The DOI matches its Crossref record; 59 of the 59 references it cites were found in a registry or an index.
Riemannian geometry of resonant optical responses
Open accessOverview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source says what the claim attributes to it.
In the citing paper
“While it is possible to frame the nonlinear optical response of topological materials in terms of local geometric quantities originating from interband Berry connection44,95”
From Ultrafast terahertz emission from emerging symmetry-broken materials (2023) · cites this paper as [95]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Generalized Gradient Approximation Made Simple
Perdew, J. P., Burke, K., & Ernzerhof, M. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.3865 · Cited by 215,824 · cited 4×
“We perform firstprinciples calculations based on the density functional theory with the generalized gradient approximation (GGA) [52].”
MethodInfluentialEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
Kresse, G., & Furthmüller, J. (1996) · Physical Review B · doi:10.1103/physrevb.54.11169 · Cited by 123,652 · cited 2×
“The electronic structure calculations are carried out using the accurate projector-augmented wave (PAW) method, as implemented in the Vienna ab-initio simulation package (VASP) [53, 54].”
MethodAb initio molecular dynamics for liquid metals
Kresse, G., & Häfner, J. (1993) · Physical review. B, Condensed matter · doi:10.1103/physrevb.47.558 · Cited by 46,097
Colloquium : Topological insulators
Hasan, M. Z., & Kane, C. L. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.3045 · Cited by 19,930
Quantized Hall Conductance in a Two-Dimensional Periodic Potential
Thouless, D. J., Kohmoto, M., Nightingale, M. P., & den Nijs, M. (1982) · Physical Review Letters · doi:10.1103/physrevlett.49.405 · Cited by 6,973 · cited 1×
“The real projective line RP(1) ' S(1) is the only compact and connected onedimensional manifold [1].”
BackgroundBerry phase effects on electronic properties
Xiao, D., Chang, M.-C., & Niu, Q. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.1959 · Cited by 5,277 · cited 2×
“As we show in Supplementary Note 3, w 1 = w n 1 − w 1 , (63) where w 1 and w m 1 are the first Stiefel-Whitney numbers of bands n and m, or equivalently, the Berry phases of bands n and m divided by π [3, 4].”
BackgroundNagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H., & Ong, N. P. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.1539 · Cited by 4,798
Self-citationWeyl and Dirac semimetals in three-dimensional solids
Armitage, N. P., Mele, E. J., & Vishwanath, A. (2018) · Reviews of Modern Physics · doi:10.1103/revmodphys.90.015001 · Cited by 4,528
Self-citationObservation of a large-gap topological-insulator class with a single Dirac cone on the surface
Xia, Y., Qian, D., Hsieh, D., Wray, L., Pal, A., Lin, H., Bansil, A., Grauer, D., Hor, Y. S., Cava, R. J., et al. (2009) · Nature Physics · doi:10.1038/nphys1274 · Cited by 3,765 · cited 1×
“These materials are Z2 topological insulators in two and three dimensions [37, 38], and a Chern insulator [39], respectively, described by a massive Dirac Hamiltonian close to the band edge.”
BackgroundMaximally localized Wannier functions: Theory and applications
Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I., & Vanderbilt, D. (2012) · Reviews of Modern Physics · doi:10.1103/revmodphys.84.1419 · Cited by 3,283
Karplus, R., & Luttinger, J. M. (1954) · Physical Review · doi:10.1103/physrev.95.1154 · Cited by 1,737
Photovoltaic Hall effect in graphene
Oka, T., & Aoki, H. (2009) · Physical Review B · doi:10.1103/physrevb.79.081406 · Cited by 1,423 · cited 2×
“Note that the response at high photon energies follows Oka and Aoki’s result [35] for massless graphene.”
BackgroundQuantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials
Sodemann, I., & Fu, L. (2015) · Physical Review Letters · doi:10.1103/physrevlett.115.216806 · Cited by 1,145 · cited 2×
“As we show in Supplementary Note 3, w 1 = w n 1 − w 1 , (63) where w 1 and w m 1 are the first Stiefel-Whitney numbers of bands n and m, or equivalently, the Berry phases of bands n and m divided by π [3, 4].”
BackgroundSecond-order optical response in semiconductors
Sipe, J. E., & Shkrebtii, A. I. (2000) · Physical Review B · doi:10.1103/physrevb.61.5337 · Cited by 910 · cited 1×
“To explain the Hermitian connection, we write it as Cbca = r nmr a mn,c = −ir nmr mnR mn, where R mn = Amm−Ann + i∂c log r a mn is the shift vector describing the shift of the electron position during the optical excitation (note that R mn ∼ r mm−r nn) [25, 34].”
MethodNonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis
Aversa, C., & Sipe, J. E. (1995) · Physical review. B, Condensed matter · doi:10.1103/physrevb.52.14636 · Cited by 903 · cited 6×
“e Q be (C )ca = (ê mn b ,∇cê a ) = r nmr mn,c, (8) where Omn,c ≡ ∂cOmn − i[Ac, O]mn is called the generalized derivative [29], where (Ac)mn = δmn 〈um|i∂c|un〉 is the U(1) Berry connection.”
BackgroundInfluentialRiemannian structure on manifolds of quantum states
Provost, J.-P., & Vallée, G. (1980) · Communications in Mathematical Physics · doi:10.1007/bf02193559 · Cited by 894 · cited 1×
“The fundamental reason for this is the absence of a proper theory explaining the geometry of optical transitions; the geometric quantities like the quantum metric and the Berry curvature are usually defined for a single quantum state [28], while optical transitions involve a pair of states.”
BackgroundQuantized circular photogalvanic effect in Weyl semimetals
de Juan, F., Grushin, A. G., Morimoto, T., & Moore, J. E. (2017) · Nature Communications · doi:10.1038/ncomms15995 · Cited by 708
Nonreciprocal responses from non-centrosymmetric quantum materials
Tokura, Y., & Nagaosa, N. (2018) · Nature Communications · doi:10.1038/s41467-018-05759-4 · Cited by 694
Self-citationSuperfluidity in topologically nontrivial flat bands
Peotta, S., & Törmä, P. (2015) · Nature Communications · doi:10.1038/ncomms9944 · Cited by 662
Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation
Wang, X., Yates, J. R., Souza, I., & Vanderbilt, D. H. (2006) · Physical Review B · doi:10.1103/physrevb.74.195118 · Cited by 596
Topological nature of nonlinear optical effects in solids
Morimoto, T., & Nagaosa, N. (2016) · Science Advances · doi:10.1126/sciadv.1501524 · Cited by 569
Self-citationGermanene: the germanium analogue of graphene
Acun, A., Zhang, L., Bampoulis, P., Farmanbar, M., van Houselt, A., Rudenko, A. N., Lingenfelder, M., Brocks, G., Bene Poelsema, B., Katsnelson, M. I., et al. (2015) · Journal of Physics Condensed Matter · doi:10.1088/0953-8984/27/44/443002 · Cited by 503 · cited 1×
“These materials are Z2 topological insulators in two and three dimensions [37, 38], and a Chern insulator [39], respectively, described by a massive Dirac Hamiltonian close to the band edge.”
BackgroundGeometry, Topology and Physics, Second Edition
Nakahara, M. (2003) · Taylor & Francis eBooks · doi:10.1201/9781420056945 · Cited by 465
Blount, E. I. (1962) · Solid state physics · doi:10.1016/s0081-1947(08)60459-2 · Cited by 461 · cited 1×
“The position operator is represented in the Bloch state basis as [31, 32] r mn(k) = δmni∂a + 〈umk|i∂a|unk〉 .”
BackgroundTopology-Bounded Superfluid Weight in Twisted Bilayer Graphene
Xie, F., Song, Z., Lian, B., et al. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.167002 · Cited by 273
Ahn, J., Guo, G.-Y., & Nagaosa, N. (2020) · Physical Review X · doi:10.1103/physrevx.10.041041 · Cited by 272 · cited 2×
“Since a large number of k-points are needed to get accurate NLO responses [19], we use the efficient Wannier function interpolation method based on maximally localized Wannier functions (MLWFs) [55–57].”
BackgroundMethodSelf-citationBand Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
Ahn, J., Kim, D., Kim, Y., & Yang, B. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.106403 · Cited by 268
Self-citationCircular photogalvanic effect on topological insulator surfaces: Berry-curvature-dependent response
Hosur, P. (2011) · Physical Review B · doi:10.1103/physrevb.83.035309 · Cited by 258
Chiral optical response of multifold fermions
Flicker, F., de Juan, F., Bradlyn, B., Morimoto, T., Vergniory, M. G., & Grushin, A. G. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.98.155145 · Cited by 205 · cited 3×
“edu cal responses in more-than-two-band systems such as chiral multifold fermions [22].”
BackgroundInfluentialAb initio calculation of the shift photocurrent by Wannier interpolation
Ibañez-Azpiroz, J., Tsirkin, S. S., & Souza, I. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.245143 · Cited by 196
The Gauss-Bonnet theorem for Riemannian polyhedra
Allendoerfer, C. B., & WEIL, A. A. (1943) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1943-0007627-9 · Cited by 188
Chiral Photocurrent in Parity-Violating Magnet and Enhanced Response in Topological Antiferromagnet
Watanabe, H., & Yanase, Y. (2021) · Physical Review X · doi:10.1103/physrevx.11.011001 · Cited by 187
Geometrical effects in orbital magnetic susceptibility
Gao, Y., Yang, S. A., & Niu, Q. (2015) · Physical Review B · doi:10.1103/physrevb.91.214405 · Cited by 174
Holder, T., Kaplan, D., & Yan, B. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033100 · Cited by 165
Quantum distance and anomalous Landau levels of flat bands
Rhim, J., Kim, K., & Yang, B. (2020) · Nature · doi:10.1038/s41586-020-2540-1 · Cited by 162
Tan, X., Zhang, D.-W., Yang, Z., Chu, J., Zhu, Y.-Q., Li, D., Yang, X., Song, S., Han, Z., Li, Z., et al. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.210401 · Cited by 154
Chiral Photocurrent in Parity-Violating Magnet and Enhanced Response in Topological Antiferromagnet
Watanabe, H., & Yanase, Y. (2021) · Institutional Repositories DataBase (IRDB) · Cited by 128
Gauge covariances and nonlinear optical responses
Ventura, G. B., Passos, D. J., dos Santos, J. M. B. L., et al. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.96.035431 · Cited by 122 · cited 6×
“Typical quantities in optical conductivity tensors: ∂a = ∂∂ka , ∫ k = ∫ ddk (2π)d , Ĥ0 = single-particle Hamiltonian of electrons with no external field , En = ⟨ ψnk|Ĥ0|ψnk ⟩ = ⟨ unk|Ĥ0(k)|unk ⟩ , vamn = ih̄ ⟨ ψmk|[̂ra, Ĥ0]|ψnk ⟩ = h̄−1 ⟨ umk|∂aĤ0(k)|unk ⟩ , fm = Fermi-Dirac distribution of them th band , ωmn = h…”
BackgroundInfluentialMeasuring the quantum geometry of Bloch bands with current noise
Neupert, T., Chamon, C., & Mudry, C. (2013) · Physical Review B · doi:10.1103/physrevb.87.245103 · Cited by 114
Stiefel–Whitney classes and topological phases in band theory
Ahn, J., Park, S., Kim, D., Kim, Y., & Yang, B.-J. (2019) · Chinese Physics B · doi:10.1088/1674-1056/ab4d3b · Cited by 100
Self-citationNonreciprocal Directional Dichroism Induced by the Quantum Metric Dipole
Gao, Y., & Xiao, D. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.227402 · Cited by 99
Hazra, T., Verma, N., & Randeria, M. (2019) · Physical Review X · doi:10.1103/physrevx.9.031049 · Cited by 97
Accurate Gap Determination in Monolayer and Bilayer Graphene/ h -BN Moiré Superlattices
Kim, H., Leconte, N., Chittari, B. L., et al. (2018) · Nano Letters · doi:10.1021/acs.nanolett.8b03423 · Cited by 94 · cited 1×
“Figure 2(b) shows the third-order photovoltaic Hall conductivity tensors calculated with ~Γ = 1 meV, v = 8 × 10(5) m/s, and 2|m| = 17 meV, which are relevant to graphene on a hexagonal boron nitride substrate [36].”
BackgroundDifference frequency generation in topological semimetals
de Juan, F., Zhang, Y., Morimoto, T., Sun, Y., Moore, J. E., & Grushin, A. G. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.012017 · Cited by 94 · cited 2×
“In particular, the non-quantized injection photocurrents in chiral semimetals due to higher-band corrections to the two-band description [16, 17] can still be regarded as geometric responses, although not as topological responses.”
BackgroundSemiclassical wave packet dynamics in nonuniform electric fields
Lapa, M. F., & Hughes, T. L. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.121111 · Cited by 89
Concept of Quantum Geometry in Optoelectronic Processes in Solids: Application to Solar Cells
Nagaosa, N., & Morimoto, T. (2017) · Advanced Materials · doi:10.1002/adma.201603345 · Cited by 84
Self-citationBulk photovoltaic effects in the presence of a static electric field
Fregoso, B. M. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.100.064301 · Cited by 84 · cited 2×
“[36], the third-order photoconductivity is classified into jerk, injection, and shift according to their dependence on the relaxation rate; they are proportional to Γ−2, Γ−1, and Γ, respectively.”
BackgroundDifference frequency generation in topological semimetals
Zhang, Y. (2020) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 75
Verma, N., Hazra, T., & Randeria, M. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2106744118 · Cited by 67
Pierre Bieliavsky, A., CAHEN, M., Gutt, S., et al. (2006) · International Journal of Geometric Methods in Modern Physics · doi:10.1142/s021988780600117x · Cited by 50
Intrinsic Anomalous Hall Conductivity in a Nonuniform Electric Field
Kozii, V., Avdoshkin, A., Zhong, S., & Moore, J. E. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.156602 · Cited by 48
The Euler number of Bloch states manifold and the quantum phases in gapped fermionic systems
Ma, Y.-Q., Gu, S.-J., Chen, S., Fan, H., & Liu, W.-M. (2013) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/103/10008 · Cited by 47
Superconductivity-induced spectral weight transfer due to quantum geometry
Ahn, J., & Nagaosa, N. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.l100501 · Cited by 24
Self-citationChandra, H. K., & Guo, G. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.95.134448 · Cited by 14 · cited 2×
“These materials are Z2 topological insulators in two and three dimensions [38, 39], and a Chern insulator [40], respectively, described by a massive Dirac Hamiltonian close to the band edge.”
BackgroundMethodSelf-citationElectric polarization in inhomogeneous crystals
Zhao, Y., Gao, Y., & Xiao, D. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.144203 · Cited by 9
Euler characteristic number of the energy band and the reason for its non-integer valuesPreprint
Ma, Y.-Q. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2001.05946 · Cited by 5
Zhu, Y.-Q., Zhang, D.-W., Tan, X., Yu, H., Yan, H., Yu, Y., & Zhu, S.-L. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1908.06462
Weyl and Dirac Semimetals in Three Dimensional Solids
Armitage, N. P., Mele, E. J., & Vishwanath, A. (2017) · arXiv
Self-citationNieves, J. F., & Pal, P. B. (2004) · American Journal of Physics · doi:10.1119/1.1757445
Research works printed but not linked: 2
- Sturman, B. I., Fridkin, V. M. & Bradley, J. E. S. The Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials 1st edn, Vol. 8 (Routledge, 1992).
- Boyd, R. W. Nonlinear Optics 4th edn (Academic Press, 2020).
Cited by
Sorted by citations
Schlawin, F., Kennes, D. M., & Sentef, M. A. (2022) · Applied Physics Reviews · doi:10.1063/5.0083825 · Cited by 250
Essay: Where Can Quantum Geometry Lead Us?
Törmä, P. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.240001 · Cited by 187
“Moreover, work on other quantum geometric quantities, such as Christoffer symbols and Riemann curvature tensors, to describe physical phenomena has already started [15] and should be continued.”
BackgroundRecent progress in the theory of bulk photovoltaic effect
Dai, Z., & Rappe, A. M. (2023) · Chemical Physics Reviews · doi:10.1063/5.0101513 · Cited by 162
Kaplan, D., Holder, T., & Yan, B. (2024) · Physical Review Letters · doi:10.1103/physrevlett.132.026301 · Cited by 156
“linear responses carry more intricate and and the same time much more interesting information about the quantum geometry than linear response functions [53, 54].”
BackgroundGeometric deep optical sensing
Yuan, S., Ma, C., Fetaya, E., Mueller, T., Naveh, D., Zhang, F., & Xia, F. (2023) · Science · doi:10.1126/science.ade1220 · Cited by 158
“Under circularly polarized light excitation, the BPVE (injection current) is governed by the interband Berry curvature dipoles or the Hermitian metrics (54).”
Method
Metadata
- Authors
- Ahn, J., Guo, G.-Y., Nagaosa, N., & Vishwanath, A.
- Affiliations
- Harvard University (US)
- National Taiwan University (TW)
- RIKEN (JP)
- Venue
- Nature Physics
- Published
- 2021-12-23
- Volume
- 18
- Issue
- 3
- Pages
- 290-295
- ISSN
- 1745-2473 (ISSN-L)
- 1745-2481
- OpenAlex
- W3134879942
- Cited by
- 252
- References
- 54
- Citations by year
- 2026: 58 · 2025: 75 · 2024: 55 · 2023: 35 · 2022: 21 · 2021: 8
- FWCI
- 12.99
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Nature Physics (publishedVersion)
- arXiv (Cornell University) (acceptedVersion)
- Digital Access to Scholarship at Harvard (DASH) (Harvard University) (acceptedVersion)
- Figshare (submittedVersion, cc-by)
- arXiv (Cornell University) (cc-by)
- Funders
- DOE | LDRD | Ames Laboratory (DE-AC02-07CH11358)
- National Research Foundation of Korea (2020R1A6A3A03037129)
- Ministry of Science and Technology, Taiwan
- MEXT | Japan Science and Technology Agency (JPMJCR1874, JPMJCR16F1)
- MEXT | Japan Society for the Promotion of Science (18H03676)
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Quantum and electron transport phenomena
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Keywords
- Riemannian geometry, quantum geometry, transition dipole moment, Riemann curvature tensor, Berry phase, quantum Hall effect, anomalous Hall effect, optical transitions
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
