- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Two-dimensional nonlinear optical response of a spiral magnet

Wolfram Brenig*

  • *Contact author: w.brenig@tu-braunschweig.de
PDF

Phys. Rev. B 112, 115132 – Published 16 September, 2025

DOI: https://doi.org/10.1103/q3ph-yf2t

Abstract

We study the dynamical response function relevant for two-dimensional coherent nonlinear optical spectroscopy of the antiferromagnetic frustrated J1−J3 Heisenberg model on the square lattice within its long-range-ordered, incommensurate diagonal spiral phase. We argue that in this phase effective dipole coupling to the electric field is important, with the spin-current coupling potentially being the dominant mechanism for spin 1/2. For this setting, we use linear spin wave theory to evaluate the leading nonlinear polarization response which is of second order in the driving field. We show that the response function features a strong antidiagonal, galvanoelectric feature. The width of this feature is set by relaxation rates beyond the noninteracting magnon picture, thereby providing access to single-magnon lifetimes within the multimagnon continuum of the response function. Moreover, the response function is shown to display various structures in the two-dimensional frequency plane related to exceptional regions of the magnon dispersion.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (81)

  1. S. Mukamel, Multidimensional femtosecond correlation spectroscopies of electronic and vibrational excitations, Annu. Rev. Phys. Chem. 51, 691 (2000).
  2. M. Cho, Coherent two-dimensional optical spectroscopy, Chem. Rev. 108, 1331 (2008).
  3. S. T. Cundiff and S. Mukamel, Optical multidimensional coherent spectroscopy, Phys. Today 66(7), 44 (2013).
  4. Y. Wan and N. P. Armitage, Resolving continua of fractional excitations by spinon echo in THz 2D coherent spectroscopy, Phys. Rev. Lett. 122, 257401 (2019).
  5. M. B. Stone, D. H. Reich, C. Broholm, K. Lefmann, C. Rischel, C. P. Landee, and M. M. Turnbull, Extended quantum critical phase in a magnetized spin-1/2 antiferromagnetic chain, Phys. Rev. Lett. 91, 037205 (2003).
  6. B. Lake, D. A. Tennant, C. D. Frost, and S. E. Nagler, Quantum criticality and universal scaling of a quantum antiferromagnet, Nat. Mater. 4, 329 (2005).
  7. Z.-L. Li, M. Oshikawa, and Y. Wan, Photon echo from lensing of fractional excitations in Tomonaga-Luttinger spin liquid, Phys. Rev. X 11, 031035 (2021).
  8. G. Sim, J. Knolle, and F. Pollmann, Nonlinear spectroscopy of bound states in perturbed Ising spin chains, Phys. Rev. B 107, L100404 (2023).
  9. Q. Gao, Y. Liu, H. Liao, and Y. Wan, Two-dimensional coherent spectrum of interacting spinons from matrix product states, Phys. Rev. B 107, 165121 (2023).
  10. G. Sim, F. Pollmann, and J. Knolle, Microscopic details of two-dimensional spectroscopy of one-dimensional quantum Ising magnets, Phys. Rev. B 108, 134423 (2023).
  11. Z.-L. Li and Y. Wan, Photon echo and fractional excitation lensing of the S=1/2 XY spin chain, Phys. Rev. B 108, 165151 (2023).
  12. M. Potts, R. Moessner, and O. Benton, Exploiting polarization dependence in two-dimensional coherent spectroscopy: Examples of Ce2Zr2O7 and Nd2Zr2O7, Phys. Rev. B 109, 104435 (2024).
  13. Y. Watanabe, S. Trebst, and C. Hickey, Exploring two-dimensional coherent spectroscopy with exact diagonalization: Spinons and confinement in one-dimensional quantum magnets, Phys. Rev. B 110, 134443 (2024).
  14. A. Srivastava, S. Birnkammer, G. Sim, M. Knap, and J. Knolle, Theory of nonlinear spectroscopy of quantum magnets, arXiv:2502.17554.
  15. W. Choi, K. H. Lee, and Y. B. Kim, Theory of two-dimensional nonlinear spectroscopy for the Kitaev spin liquid, Phys. Rev. Lett. 124, 117205 (2020).
  16. M. K. Negahdari and A. Langari, Nonlinear response of the Kitaev honeycomb lattice model in a weak magnetic field, Phys. Rev. B 107, 134404 (2023).
  17. M. Kanega, T. N. Ikeda, and M. Sato, Linear and nonlinear optical responses in Kitaev spin liquids, Phys. Rev. Res. 3, L032024 (2021).
  18. O. Krupnitska and W. Brenig, Finite-temperature second harmonic generation in Kitaev magnets, Phys. Rev. B 108, 075120 (2023).
  19. W. Brenig and O. Krupnitska, Response functions for electric field induced two-dimensional nonlinear spectroscopy in a Kitaev magnet, J. Phys.: Condens. Matter 36, 505806 (2024).
  20. Y. Qiang, V. L. Quito, T. V. Trevisan, and P. P. Orth, Probing Majorana wave functions in Kitaev honeycomb spin liquids with second-order two-dimensional spectroscopy, Phys. Rev. Lett. 133, 126505 (2024).
  21. D. A. S. Kaib, M. Möller, and R. Valentí, Nonlinear spectroscopy as a magnon breakdown diagnosis and its efficient simulation, arXiv:2502.01746.
  22. R. M. Nandkishore, W. Choi, and Y. B. Kim, Spectroscopic fingerprints of gapped quantum spin liquids, both conventional and fractonic, Phys. Rev. Res. 3, 013254 (2021).
  23. J. Lu, X. Li, H. Y. Hwang, B. K. Ofori-Okai, T. Kurihara, T. Suemoto, and K. A. Nelson, Coherent two-dimensional terahertz magnetic resonance spectroscopy of collective spin waves, Phys. Rev. Lett. 118, 207204 (2017).
  24. I. E. Dzyaloshinskii, On the magneto-electrical effect in antiferromagnets, Sov. Phys. JETP 10, 628 (1960).
  25. M. Fiebig, Revival of the magnetoelectric effect, J. Phys. D 38, R123 (2005).
  26. Y. Tokura, S. Seki, and N. Nagaosa, Multiferroics of spin origin, Rep. Prog. Phys. 77, 076501 (2014).
  27. S. Dong, J.-M. Liu, S.-W. Cheong, and Z. Ren, Multiferroic materials and magnetoelectric physics: Symmetry, entanglement, excitation, and topology, Adv. Phys. 64, 519 (2015).
  28. S. Dong, H. Xiang, and E. Dagotto, Magnetoelectricity in multiferroics: A theoretical perspective, Natl. Sci. Rev. 6, 629 (2019).
  29. V. G. Baryakhtar and I. E. Chupis, Quantum theory of oscillations in a ferroelectric ferromagnet, Sov. Phys. Solid State 11, 2628 (1970).
  30. G. A. Smolenskii and I. E. Chupis, Ferroelectromagnets, Usp. Fiziol. Nauk 137, 415 (1982) [Ferroelectromagnets, Sov. Phys. Usp. 25, 475 (1982)].
  31. Y. Takahashi, R. Shimano, Y. Kaneko, H. Murakawa, and Y. Tokura, Magnetoelectric resonance with electromagnons in a perovskite helimagnet, Nat. Phys. 8, 121 (2012).
  32. M. Azimi, M. Sekania, S. K. Mishra, L. Chotorlishvili, Z. Toklikishvili, and J. Berakdar, Pulse and quench induced dynamical phase transition in a chiral multiferroic spin chain, Phys. Rev. B 94, 064423 (2016).
  33. Y. Zhang, J. Dai, X. Zhong, D. Zhang, G. Zhong, and J. Li, Probing ultrafast dynamics of ferroelectrics by time-resolved pump-probe spectroscopy, Adv. Sci. 8, 2102488 (2021).
  34. H. Ueda, H. Jang, S. H. Chun, H.-D. Kim, M. Kim, S.-Y. Park, S. Finizio, N. Ortiz Hernandez, V. Ovuka, M. Savoini, T. Kimura, Y. Tanaka, A. Doll, and U. Staub, Optical excitation of electromagnons in hexaferrite, Phys. Rev. Res. 4, 023007 (2022).
  35. C. Jia, S. Onoda, N. Nagaosa, and J. H. Han, Microscopic theory of spin-polarization coupling in multiferroic transition metal oxides, Phys. Rev. B 76, 144424 (2007).
  36. Nonetheless, our calculations will be performed for arbitrary spin.
  37. H. Katsura, M. Sato, T. Furuta, and N. Nagaosa, Theory of the optical conductivity of spin liquid states in one-dimensional Mott insulators, Phys. Rev. Lett. 103, 177402 (2009).
  38. J. Lorenzana and G. A. Sawatzky, Phonon assisted multimagnon optical absorption and long lived two-magnon states in undoped lamellar copper oxides, Phys. Rev. Lett. 74, 1867 (1995).
  39. H. Katsura, N. Nagaosa, and A. V. Balatsky, Spin current and magnetoelectric effect in noncollinear magnets, Phys. Rev. Lett. 95, 057205 (2005).
  40. J. Richter, V. Ohanyan, J. Schulenburg, and J. Schnack, Electric field driven flat bands: Enhanced magnetoelectric and electrocaloric effects in frustrated quantum magnets, Phys. Rev. B 105, 054420 (2022).
  41. I. A. Sergienko and E. Dagotto, Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites, Phys. Rev. B 73, 094434 (2006).
  42. S.-W. Cheong and M. Mostovoy, Multiferroics: A magnetic twist for ferroelectricity, Nat. Mater. 6, 13 (2007).
  43. M. P. Gelfand, R. R. P. Singh, and D. A. Huse, Zero-temperature ordering in two-dimensional frustrated quantum Heisenberg antiferromagnets, Phys. Rev. B 40, 10801 (1989).
  44. F. Figueirido, A. Karlhede, S. Kivelson, S. Sondhi, M. Rocek, and D. S. Rokhsar, Exact diagonalization of finite frustrated spin-1/2 Heisenberg models, Phys. Rev. B 41, 4619 (1990).
  45. A. Moreo, E. Dagotto, T. Jolicoeur, and J. Riera, Incommensurate correlations in the t−J and frustrated spin-1/2 Heisenberg models, Phys. Rev. B 42, 6283 (1990).
  46. A. Chubukov, First-order transition in frustrated quantum antiferromagnets, Phys. Rev. B 44, 392 (1991).
  47. W.-Y. Liu, D. Poilblanc, S.-S. Gong, W.-Q. Chen, and Z.-C. Gu, Tensor network study of the spin-1/2 square-lattice J1−J2−J3 model: Incommensurate spiral order, mixed valence-bond solids, and multicritical points, Phys. Rev. B 109, 235116 (2024).
  48. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, Cambridge, 1990).
  49. W. A. B. Evans, A general analytic continuation response formalism for large quantum systems, Proc. Phys. Soc. 88, 723 (1966).
  50. H. Rostami, M. I. Katsnelson, G. Vignale, and M. Polini, Gauge invariance and Ward identities in nonlinear response theory, Ann. Phys. (Amsterdam, Neth.) 431, 168523 (2021).
  51. B. I. Halperin and P. C. Hohenberg, Generalization of scaling laws to dynamical properties of a system near its critical point, Phys. Rev. Lett. 19, 700 (1967); Scaling laws for dynamic critical phenomena, Phys. Rev. 177, 952 (1969).
  52. S. Chakravarty, B. I. Halperin, and D. R. Nelson, Two-dimensional quantum Heisenberg antiferromagnet at low temperatures, Phys. Rev. B 39, 2344 (1989).
  53. D. R. Grempel, Comment on “Low-temperature behavior of two-dimensional quantum antiferromagnets,” Phys. Rev. Lett. 61, 1041 (1988).
  54. S. Tyč, B. I. Halperin, and S. Chakravarty, Dynamic properties of a two-dimensional Heisenberg antiferromagnet at low temperatures, Phys. Rev. Lett. 62, 835 (1989).
  55. M. Makivić and M. Jarrell, Low-temperature dynamics of the 2D spin-1/2 Heisenberg antiferromagnet: A quantum Monte Carlo study, Phys. Rev. Lett. 68, 1770 (1992).
  56. P. Hasenfratz, The correlation length of the Heisenberg antiferromagnet with arbitrary spin S, Eur. Phys. J. B 13, 11 (2000).
  57. O. P. Vajk, P. K. Mang, M. Greven, P. M. Gehring, and J. W. Lynn, Quantum impurities in the two-dimensional spin one-half Heisenberg antiferromagnet, Science 295, 1691 (2002).
  58. M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. Endoh, Magnetic, transport, and optical properties of monolayer copper oxides, Rev. Mod. Phys. 70, 897 (1998).
  59. H. M. Rønnow, D. F. McMorrow, R. Coldea, A. Harrison, I. D. Youngson, T. G. Perring, G. Aeppli, O. Syljuåsen, K. Lefmann, and C. Rischel, Spin dynamics of the 2D spin 1/2 quantum antiferromagnet copper deuteroformate tetradeuterate (CFTD), Phys. Rev. Lett. 87, 037202 (2001).
  60. A. M. Toader, J. P. Goff, M. Roger, N. Shannon, J. R. Stewart, and M. Enderle, Spin correlations in the paramagnetic phase and ring exchange in La2CuO4, Phys. Rev. Lett. 94, 197202 (2005).
  61. M. Takahashi, Modified spin-wave theory of a square-lattice antiferromagnet, Phys. Rev. B 40, 2494 (1989).
  62. A. Auerbach and D. P. Arovas, Spin dynamics in the square-lattice antiferromagnet, Phys. Rev. Lett. 61, 617 (1988).
  63. A. F. Barabanov and O. A. Starykh, Spherical symmetric spin wave theory of Heisenberg model, J. Phys. Soc. Jpn. 61, 704 (1992).
  64. D. E. Parker, T. Morimoto, J. Orenstein, and J. E. Moore, Diagrammatic approach to nonlinear optical response with application to Weyl semimetals, Phys. Rev. B 99, 045121 (2019).
  65. J. E. Sipe and A. I. Shkrebtii, Second-order optical response in semiconductors, Phys. Rev. B 61, 5337 (2000).
  66. R. Fei, W. Song, and L. Yang, Giant photogalvanic effect and second-harmonic generation in magnetic axion insulators, Phys. Rev. B 102, 035440 (2020).
  67. H. Ishizuka and M. Sato, Large photogalvanic spin current by magnetic resonance in bilayer Cr trihalides, Phys. Rev. Lett. 129, 107201 (2022).
  68. A. Raj, S. Chaudhary, and G. A. Fiete, Photogalvanic response in multi-Weyl semimetals, Phys. Rev. Res. 6, 013048 (2024).
  69. K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, and T. Arima, Ferroelectric polarization flop in a frustrated magnet MnWO4 induced by a magnetic field, Phys. Rev. Lett. 97, 097203 (2006).
  70. S. Park, Y. J. Choi, C. L. Zhang, and S.-W. Cheong, Ferroelectricity in an S=1/2 chain cuprate, Phys. Rev. Lett. 98, 057601 (2007).
  71. Y. Naito, K. Sato, Y. Yasui, Y. Kobayashi, Y. Kobayashi, and M. Sato, Ferroelectric transition induced by the incommensurate magnetic ordering in LiCuVO4, J. Phys. Soc. Jpn. 76, 023708 (2007).
  72. S. Seki, Y. Onose, and Y. Tokura, Spin-driven ferroelectricity in triangular lattice antiferromagnets ACrO2 (A = Cu, Ag, Li, or Na), Phys. Rev. Lett. 101, 067204 (2008).
  73. Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, and Y. Tokura, Magnetic reversal of the ferroelectric polarization in a multiferroic spinel oxide, Phys. Rev. Lett. 96, 207204 (2006).
  74. A. Pimenov, A. M. Shuvaev, A. A. Mukhin, and A. Loidl, Electromagnons in multiferroic manganites, J. Phys.: Condens. Matter 20, 434209 (2008).
  75. H. Ju, Y. Lee, K.-T. Kim, I. H. Choi, C. J. Roh, S. Son, P. Park, J. H. Kim, T. S. Jung, J. H. Kim, K. H. Kim, J.-G. Park, and J. S. Lee, Possible persistence of multiferroic order down to bilayer limit of van der Waals material NiI2, Nano Lett. 21, 5126 (2021).
  76. Q. Song, C. A. Occhialini, E. Ergeçen, B. Ilyas, D. Amoroso, P. Barone, J. Kapeghian, K. Watanabe, T. Taniguchi, A. S. Botana, S. Picozzi, N. Gedik, and R. Comin, Evidence for a single-layer van der Waals multiferroic, Nature (London) 602, 601 (2022).
  77. Y. Jiang, Y. Wu, J. Zhang, J. Wei, B. Peng, and C.-W. Qiu, Dilemma in optical identification of single-layer multiferroics, Nature (London) 619, E40 (2023).
  78. F. Y. Gao, X. Peng, X. Cheng, E. Viñas Boström, D. S. Kim, R. K. Jain, D. Vishnu, K. Raju, R. Sankar, S.-F. Lee, M. A. Sentef, T. Kurumaji, X. Li, P. Tang, A. Rubio, and E. Baldini, Giant chiral magnetoelectric oscillations in a van der Waals multiferroic, Nature (London) 632, 273 (2024).
  79. N. Liu, C. Wang, C. Yan, C. Xu, J. Hu, Y. Zhang, and W. Ji, Competing multiferroic phases in monolayer and few-layer NiI2, Phys. Rev. B 109, 195422 (2024).
  80. A. Edström, D. Amoroso, S. Picozzi, P. Barone, and M. Stengel, Curved magnetism in CrI3, Phys. Rev. Lett. 128, 177202 (2022).
  81. W. Brenig, Data underpinning “Two-dimensional nonlinear optical response of a spiral magnet” [Data set] Zenodo (2025), https://doi.org/10.5281/zenodo.15367726.

Outline

Information

Phys. Rev. B 112, 115132– Published 16 September, 2025
  • Received 9 May 2025
  • Revised 21 August 2025
  • Accepted 25 August 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation