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Riemannian geometry of resonant optical responses

Nature Physics volume 18, pages 290–295 (2022)Cite this article

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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.

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Fig. 1: Geometry of the cell-periodic Bloch state and optical transitions.
Fig. 2: Third-order photovoltaic Hall conductivity of a Dirac fermion.
Fig. 3: First-principles calculations on massive Dirac materials.

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

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

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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.

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Authors and Affiliations

  1. Department of Physics, Harvard University, Cambridge, MA, USA

    Junyeong Ahn & Ashvin Vishwanath

  2. Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei, Taiwan

    Guang-Yu Guo

  3. Physics Division, National Center for Theoretical Sciences, Taipei, Taiwan

    Guang-Yu Guo

  4. RIKEN Center for Emergent Matter Science (CEMS), Wako, Japan

    Naoto Nagaosa

  5. Department of Applied Physics, The University of Tokyo, Tokyo, Japan

    Naoto Nagaosa

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  1. Junyeong Ahn
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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.

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Correspondence to Junyeong Ahn, Guang-Yu Guo, Naoto Nagaosa or Ashvin Vishwanath.

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Ahn, J., Guo, GY., Nagaosa, N. et al. Riemannian geometry of resonant optical responses. Nat. Phys. 18, 290–295 (2022). https://doi.org/10.1038/s41567-021-01465-z

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