-Heusler RPtBi is another group of materials that exhibits the magnetic fieldinduced chiral magnetic effect (109, 110).”

• cites [109] 109. Shekhar C, Nayak AK, Singh S, Kumar N, Wu S-C, et al. 2016. Observation of chiral magneto-transport in RPtBi topological Heusler compounds. arXiv:1604.01641 [cond-mat.mtrl-sci] 111. sourcePartly supported

  The Abstract (page not specified) says that magnetic-field-induced Weyl points in GdPtBi and NdPtBi are the source of a chiral anomaly, and the paper reports associated negative magnetoresistance. It does not identify the observed phenomenon as the chiral magnetic effect, so that specific attribution is not established by our copy. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [110] 110. Wu Y, Mou D, Jo NH, Sun K, Huang L, et al. 2016. Observation of Fermi arcs in type-II Weyl semimetal candidate WTe 2 . Phys. Rev. B 94:121113(R) 112.Not checked — no full text of the source

For coplanar electric and magnetic fields, the planar Hall transverse resistance follows a sin 2φ dependence with amplitude half the difference between parallel and perpendicular resistivities.

“With coplanar electric and magnetic fields, the transverse resistance ρ xy of the PHE is (309) ρ xy = (ρ || -ρ ⊥ ) 2 sin 2ϕ, 17.”

• cites [309] 309. Li P, Zhang CH, Zhang JW, Wen Y, Zhang XX. 2018. Giant planar Hall effect in the Dirac semimetal ZrTe 5-δ . Phys. Rev. B 98:121108(R) 312. sourcePartly supported

  On p. 3, the paper defines the resistivity difference between directions parallel and perpendicular to the magnetic field and identifies the off-diagonal resistivity in Eq. (15) as the planar Hall effect. The equation itself is not decoded in our copy, so the claimed sin 2φ dependence and factor of one-half cannot be verified from it. (LLM, conf=0.80)

  Experimental — accuracy not yet measured

Longitudinal resistivity can be expressed as ρxx = ρ⊥ + (ρ|| - ρ⊥) cos^2 φ.

“With the above definition of ρ and ρ ⊥ , longitudinal resistivity can be expressed as ( 309) ρ xx = ρ ⊥ + (ρ -ρ ⊥ )cos 2 ϕ. 18.”

• cites [309] 309. Li P, Zhang CH, Zhang JW, Wen Y, Zhang XX. 2018. Giant planar Hall effect in the Dirac semimetal ZrTe 5-δ . Phys. Rev. B 98:121108(R) 312. sourcePartly supported

  On p. 3, the paper defines ρ‖ and ρ⊥ as resistivities parallel and perpendicular to the magnetic field and identifies the first equation of Eq. (15) as the anisotropic-magnetoresistance relation. The equation itself is not decoded in our copy, so the specific cos²φ dependence and coefficient cannot be verified. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Using WHMs as a platform for quantum spin Hall insulators may offer a variable spin-orbit-coupling gap with different W, H, and M combinations, enabling design of different quantum spin Hall insulators.

“One possible advantage of using WHMs as a platform for realizing QSHIs is the variable SOC gap with various combinations of W, H, and M (226); such a gap offers the opportunity to design different QSHIs.”

• cites [226] 226. Hu J, Zhu Y, Gui X, Graf D, Tang Z, et al. 2018. Quantum oscillation evidence of a topological semimetal phase in ZrSnTe. Phys. Rev. B 97:155101 229. sourcePartly supported

  The discussion on p. 9 says that changing the W, H, and M elements can fine-tune spin–orbit coupling in WHM compounds and that their symmetry allows an SOC gap. It discusses WHMs as topological semimetals, however, and does not propose using that gap to realize quantum spin Hall insulators. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

The high temperature at which the conductance plateau develops in WTe2 monolayer is ascribed to the large bulk band gap of the 1T'-WTe2 monolayer.

“and could be ascribed to the large bulk band gap of the 1T -WTe 2 monolayer [which was predicted to be 100 meV (72) and found to be 55 ± 20 meV for MBE-grown samples (73)].”

• cites [72] 72. Tang S, Zhang C, Wong D, Pedramrazi Z, Tsai H-Z, et al. 2017. Quantum spin Hall state in monolayer 1T -WTe 2 . Nat. Phys. 13:683-87 sourceSupported
• cites [73] 73. Fei Z, Palomaki T, Wu S, Zhao W, Cai X, et al. 2017. Edge conduction in monolayer WTe 2 . Nat. Phys. 13:677-82 sourcePartly supported

  The main text (p. 2) reports that the monolayer conductance develops a plateau below about 100 K as the bulk becomes insulating. It does not ascribe that temperature to a large bulk band gap; instead, the discussion (p. 7) considers either a small band gap or a small band overlap, and SI-5 (p. 15) reports an approximate 5 meV activation energy for bulk conduction. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

Tetragonal layered WHM compounds have been predicted to become quantum spin Hall insulators in monolayer form.

“The tetragonal layered WHM compounds have also been predicted to become QSHIs in the monolayer form (322).”

• cites [322] 322. König M, Wiedmann S, Brüne C, Roth A, Buhmann H, et al. 2007. Quantum spin Hall insulator state in HgTe quantum wells. Science 318:766-70 325. sourceCould not verify

  The paper discusses quantum spin Hall behavior in HgTe/(Hg,Cd)Te quantum wells (p. 3) and reports evidence from inverted HgTe wells (p. 8). Searches for WHM, tetragonal layered compounds, and monolayers found no passage addressing the claimed prediction; the HgTe results neither establish nor contradict it. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Transport and spectroscopic evidence supports the quantum spin Hall insulator state in WTe2 monolayers prepared by mechanical exfoliation or molecular beam epitaxy growth.

“There is transport (74,75) and spectroscopic (73) evidence of the QSHI state in WTe 2 monolayers prepared using mechanical exfoliation or molecular beam epitaxy (MBE) growth.”

• cites [74] 74. Wu S, Fatemi V, Gibson QD, Watanabe K, Taniguchi T, et al. 2018. Observation of the quantum spin Hall effect up to 100 Kelvin in a monolayer crystal. Science 359:76-79 sourceSupported
• cites [75] 75. Weng H, Yu R, Hu X, Dai X, Fang Z. 2015. Quantum anomalous Hall effect and related topological electronic states. Adv. Phys. 64:227-82 sourceNot checked — no full text of the source
• cites [73] 73. Fei Z, Palomaki T, Wu S, Zhao W, Cai X, et al. 2017. Edge conduction in monolayer WTe 2 . Nat. Phys. 13:677-82 sourceSupported

The quantum spin Hall insulator state has been proposed in monolayer 1T'-transition metal dichalcogenides MX2 (M = W, Mo; X = S, Se, Te) and in WHM.

“The QSHI state has been proposed in the monolayer form of the layered 1T -transition metal dichalcogenides MX 2 (M = W, Mo; X = S, Se, Te) (72) and WHM (322).”

• cites [72] 72. Tang S, Zhang C, Wong D, Pedramrazi Z, Tsai H-Z, et al. 2017. Quantum spin Hall state in monolayer 1T -WTe 2 . Nat. Phys. 13:683-87 sourceSupported
• cites [322] 322. König M, Wiedmann S, Brüne C, Roth A, Buhmann H, et al. 2007. Quantum spin Hall insulator state in HgTe quantum wells. Science 318:766-70 325. sourceCould not verify

  The citation (322) is attached to the proposed QSHI state in WHM, not to the preceding claim about monolayer transition-metal dichalcogenides. Our copy discusses a proposed QSH state in HgTe/(Hg,Cd)Te quantum wells (p. 3), but the passages found when searching for WHM do not address a proposal in WHM. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

The quantum Hall effect in the 3D layered topological semimetal EuMnBi2 is caused by the formation of 2D electronic states due to restriction of electron motion in the 2D Bi plane.

“In Section 3.2.8,we mention that the QHE in the 3D layered topological semimetal EuMnBi 2 is caused by the formation of 2D electronic states due to restriction of electron motion in the 2D Bi plane (177).”

• cites [177] 177. Wang Z, Zheng Y, Shen Z, Lu Y, Fang H, et al. 2016. Helicity-protected ultrahigh mobility Weyl fermions in NbP. Phys. Rev. B 93:121112 180. sourceCould not verify

  The paper’s discussion on p. 1 concerns Weyl fermions and carrier mobility in NbP, not the quantum Hall effect in EuMnBi2. Searches for EuMnBi2, the Bi plane, and restricted electron motion found no passage addressing the claimed mechanism. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Because the Fermi level crosses the gapped cones and the band dispersion is extremely linear over a wide energy range, WHMs are established as topological nodal-line semimetals.

“Because the Fermi level crosses the gapped cones and the band dispersion is extremely linear over a wide energy range, WHMs have been established as topological NLSMs (78,85,154).”

• cites [78] 78. Liu C-X, Zhang S-C, Qi X-L. 2016. The quantum anomalous Hall effect: theory and experiment. Annu. Rev. Condens. Matter Phys. 7:301-21 sourceCould not verify

  Section 3.1 discusses the Fermi energy and band dispersion of HgTe and InAs/GaSb quantum wells, not WHMs. Searches for WHMs, gapped cones, and nodal-line semimetals found no passage in our copy addressing the claim; page numbers were not provided. (LLM, conf=0.78)

  Experimental — accuracy not yet measured
• cites [85] 85. Xu S-Y, Alidoust N, Belopolski I, Yuan Z, Bian G, et al. 2015. Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nat. Phys. 11:748-54Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [154] 154. Hu J, Tang Z, Liu J, Liu X, Zhu Y, et al. 2016. Evidence of topological nodal-line fermions in ZrSiSe and ZrSiTe. Phys. Rev. Lett. 117:016602 157. sourceNot checked — no full text of the source

A quantum anomalous Hall insulator system provides a promising platform for creation, manipulation, and utilization of Majorana fermions, hypothetical particles that are their own antiparticles.

“Furthermore, a QAHI system also provides a promising platform for the creation, manipulation, and utilization of Majorana fermions, the hypothetical particles that are their own antiparticles (328, 329).”

• cites [328] 328. Chang C-Z, Zhang J, Feng X, Shen J, Zhang Z, et al. 2013. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340:167-70 331. sourcePartly supported

  The paper proposes a QAH-insulator–superconductor hybrid that generates Majorana edge and vortex modes (p. 1), and discusses detecting vortex modes and their potential use in topological quantum computing (pp. 1, 4). It does not describe how to manipulate the modes, and its proposal requires superconducting proximity rather than a QAH insulator alone. (LLM, conf=0.83)

  Experimental — accuracy not yet measured
• cites [329] 329. Checkelsky JG, Yoshimi R, Tsukazaki A, Takahashi KS, Kozuka Y, et al. 2014. Trajectory of the anoma- lous Hall effect towards the quantized state in a ferromagnetic topological insulator. Nat. Phys. 10:731-36 332. sourceCould not verify

  The discussion on p. 2 describes the quantum anomalous Hall state and possible adjacent topological states, but does not connect them to Majorana fermions. Searches for Majorana fermions and superconducting-proximity platforms found no passage addressing their creation, manipulation, or use. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Table 1 shows mobilities of representative topological semimetals acquired from two-band model analysis are high, in the range 10^3-10^6 cm^2/(V·s).

“Table 1 shows the mobilities of some representative topological semimetals acquired from two-band model analysis; the mobilities are indeed high, in the range of 10 3 -10 6 cm 2 /(V•s).”

Cyclotron effective masses derived from quantum oscillations are small for many topological semimetals, reaching as low as 0.02me for some materials.

“As shown in Section 3.2.2, the cyclotron effective masses derived from quantum oscillations are indeed small for many topological semimetals, reaching as low as 0.02m e (where m e is the free electron mass) for some materials.”

Topological semimetals support topologically protected boundary modes such as Fermi arcs and drumhead surface states.

“They support exotic, topologically protected boundary modes such as the topological Fermi arcs and drumhead surface states.”

The Weyl fermion has never been observed as a fundamental particle.

“although the Weyl fermion played a crucial rule in the Standard Model (15), it has never been observed as a fundamental particle.”

• cites [15] 15. Weyl H. 1929. Elektron und Gravitation. I. Z. Phys. 56:330-52 sourceNot checked — no full text of the source

These surface states have been directly observed in spectroscopic measurements.

“These surface states have been directly observed in spectroscopic measurements (19,25,27,(38)(39)(40)(41)(42).”

• cites (19,25,27,(38)(39)(40)(41)(42)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Relativistic fermions in topological semimetals can produce exotic transport properties such as extremely large magnetoresistance, chiral anomaly, and intrinsic anomalous Hall effect.

“Experimental studies have shown that relativistic fermions can result in a rich variety of exotic transport properties, e.g., extremely large magnetoresistance, the chiral anomaly, and the intrinsic anomalous Hall effect.”

Realization of the topological WSM state enables observation of the Weyl fermion in physics.

“The realization of the topological WSM state (22,23,(25)(26)(27) enables the observation of this elusive particle in physics.”

• cites (22,23,(25)(26)(27)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Separation of opposite chiral charges in momentum space leads to surface Fermi arcs connecting Weyl nodes of opposite chirality.

“The separation of the opposite chiral charges in momentum space leads to surface Fermi arcs, whose constant energy contours are open arcs that connect the Weyl nodes of opposite chiralities on the surface.”

Electron transport in topological semimetals is usually strongly affected by external magnetic field.

“Electron transport in topological semimetals is usually strongly affected by external magnetic field.”

In a triple-point semimetal, three singly degenerate bands cross at discrete triple points.

“In such a triple-point semimetal, three singly degenerate bands cross at discrete points, the triple points (Figure 1d).”

Thinning a 3D topological semimetal into 2D may give rise to new 2D topology including quantum spin Hall insulator and quantum anomalous Hall insulator.

“Furthermore, thinning down a 3D topological semimetal into 2D may give rise to new 2D topology, including the quantum spin Hall insulator (QSHI) and the quantum anomalous Hall insulator (QAHI) (14,20,21,(72)(73)(74)(75)(76).”

• cites (14,20,21,(72)(73)(74)(75)(76)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

WSMs lack a global band gap because of Weyl nodes.

“Because of the existence of Weyl nodes, WSMs lack a global band gap.”

A power law field dependence (MR ∝ B^n) is usually seen in various topological semimetals, with exponent n ranging from 1 to 2.

“A power law field dependence (MR ∝ B n ) is usually seen in various topological semimetals, with the exponent n ranging from 1 to 2 (45,46,48,107,(171)(172)(173)(174)(175)(176)(177)(178)(179)(180)(181)(182)(183)(184)(185)(186)(187).”

• cites (45,46,48,107,(171)(172)(173)(174)(175)(176)(177)(178)(179)(180)(181)(182)(183)(184)(185)(186)(187)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Characterization of relativistic fermions through transport measurements provides a convenient approach for verifying a nontrivial topological state, complementary to direct observation by ARPES.

“Characterization of relativistic fermions through transport measurements provides a convenient approach for verifying a nontrivial topological state, complementary to the direct observation of nontrivial band topology by ARPES experiments.”

The two-band model provides only an approximate description for magnetotransport properties of multiple-band materials.

“The two-band model, while widely used, provides only an approximate description for the magnetotransport properties of multiple-band materials.”

Weyl nodes are topologically stable under translational symmetry and cannot be removed without pair annihilation.

“In the presence of translational symmetry, these Weyl nodes are topologically stable and cannot be removed without pair annihilation.”

The Chern number associated with the 2D closed surface corresponds to the topological invariant of a Weyl node (chiral charge).

“Specifically, the Chern number associated with the 2D closed surface directly corresponds to the topological invariant of a Weyl node (i.e., the chiral charge).”

Triple points are not associated with any topological invariant due to the lack of a global band gap on any 2D closed surface enclosing the triple point.

“These triple points are not associated with any topological invariant due to the lack of a global band gap on any 2D closed surface that encloses the triple point.”

Under electron-hole compensation, large MR is expected when mobility is high.

“Under such a circumstance, large MR is expected when mobility is high.”

Chiral anomaly leads to enhanced conductivity with increasing magnetic field.

“Such chiral anomaly leads to enhanced conductivity with an increasing magnetic field.”

WSMs host Weyl fermions as low-energy quasiparticle excitations.

“WSMs are a class of topological semimetals that host Weyl fermions as low-energy quasiparticle excitations (2,(6)(7)(8)(9)(10)(11)(15)(16)(17)(18)(19)(20)(21).”

• cites (2,(6)(7)(8)(9)(10)(11)(15)(16)(17)(18)(19)(20)(21)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Triple points are protected by the combination of a uniaxial rotational axis, mirror planes, and time-reversal symmetry.

“These triple points are protected by the combination of a uniaxial rotational axis, mirror planes, and TRS.”

Type II WSM Fermi surface consists of electron and hole pockets that touch at the type II Weyl nodes.

“Their Fermi surface consists of electron and hole pockets that touch at the type II Weyl nodes.”

In multiple-band systems, the field dependence of Hall resistivity ρxy deviates from linearity.

“However, in multiple-band systems, the field dependence of Hall resistivity ρ xy deviates from linearity.”

In NLSMs, conduction and valence bands cross at 1D lines in k space.

“In NLSMs, conduction and valence bands cross at 1D lines in k space (Figure 1a) (30,40,78,85,133,134,(145)(146)(147)(148)(149)(150)(151)(152)(153)(154)(155)(156)(157)(158)(159)(160)(161).”

• cites (30,40,78,85,133,134,(145)(146)(147)(148)(149)(150)(151)(152)(153)(154)(155)(156)(157)(158)(159)(160)(161)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Absence of a global band gap prevents definition of a topological invariant for the entire 3D bulk Brillouin zone.

“The absence of a global band gap prevents the definition of a topological invariant for the entire 3D bulk Brillouin zone (BZ).”

Existence of Weyl nodes does not rely on additional crystalline point group symmetries.

“The existence of Weyl nodes does not rely on any additional crystalline point group symmetries.”

For nonrelativistic electrons in three dimensions, motion along the magnetic field is not quantized, adding a kz-squared over two-m term to the Landau levels.

“For nonrelativistic electrons in 3D, the motion along the magnetic field direction is not quantized, leading to additional energy of ( k z ) 2 /2m (where k z is the momentum along the magnetic field direction) for LLs:”

In three dimensions, the additional unquantized kz energy makes different Landau levels overlap in energy, producing mixed Landau bands and a continuous energy spectrum.

“For the 3D case, due to the additional energy related to unquantized k z as shown in Equations 6 and 7, different LLs overlap in energy space, leading to a mixture of Landau bands for particular energy (Figure 3e,f ) and a continuous energy spectrum.”

According to Equation 3, even approximate electron-hole compensation should lead to quadratic or nearly-quadratic field dependence for ρxx.

“Third, according to Equation 3, even approximate electron-hole compensation should be able to lead to a quadratic or nearly-quadratic field dependence for ρ xx .”

Massless behavior is naturally expected for ideal topological fermions because they are hosted by linearly dispersed bands crossing near the Fermi level, requiring zero mass in the Hamiltonian.

“Such massless behavior is naturally expected for ideal topological fermions since they are hosted by linearly dispersed bands crossing near the Fermi level, which requires zero mass in the Hamiltonian (11).”

• cites [11] 11. Bernevig A, Weng H, Fang Z, Dai X. 2018. Recent progress in the study of topological semimetals. J. Phys. Soc. Jpn. 87:041001 sourceNot checked — no full text of the source

A nonzero Berry phase originates from a band touching point such as a Dirac node.

“A nonzero Berry phase φ B originates from the band touching point, such as Dirac nodes.”

Under magnetic fields, the cyclotron motion of Dirac fermions induces a Berry phase that changes the phase of quantum oscillations.

“Under magnetic fields, the cyclotron motion of Dirac fermions, i.e., the closed trajectory in momentum space, induces a Berry phase that changes the phase of quantum oscillations.”

For three-dimensional relativistic fermions, an additional energy term from unquantized kz also occurs.

“Similarly, an additional energy term due to unquantized k z also occurs for 3D relativistic fermions:”

Different models have been derived for three-dimensional and two-dimensional quantum oscillations.

“Therefore, different models have been derived for 3D and 2D quantum oscillations.”

LL quantization is completely different for relativistic fermions with linear dispersion.

“The LL quantization is completely different for the relativistic fermions with linear dispersion (Figure 3c).”

The two-band model appears applicable only for a limited field range or at higher temperatures at which quantum effects are not significant.

“With these considerations, the two-band model appears to be applicable only for a limited field range or at higher temperatures at which quantum effects are not significant.”

The chiral zeroth Landau level leads to the chiral anomaly for Weyl fermions.

“As discussed in Section 3.4, the chiral zeroth LL leads to one important effect for Weyl fermions, i.e., the chiral anomaly.”

For relativistic fermions, Landau levels are not equally spaced because the energy depends on the square root of the absolute level index.

“As illustrated in Figure 3d, LLs are no longer equally spaced for relativistic fermions given ε n ∝ √ |n|.”

Greater relaxation time in topological materials may be associated with symmetry protection in many cases.

“Greater relaxation time in topological materials may be associated with symmetry protection in many cases.”

The two-band model successfully explains the extremely large MR arising from high mobility and approximate carrier compensation.

“This model successfully explains the extremely large MR arising from high mobility and approximate carrier compensation.”

The two-band model still provides an effective approach for approximate description of magnetotransport properties of multiband materials.

“Although obtaining the precise value of carrier mobility for individual bands might be challenging, the two-band model still provides an effective approach for the approximate description of magnetotransport properties of multiband materials.”

The Berry phase can deviate from the ideal pi value if the band dispersion is not perfectly linear.

“First, the Berry phase can deviate from an ideal value of π if the band dispersion is not perfectly linear (210).”

• cites [210] 210. Lifshitz IM, Kosevich AM. 1956. Theory of magnetic susceptibility in metals at low temperatures. Sov. Phys. JETP 2:636-45 213.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Reading the Berry phase from a Landau-level fan diagram may have large uncertainty in some cases.

“Furthermore, from the aspect of data analysis, reading the Berry phase from a LL fan diagram may bear large uncertainty in some cases.”

Reaching low Landau-level indices under high magnetic fields is necessary for a reliable Berry phase from a Landau-level fan diagram.

“Therefore, reaching low-LL indices under high magnetic fields is necessary for obtaining a reliable Berry phase from a LL fan diagram.”

Within the LK model based on an oscillating scattering rate, conductivity should show maxima when the scattering rate reaches minima at minimal density of states at the Fermi energy.

“Clearly, within the framework of this LK model based on the oscillation scattering rate, conductivity should exhibit maxima when the scattering rate reaches minima that occur at minimal DOS(E F ).”

Quantum mobility obtained from quantum oscillation is usually less than transport mobility derived from magnetotransport, as observed in various topological semimetals.

“Consequently, μ q obtained from quantum oscillation is usually less than μ t derived from magnetotransport, as observed in various topological semimetals (see Table 1).”

The requirement of a high potential barrier (~mc^2) imposes a great challenge for experimental observation of this phenomenon in particle physics.

“However, the requirement of the high potential barrier (∼mc 2 ) imposes a great challenge for the experimental observation of this phenomenon in particle physics.”

A similar effect is expected in topological semimetals with massless relativistic fermions.

“A similar effect is expected in topological semimetals with massless relativistic fermions.”

For a Landau-level fan diagram, linear extrapolation to zero inverse field gives an intercept of zero for nonrelativistic electrons and one half for relativistic fermions because the zeroth Landau level is pinned at zero energy.

“For a LL fan diagram established with such a definition of the LL index, the linear extrapolation of the linear fit of n(1/B) to the 1 B → 0 limit must lead to n = 0 for nonrelativistic electrons, but n = 1/2 for relativistic fermions due to the zeroth LL pinned at the zero energy.”

The one-half intercept corresponds to an ideal Berry phase of pi.

“This n = 1/2 intercept corresponds to an ideal Berry phase of π.”

To observe such a Weyl orbit, it is necessary to reduce the sample size to suppress the contribution of bulk states.

“To observe such a Weyl orbit, it is necessary to reduce the sample size to suppress the contribution of the bulk states.”

Transport relaxation time is usually larger or much larger than quantum relaxation time.

“Therefore, τ t is usually larger or even much larger than τ q .”

Dingle temperature and Berry phase can be extracted by fitting the oscillation pattern to a generalized multiband LK formula, assuming that quantum oscillations of different bands are additive.

“The Dingle temperature and Berry phase can be extracted through fitting the oscillation pattern to the generalized multiband LK formula, with the assumption that the quantum oscillations of different bands are additive.”

• AssumptionThe authors state this as an assumption: Quantum oscillations from different bands are additive.

Because the Shubnikov-de Haas effect originates from Landau quantization, the nontrivial Berry phase associated with the zeroth Landau level manifests as a phase shift in the Shubnikov-de Haas oscillation.

“Since the SdH effect also originates from Landau quantization, the nontrivial Berry phase associated with the zeroth LL also manifests itself by a phase shift in the SdH oscillation.”

Matching electron and positron wave functions across the barrier leads to high-probability tunneling.

“The matching between electron and positron wave functions across the barrier leads to high-probability tunneling (257).”

• cites [257] 257. O'Brien TE, Diez M, Beenakker CWJ. 2016. Magnetic breakdown and Klein tunneling in a type-II Weyl semimetal. Phys. Rev. Lett. 116:236401 260. sourceNot checked — no full text of the source

Quantum oscillations due to Weyl orbits exhibit anomalous properties such as a sample thickness-dependent phase shift.

“Quantum oscillations due to Weyl orbits exhibit anomalous properties such as a sample thickness-dependent phase shift.”

For a DSM whose Dirac node can be viewed as the superposition of two Weyl nodes with opposite chirality, its surface state exhibits two sets of Fermi arcs curving in opposite directions on two opposite surfaces.

“For a DSM whose Dirac node can be viewed as the superposition of two Weyl nodes with opposite chirality, its surface state exhibits two sets of Fermi arcs curving in opposite directions on two opposite surfaces, as shown in Figure 4d.”

The Zeeman effect also causes the Berry phase obtained from a Landau-level fan diagram to deviate.

“Second, the Zeeman effect, which has not been considered so far, also leads to a deviation of the Berry phase obtained from a LL fan diagram (210).”

• cites [210] 210. Lifshitz IM, Kosevich AM. 1956. Theory of magnetic susceptibility in metals at low temperatures. Sov. Phys. JETP 2:636-45 213.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Magnetic breakdown becomes more pronounced at high fields because the tunneling probability scales exponentially with inverse field as e^{-α/B}, where α depends on the k-space separation of the orbits.

“This phenomenon, termed magnetic breakdown, becomes more pronounced at high fields because the tunneling probability scales exponentially with the inverse field 1/B as e -α /B , where α is a material-dependent parameter relevant to the k-space separation of the orbits (203).”

• cites [203] 203. McClure JW. 1956. Diamagnetism of graphite. Phys. Rev. 104:666-71 206. sourceNot checked — no full text of the source

The spin-damping factor can be used to extract the Lande g-factor of a two-dimensional or quasi-two-dimensional system via the spin-zero method, where the oscillation amplitude vanishes at certain field orientations due to interference of spin-split Fermi surfaces.

“Furthermore, R S can be used to extract the Landé g-factor of a 2D/quasi-2D system via the spin-zero method; that is, the oscillation amplitude vanishes at some field orientation due to the interference of spin split Fermi surfaces.”

Because integer Landau-level indices correspond to minima in the density of states at the Fermi energy, maxima of conductivity oscillation should be assigned integer Landau-level indices.

“Given that integer LL indices should correspond to DOS(E F ) minima as indicated above, the maxima of conductivity oscillation should be assigned with integer LL indices.”

A simple connection between integer Landau-level indices and Shubnikov-de Haas oscillation extrema may be problematic in some cases.

“As a result, a simple connection between the integer LL indices and the SdH oscillation extrema may be problematic in some cases.”

Although the integer quantum Hall effect has a semiclassical interpretation based on Landau quantization, its transport mechanism is distinct from the Shubnikov-de Haas effect because of its nonlocal character.

“Although the integer quantum Hall effect (QHE) also has a semiclassical interpretation based on Landau quantization, its underlying transport mechanism is distinct from the SdH effect due to its nonlocal character.”

Given the difference in Landau quantization in 2D and 3D systems, it is challenging to probe the half-integer QHE in 3D topological semimetals.

“Given the difference in Landau quantization in 2D and 3D systems as mentioned in Section 3.2.1, it is challenging to probe the half-integer QHE in 3D topological semimetals.”

This gives rise to quantized Hall conductance Gxy = nG0, where G0 = e^2/h.

“This gives rise to quantized Hall conductance of G xy = nG 0 , where G 0 = e 2 /h is the conductance quantum.”

The quantum limit corresponding to (1/ρxy)/(1/ρxy0) = 1/2 could not be reached in EuMnBi2 because the canted AFM state of the Eu sublattice exists only in a limited field range.

“However, the quantum limit corresponding to (1/ρ xy )/(1/ρ xy 0 ) = 1/2 could not be reached in this system because the canted AFM state of Eu sublattice exists only in a limited field range.”

The quantum limit is not accessible under a moderate magnetic field for most materials with high carrier density (large Fermi surface and large quantum oscillation frequency).

“The quantum limit is not accessible under a moderate magnetic field for most materials with high carrier density (i.e., large Fermi surface and large quantum oscillation frequency).”

At the quantum limit, magnetization is entirely due to the zeroth LL states, with M_{n=0} = -∂ε_{n=0,k}/∂B.

“At the quantum limit, magnetization is entirely due to the zeroth LL states, with M n=0 = -∂ε n=0,k /∂B.”

In EuMnBi2, 1/ρxy normalized by 1/ρxy0 displays quantized plateaus with half-integers.

“As seen in Figure 5c, 1/ρ xy normalized by 1/ρ xy 0 (where ρ xy 0 is the step size of successive plateaus) displays quantized plateaus with half-integers.”

Given the existence of the field-independent zeroth LL pinned at the band crossing point, there is always an edge channel formed by the zeroth LL.

“Given the existence of the field-independent zeroth LL pinned at the band crossing point (Figure 3d,f ), there is always an edge channel formed by the zeroth LL, as shown in Figure 5a.”

EuMnBi2 exhibits coexistence of two AFM orders: one formed by the Mn sublattice and the other by the Eu sublattice.

“This material exhibits the coexistence of two AFM orders: one formed by the Mn sublattice and the other by the Eu sublattice.”

The Landau quantization for a 3D WSM yields energy spectra ε_n,k = v_F sgn(n) sqrt(2e|B||n| + 2k_z^2) + χ v_F k_z for n ≠ 0, and ε_0 = χ v_F k_z for n = 0.

“The Landau quantization for a 3D WSM yields energy spectra of ε n,k = v F sgn(n) 2e |B||n| + 2 k 2 z , χ v F k z , n = 0, n = 0,16.”

In the quantum limit, MR grows linearly with B.

“In the quantum limit, however, MR grows linearly with B (195).”

• cites [195] 195. Chen YL, Chu J-H, Analytis JG, Liu ZK, Igarashi K, et al. 2010. Massive Dirac fermion on the surface of a magnetically doped topological insulator. Science 329:659-62 198. sourceNot checked — no full text of the source

The critical field needed to reach a quantum limit is at least comparable to the quantum oscillation frequency.

“From this definition, one can find that the critical field needed to reach a quantum limit is at least comparable to the quantum oscillation frequency.”

In addition to the above phenomena, new quantum states in the quantum limit regime have been proposed.

“In addition to the above phenomena associated with the properties of the relativistic Dirac or Weyl fermions on the zeroth LL, new quantum states in the quantum limit regime have been proposed (274, 275).”

• cites [274] 274. Xu R, Husmann A, Rosenbaum TF, Saboungi ML, Enderby JE, Littlewood PB. 1997. Large magne- toresistance in non-magnetic silver chalcogenides. Nature 390:57-60 277. sourceNot checked — no full text of the source
• cites [275] 275. Hu J, Liu TJ, Qian B, Mao ZQ. 2013. Coupling of electronic and magnetic properties in Fe 1+y (Te 1-x Se x ). Phys. Rev. B 88:094505 278.Not matched

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Under sufficiently strong field, electrons are pinned to quantized small-radius orbits, causing a bulk insulating state.

“Semiclassically, under sufficiently strong field, the electrons are pinned to these quantized small radii orbits, which causes a bulk insulating state.”

At low field when LLs are not well separated, LL broadening and thermal excitations smear out discrete LLs, leading to conventional (sinθ)^2 dependence for the angular-dependent interlayer resistance (AMR).

“For example, for the experimental setup shown in Figure 5e, at low field when LLs are not well separated, LL broadening and thermal excitations smear out discrete LLs, which leads to conventional (sinθ ) 2 dependence for the angular-dependent interlayer resistance (AMR) (Figure 5f, inset).”

For ZrTe5, whose carrier density varies with different crystal growth techniques, its quantum limit can be reached under a very small magnetic field (~0.2 T) for low-carrier-density samples.

“For ZrTe 5 , whose carrier density varies with different crystal growth techniques, its quantum limit can be reached under a very small magnetic field (∼0.2 T) for low-carrier-density samples.”

High magnetic field may also lead to annihilation of a Weyl state.

“High magnetic field may also lead to annihilation of a Weyl state.”

Therefore, one can expect a collapse of magnetization for topological semimetals crossing the quantum limit.

“Therefore, one can expect a collapse of magnetization for topological semimetals crossing the quantum limit.”

The 3D Landau quantization of a WSM leads to counterpropagating zeroth LLs for a pair of Weyl cones, which disperse only along the magnetic field direction.

“As described in Equation 16and illustrated in Figure 8a, the 3D Landau quantization of a WSM leads to counterpropagating zeroth LLs for a pair of Weyl cones, which disperse only along the magnetic field direction.”

As a result, the chirality becomes imbalanced.

“As a result, the chirality becomes imbalanced.”

Negative longitudinal MR related to the chiral anomaly has been discovered in several topological semimetal systems.

“Negative longitudinal MR (i.e., the increase in magnetic field parallel to the electrical current leading to a decrease of resistivity) related to the chiral anomaly has been discovered in several topological semimetal systems, as shown below.”

Mn3Sn is the first antiferromagnet discovered to exhibit such a surprisingly large room temperature AHE.

“In particular, Mn 3 Sn is the first antiferromagnet to be discovered to exhibit such a surprising large room temperature AHE (100).”

• cites [100] 100. Nayak AK, Fischer JE, Sun Y, Yan B, Karel J, et al. 2016. Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn 3 Ge. Sci. Adv. 2:e1501870 102. sourceNot checked — no full text of the source

In a magnetic WSM, a pair of Weyl nodes with opposite chirality can be seen as monopole sources of Berry curvature.

“In a magnetic WSM, a pair of Weyl nodes with opposite chirality can be seen as monopole sources of Berry curvature.”

In real materials, the situation can be further complicated by positive orbital MR due to the Lorentz effect, which is determined by the magnetic field component perpendicular to current.

“In real materials, the situation can be further complicated by positive orbital MR due to the Lorentz effect, which is determined by the magnetic field component perpendicular to current, as discussed in Section 3.1.”

If the negative MR is too sensitive to field orientation (e.g., it disappears when the magnetic field is deviated by 1° or 2° from the parallel direction), it may suggest a classical origin of current jetting.

“If the negative MR is too sensitive to field orientation (e.g., it disappears when the magnetic field is deviated by 1° or 2° from the parallel direction), it may suggest a classical origin of current jetting, which is discussed below.”

This direction is also the direction for electrons to have coherent motion when an external electric field E is applied.

“This direction is also the direction for electrons to have coherent motion when an external electric field E is applied.”

This individual conservation of particles with opposite chirality is violated in the presence of parallel electric and magnetic fields.

“This individual conservation of particles with opposite chirality is violated in the presence of parallel electric and magnetic fields.”

Clearly, this chiral current contributes to negative MR when E//B.

“Clearly, this chiral current contributes to negative MR when E//B.”

The chiral magnetic effect is not limited to the case of exact E//B, since the chiral charge pumping rate is finite for nonorthogonal electric and magnetic fields.

“It is also worth noting that the chiral magnetic effect is not limited to the case of exact E//B, since the chiral charge pumping rate is finite for nonorthogonal electric and magnetic fields.”

Such an intrinsic Hall component can be understood in terms of the Berry curvature of the electronic Bloch states, which leads to an anomalous electron group velocity perpendicular to the longitudinal electric field.

“Such an intrinsic Hall component can be understood in terms of the Berry curvature of the electronic Bloch states, which leads to an anomalous electron group velocity perpendicular to the longitudinal electric field [(e/ ) E × ] (288).”

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One important feature of magnetic WSMs is their intrinsic AHE.

“One important feature of magnetic WSMs is their intrinsic AHE.”

This effect can be attributed to the fact that the Dirac point in a 3D DSM can be viewed as a superposition of two paired Weyl nodes with opposite chirality.

“This effect can be attributed to the fact that the Dirac point in a 3D DSM can be viewed as a superposition of two paired Weyl nodes with opposite chirality.”

Such two overlapping Weyl nodes can be separated in momentum space by magnetic field, which breaks TRS.

“Such two overlapping Weyl nodes can be separated in momentum space by magnetic field, which breaks TRS (Figure 8b).”

At low fields, Landau levels are wiped out by energy level broadening and thermal excitation.

“At low fields, LLs are wiped out by energy level broadening and thermal excitation.”

When a strong magnetic field drives the system to the quantum limit, the field dependence of each parameter in the width expression leads to saturation of φ.

“When a strong magnetic field drives the system to the quantum limit, the field dependence of each parameter in Equation 19leads to the saturation of ϕ, as shown in Figure 9e,f (176).”

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The difference between parallel and perpendicular resistivity represents the resistivity anisotropy caused by the chiral anomaly.

“Therefore, ρ || -ρ ⊥ represents the resistivity anisotropy caused by the chiral anomaly.”

Negative longitudinal magnetoresistance has been observed in WTe2.

“Therefore, their negative longitudinal MR is strongly anisotropic (28,306,307); this has been observed in WTe 2 (302,303).”

• cites [302] 302. Liang S, Lin J, Kushwaha S, Xing J, Ni N, et al. 2018. Experimental tests of the chiral anomaly magne- toresistance in the Dirac-Weyl semimetals Na 3 Bi and GdPtBi. Phys. Rev. X 8:031002 305.Not matched

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Current jetting is simply due to the rule that the current flows predominately along the high-conductance direction.

“Current jetting is simply due to the rule that the current flows predominately along the high-conductance direction.”

The chiral anomaly produces a nontrivial transverse Hall signal when the magnetic field is in the plane.

“In addition to generating negative MR in longitudinal transport, the chiral anomaly also leads to a nontrivial transverse (Hall) signal under in-plane magnetic field (Figure 9a).”

The Hall conductance is given by Gxy = C e^2/h, where C is the Chern number.

“The Hall conductance can be obtained from the Chern number C by G xy = Ce 2 /h; a half-quantized Hall conductance is also expected for massless relativistic fermions, as discussed in Section 3.2.8(Equation 15).”

One possible route to realize the predicted quantum spin Hall insulator state is to exfoliate bulk WHMs to their monolayers.

“To realize the predicted QSHI state, one possible route is to exfoliate the bulk WHMs to their monolayers.”

For a quantum spin Hall insulator, Gxy = 0 because C = 0 in a time-reversal-symmetric system, attributed to cancellation of the pair of time-reversed chiral edge states.

“For a QSHI, G xy = 0 due to C = 0 in a TRS system, which can be attributed to the fact that the pair of time-reversed chiral edge states cancels each other (Figure 10c).”

The topological classification of a quantum spin Hall insulator can be extended to 3D to create a strong 3D topological insulator.

“It is also possible to extend the topological classification of a QSHI to 3D and create a strong 3D topological insulator (319).”

• cites [319] 319. Kou X, Fan Y, Wang KL. 2017. Review of quantum Hall trio. J. Phys. Chem. Solids In press 322. sourceNot checked — no full text of the source

Stacking these 2D building blocks yields a 3D weak Chern insulator or weak topological insulator that is not robust against disorder.

“Simple stacking of these 2D building blocks leads to a 3D weak Chern insulator or a weak topological insulator that is not robust against disorder (319).”

• cites [319] 319. Kou X, Fan Y, Wang KL. 2017. Review of quantum Hall trio. J. Phys. Chem. Solids In press 322. sourceNot checked — no full text of the source

The quantum anomalous Hall insulator state was first experimentally demonstrated in magnetically doped topological insulators.

“The QAHI state was first experimentally demonstrated in magnetically doped topological insulators (330-332).”

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The quantum anomalous Hall insulator state has so far been realized only at very low temperatures below 1 K.

“However, it has so far been realized only at very low temperatures (<1 K) (330-332).”

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Topological semimetals exhibit a rich variety of exotic properties that are not seen in nonrelativistic electron systems.

“The above discussion shows that topological semimetals exhibit a rich variety of exotic properties that are not seen in nonrelativistic electron systems.”

Recent studies revealed a possible route to high-temperature quantum anomalous Hall insulators: 3D ferromagnetic Weyl semimetals can evolve into large-gap quantum anomalous Hall insulators when dimensionality is reduced from 3D to 2D, due to confinement-induced quantization of low-energy states.

“Recent studies have revealed a new possible route to the realization of high-temperature QAHIs: 3D FM WSMs can evolve into large-gap QAHIs when the dimensionality is reduced from 3D to 2D, due to the confinementinduced quantization of low-energy states (21).”

• cites [21] 21. Xu G, Weng H, Wang Z, Dai X, Fang Z. 2011. Chern semimetal and the quantized anomalous Hall effect in HgCr 2 Se 4 . Phys. Rev. Lett. 107:186806Not matched

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There have been recent breakthroughs in topological phase screening and database development for topological quantum materials.

“There have been recent breakthroughs in topological phase screening and database development for topological quantum materials (37, 325, 333-336a).”

• cites 37,325,333-336aNot matched

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