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Riemannian geometry of resonant optical responses
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“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]
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“linear responses carry more intricate and and the same time much more interesting information about the quantum geometry than linear response functions [53, 54].”
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“Under circularly polarized light excitation, the BPVE (injection current) is governed by the interband Berry curvature dipoles or the Hermitian metrics (54).”
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“The quantum geometric connection is decomposed into real and imaginary parts C mp = Γ mp−iΓ̃ mp, where Γ (Γ̃) is the metric (symplectic) connection [33].”
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Bound on resistivity in flat-band materials due to the quantum metric
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Light-matter coupling and quantum geometry in moiré materials
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“Following earlier work for first-principles computations of magnetic susceptibilities in insulators [133–135], it has been discussed how the quantum geometry of Bloch wave functions affects the orbital magnetic susceptibility [14–16] and nonlinear optical responses [136, 137], contributes to the pair mass and Goldston…”
BackgroundAxion optical induction of antiferromagnetic order
Qiu, J.-X., Tzschaschel, C., Ahn, J., et al. (2023) · Nature Materials · doi:10.1038/s41563-023-01493-5 · Cited by 56
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“As mentioned, P symmetry renders the second order current j c (¯ ω ) = σ ab ; c (¯ ω ; ω 1 ,ω 2 ) E a ( ω 1 ) E b ( ω 2 ) zero since P changes the sign of current j c (¯ ω ) yet leaves E a E b invariant.”
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“A unified formula that captures both intra-cell optical magneto-electric coupling and topological magnetoelectric effect is desired, and it is likely to require extending the quantum geometric formulation for electric dipole moments [47] to magnetic dipole and electric quadrupole moments and defining a proper optical…”
BackgroundSelf-citationNontrivial quantum geometry of degenerate flat bands
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Probing quantum geometry through optical conductivity and magnetic circular dichroism
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“The connection between linear optical conductivity and ground state quantum geometry has remained largely unexplored.”
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“…we show that (i) the shift response in these systems is purely governed by the virtual transitions in the topological multiband subspaces, which (ii) we identify with the nontriviality of the torsion tensor introduced in the Riemannian-geometric reformulation of the optical responses [53].”
BackgroundMethodInfluentialMapping quantum geometry and quantum phase transitions to real space by a fidelity marker
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“This form highlights the connection to higher-order optical responses as presented by Ahn et al. [23], where they interpreted transition dipole moment matrix elements as tangent vectors ˆ e mnα with projector form given in Eq.”
InfluentialControllable orbital angular momentum monopoles in chiral topological semimetals
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“In this regard, we note that the quantity − ¯ q bcln /e is distinct from the “band-resolved quantum metric” g bcln = (cid:0) A A + A A / 2 that has been introduced in connection with nonlinear optical responses [51–53].”
Non-Abelian Hopf-Euler insulators
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“The reason for such distinction is that due to the PT symmetry, the linear shift photoconductivities [65, 66] completely vanish in the Hopf-Euler insulators, unlike in the Hopf insulators where the linear shift response is reflected by the RTP [67].”
BackgroundInfluentialMultistate Geometry of Shift Current and Polarization
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“Linear and non-linear optical responses are natural candidates for the physical manifestation of multi-state quantum geometry due to their momentum-local coupling of the Bloch states of multiple bands [21, 26–28].”
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“51 The KBSS formula has been derived in a variety of models and methods,[4, 6, 7, 12, 13, 15, 16, 18, 19, 85] which may have created an impression that the KBSS formula is universally truthful.”
BackgroundMethodQuantum geometry induced third-order nonlinear transport responses
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“Recent studies by the present authors [27, 28] allow a stronger quantum geometric interpretation that the interband superfluid weight is quantum geometric even away from the flat-band limit.”
BackgroundSelf-citationQuantum geometry, flat Chern bands, and Wannier orbital quantization
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Extrinsic geometry of quantum states
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Linear displacement current solely driven by the quantum metric
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Strongly enhanced shift current at exciton resonances in a noncentrosymmetric wide-gap semiconductor
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“Although shift current is not determined solely by shift vector 35,36 , it is a good quantity to roughly estimate the sign of shift current.”
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“The modern quantum geometric perspective on D ( k ) establishes a link between dipole transitions and the Berry connection 72 , thus to topological properties.”
Nonlinear photoconductivities and quantum geometry of chiral multifold fermions
Hsu, H.-C., You, J.-S., Ahn, J., & Guo, G. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.107.155434 · Cited by 19
“The injection current is related to Berry curvature [5, 12] and quantum metric [6, 7], while the shift current is related to Hermitian connections [1, 9, 24].”
BackgroundSelf-citationPhoton-drag photovoltaic effects and quantum geometric nature
Xie, Y.-M., & Nagaosa, N. (2025) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2424294122 · Cited by 17
Self-citationBulk photovoltaic effects in altermagnets
Ezawa, M. (2025) · Physical review. B./Physical review. B · doi:10.1103/physrevb.111.l201405 · Cited by 16
Local geometry and quantum geometric tensor of mixed states
Hou, X.-Y., Zhou, Z., Wang, X., Guo, H., & Chien, C.-C. (2024) · Physical review. B./Physical review. B · doi:10.1103/physrevb.110.035144 · Cited by 20
“…state tomography of cold atoms [29], and transmission measurements of plasmonic lattices [30], as well as theoretical works on optical response [31], quantum fluctuations of the QGT [32], quantum phase transitions [33], superfluidity in flat bands [34], generalizations to N -band systems [35],…”
BackgroundQuantum entanglement and quantum geometry measured with inelastic x-ray scattering
Bałut, D., Bradlyn, B., & Abbamonte, P. M. (2025) · Physical review. B./Physical review. B · doi:10.1103/physrevb.111.125161 · Cited by 16
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“Very recently, nonlinear optical transitions have been related to the Riemannian geometry of the energy bands [17, 35].”
BackgroundQiang, Y., Quito, V. L., Trevisan, T. V., & Orth, P. P. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.126505 · Cited by 16
Quantum geometry probed by chiral excitonic optical response of Chern insulators
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“In the complete Bloch basis, the position operator ˆ r in the second quantized form is [23, 44], where | u n k ⟩ is the periodic part of the Bloch state | n k ⟩ and A mn ( k ) is the Berry connection.”
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Spectral density and sum rules for second-order response functions
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Quantum geometry and the hidden scales in materials
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Drude weight of an interacting flat-band metal
Antebi, O., Mitscherling, J., & Holder, T. (2024) · Physical review. B./Physical review. B · doi:10.1103/physrevb.110.l241111 · Cited by 13
Intrinsic Gyrotropic Magnetic Current from Zeeman Quantum Geometry
Xiang, L., Jia, J., Xu, F., et al. (2025) · Physical Review Letters · doi:10.1103/physrevlett.134.116301 · Cited by 13
Quantum metric nonlinear spin-orbit torque enhanced by topological bands
Feng, X., Wu, W., Wang, H., et al. (2025) · Materials Today Quantum · doi:10.1016/j.mtquan.2025.100040 · Cited by 13
Photocurrents, inverse Faraday effect, and photospin Hall effect in Mn2Au
Merte, M., Freimuth, F., Go, D., et al. (2023) · APL Materials · doi:10.1063/5.0149955 · Cited by 13
Tiwari, P., Sahani, D., Chakraborty, A., et al. (2023) · Nano Letters · doi:10.1021/acs.nanolett.3c00045 · Cited by 12
McKay, R. C., Mahmood, F., & Bradlyn, B. (2024) · Physical Review X · doi:10.1103/physrevx.14.011058 · Cited by 13
“For noninteracting electron systems, recent work has shown how the Kubo formula for the uniform optical conductivities can be interpreted in terms of the quantum geometry of electronic wavefunctions in the systems [34–36].”
BackgroundQuantum geometric superfluid weight in multiband superconductors: A microscopic interpretation
Hu, Y., & Huang, W. (2025) · Physical review. B./Physical review. B · doi:10.1103/physrevb.111.134511 · Cited by 12
Magnetic bulk photovoltaic effect as a probe of magnetic structures of $\mathrm{{EuSn_{2}As_{2}}}$
Pi, H., Zhang, S., & Weng, H. (2023) · Quantum Frontiers · doi:10.1007/s44214-023-00029-x · Cited by 12
Photogalvanic response in multi-Weyl semimetals
Raj, A., Chaudhary, S., & Fiete, G. A. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.013048 · Cited by 15
“These mechanisms for a BPVE are intimately related to the quantum geometry of the electronic wavefunction [14, 15], and thus are proving to be reliable tools to probe and utilize the band topology [6, 7].”
MethodPang, H., Jin, G., & He, L. (2024) · Physical Review Materials · doi:10.1103/physrevmaterials.8.043403 · Cited by 13
Quantum geometry and nonlinear optical responses in rhombohedral trilayer graphene
Postlewaite, A., Raj, A., Chaudhary, S., & Fiete, G. A. (2024) · Physical review. B./Physical review. B · doi:10.1103/physrevb.110.245122 · Cited by 11
“Third-order responses were also studied and correlated with the Riemannian curvature tensor [60], and the superconducting quasi-particle Berry curvature was found to play a key role in the nonlinear optical response of a noncentrosymmetric superconductor [68].”
Fregoso, B. M. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.106.195108 · Cited by 11
Spin Transport Revealed by Spin Quantum Geometry
Xiang, L., Jin, H., & Wang, J. (2025) · Physical Review Letters · doi:10.1103/14mp-263q · Cited by 12
Nonlinear Hall Effect from Long-Lived Valley-Polarizing Relaxons
Lihm, J.-M., & Park, C.-H. (2024) · Physical Review Letters · doi:10.1103/physrevlett.132.106402 · Cited by 11
“By combining the relaxon analysis and the theories of nonlinear transport including band-geometric effects [7, 9, 57–59] using ab initio scattering matrices [60, 61], one can reveal novel response properties of real materials beyond the RTA, as demonstrated in this work for the NLH effect.”
MethodAnomalous geometric transport signatures of topological Euler class
Jain, A., Jankowski, W. J., & Slager, R.-J. (2025) · Physical review. B./Physical review. B · doi:10.1103/5gmg-q1z5 · Cited by 10
Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
Shinada, K., & Nagaosa, N. (2025) · Physical review. B./Physical review. B · doi:10.1103/qxbl-qd4f · Cited by 10
“Furthermore, the extension to higher-order geometric quantities such as connection and its relation with nonlinear responses have recently been discussed [34–37].”
Self-citation
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
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