Riemannian geometry of resonant optical responses

ABSTRACT The geometry of quantum states is well established as a basis for understanding the response of electronic systems to static electromagnetic fields, as exemplified by the theory of

the quantum and anomalous Hall effects. However, it has been challenging to relate quantum geometry to resonant optical responses. The main obstacle is that optical transitions involve a

pair of states, whereas existing geometrical properties are defined for a single state. As a result, a concrete geometric understanding of optical responses has so far been limited to

two-level systems, where the Hilbert space is completely determined by a single state and its orthogonal complement. Here, we construct a general theory of Riemannian geometry for resonant

optical processes by identifying transition dipole moment matrix elements as tangent vectors. This theory applies to arbitrarily high-order responses, suggesting that optical responses can

generally be thought of as manifestations of the Riemannian geometry of quantum states. We use our theory to show that third-order photovoltaic Hall effects are related to the Riemann

curvature tensor and demonstrate an experimentally accessible regime where they dominate the response. Access through your institution Buy or subscribe This is a preview of subscription

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INTERACTION Article Open access 09 June 2022 ALL ELECTROMAGNETIC SCATTERING BODIES ARE MATRIX-VALUED OSCILLATORS Article Open access 24 November 2023 THE COMPLEX MAXWELL STRESS TENSOR

THEOREM: THE IMAGINARY STRESS TENSOR AND THE REACTIVE STRENGTH OF ORBITAL MOMENTUM. A NOVEL SCENERY UNDERLYING ELECTROMAGNETIC OPTICAL FORCES Article Open access 12 October 2022 DATA

AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. CODE AVAILABILITY The codes that support the findings of

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Scholar  Download references ACKNOWLEDGEMENTS We appreciate E. Khalaf and D. Parker for helpful discussions and thank M. Christos for useful comments on the manuscript. J.A. was supported

by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (grant no. 2020R1A6A3A03037129). J.A. and A.V. were supported by

the Center for Advancement of Topological Semimetals, an Energy Frontier Research Center funded by the United States Department of Energy Office of Science, Office of Basic Energy Sciences,

through the Ames Laboratory under contract no. DE-AC02-07CH11358. G.-Y.G. acknowledges the support from the Ministry of Science and Technology and National Center for Theoretical Sciences in

Taiwan and thanks the National Center for High-performance Computing in Taiwan for the computing time. N.N. was supported by Japan Science and Technology Agency CREST grant nos. JPMJCR1874

and JPMJCR16F1 and by Japan Society for the Promotion of Science KAKENHI grant no. 18H03676. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Physics, Harvard University,

Cambridge, MA, USA Junyeong Ahn & Ashvin Vishwanath * Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei, Taiwan Guang-Yu Guo * Physics

Division, National Center for Theoretical Sciences, Taipei, Taiwan Guang-Yu Guo * RIKEN Center for Emergent Matter Science (CEMS), Wako, Japan Naoto Nagaosa * Department of Applied Physics,

The University of Tokyo, Tokyo, Japan Naoto Nagaosa Authors * Junyeong Ahn View author publications You can also search for this author inPubMed Google Scholar * Guang-Yu Guo View author

publications You can also search for this author inPubMed Google Scholar * Naoto Nagaosa View author publications You can also search for this author inPubMed Google Scholar * Ashvin

Vishwanath View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS J.A. conceived the original idea and performed the theoretical analysis. G.-Y.G.

performed first-principles calculations. N.N. and A.V. supervised the project. All authors discussed results and contributed to the formulation of the theory and writing of the manuscript.

CORRESPONDING AUTHORS Correspondence to Junyeong Ahn, Guang-Yu Guo, Naoto Nagaosa or Ashvin Vishwanath. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing financial

interests. ADDITIONAL INFORMATION PEER REVIEW INFORMATION _Nature Physics_ thanks the anonymous reviewers for their contribution to the peer review of this work. PUBLISHER’S NOTE Springer

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RIGHTS AND PERMISSIONS Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Ahn, J., Guo, GY., Nagaosa, N. _et al._ Riemannian geometry of resonant optical responses. _Nat. Phys._

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