Field-tunable quadruple-Q states driven by momentum-space frustration
DOI:
https://doi.org/10.5488/cmp.29.23705Keywords:
multiple-Q states, square lattice, biquadratic interaction, momentum-space frustration, RKKY interaction, centrosymmetric magnetsAbstract
Multiple-Q magnetism in itinerant electron systems enables complex spin crystals and noncoplanar textures even in centrosymmetric settings. We study a minimal momentum-space spin model on a square lattice with four symmetry-related ordering wave vectors, including bilinear and biquadratic interactions under an out-ofplane magnetic field. Using simulated annealing, we obtain the field-dependent phase diagram and identify successive transitions among single-Q, double-Q, and multiple inequivalent quadruple-Q states. The quadruple-Q manifold exhibits rich internal structures: the states sharing the same wave vectors differ in phase locking, amplitude distribution, and noncoplanarity, leading to distinct real-space textures and scalar spin chirality patterns. Our results demonstrate that momentum-space frustration and biquadratic coupling provide an efficient route to stabilizing diverse quadruple-Q spin crystals, offering a general framework for higher-order spin textures in centrosymmetric itinerant magnes.
References
Grüner G., Rev. Mod. Phys., 1994, 66, 1–24.
Baltz V., Manchon A., Tsoi M., Moriyama T., Ono T., Tserkovnyak Y., Rev. Mod. Phys., 2018, 90, 015005.
Yambe R., Hayami S., Phys. Rev. B, 2022, 106, 174437.
Bak P., Lebech B., Phys. Rev. Lett., 1978, 40, 800–803.
McEwen K. A., Walker M. B., Phys. Rev. B, 1986, 34, 1781–1783.
Zochowski S., McEwen K., J. Magn. Magn. Mater., 1986, 54, 515–516.
Forgan E., Rainford B., Lee S., Abell J., Bi Y., J. Phys.: Condens. Matter, 1990, 2, No. 50, 10211.
Longfield M. J., Paixão J. A., Bernhoeft N., Lander G. H., Phys. Rev. B, 2002, 66, 054417.
Bernhoeft N., Paixão J. A., Detlefs C., Wilkins S. B., Javorský P., Blackburn E., Lander G. H., Phys. Rev. B, 2004, 69, 174415.
Stewart J., Ehlers G., Wills A., Bramwell S. T., Gardner J., J. Phys.: Condens. Matter, 2004, 16, No. 28, L321.
Watson D., Forgan E. M., Nuttall W. J., Stirling W. G., Fort D., Phys. Rev. B, 1996, 53, 726–730.
Harris A. B., Schweizer J., Phys. Rev. B, 2006, 74, 134411.
Schweizer J., Givord F., Boucherle J., Bourdarot F., Ressouche E., J. Phys.: Condens. Matter, 2008, 20, No. 13, 135204.
Szabó A., Orlandi F., Manuel P., Phys. Rev. Lett., 2022, 129, 247201.
Skyrme T. H. R., Nucl. Phys., 1962, 31, 556–569.
Bogdanov A. N., Yablonskii D. A., Sov. Phys. JETP, 1989, 68, 101.
Bogdanov A., Hubert A., J. Magn. Magn. Mater., 1994, 138, No. 3, 255–269.
Rößler U. K., Bogdanov A. N., Pfleiderer C., Nature, 2006, 442, No. 7104, 797–801.
Mühlbauer S., Binz B., Jonietz F., Pfleiderer C., Rosch A., Neubauer A., Georgii R., Böni P., Science, 2009, 323, No. 5916, 915–919.
Yi S. D., Onoda S., Nagaosa N., Han J. H., Phys. Rev. B, 2009, 80, 054416.
Yu X. Z., Onose Y., Kanazawa N., Park J. H., Han J. H., Matsui Y., Nagaosa N., Tokura Y., Nature, 2010, 465, No. 7300, 901–904.
Butenko A. B., Leonov A. A., Rößler U. K., Bogdanov A. N., Phys. Rev. B, 2010, 82, 052403.
Münzer W., Neubauer A., Adams T., Mühlbauer S., Franz C., Jonietz F., Georgii R., Böni P., Pedersen B., Schmidt M., Rosch A., Pfleiderer C., Phys. Rev. B, 2010, 81, 041203.
Adams T., Mühlbauer S., Neubauer A., Münzer W., Jonietz F., Georgii R., Pedersen B., Böni P., Rosch A., Pfleiderer C., J. Phys. Conf. Ser., 2010, 200, No. 3, 032001.
Yu X. Z., Kanazawa N., Onose Y., Kimoto K., Zhang W., Ishiwata S., Matsui Y., Tokura Y., Nat. Mater., 2011, 10, No. 2, 106–109.
Heinze S., von Bergmann K., Menzel M., Brede J., Kubetzka A., Wiesendanger R., Bihlmayer G., Blügel S., Nat. Phys., 2011, 7, No. 9, 713–718.
Adams T., Mühlbauer S., Pfleiderer C., Jonietz F., Bauer A., Neubauer A., Georgii R., Böni P., Keiderling U., Everschor K., Garst M., Rosch A., Phys. Rev. Lett., 2011, 107, 217206.
Nagaosa N., Tokura Y., Nat. Nanotechnol., 2013, 8, No. 12, 899–911.
Hayami S., Phys. Rev. B, 2022, 105, 014408.
Dzyaloshinsky I., J. Phys. Chem. Solids, 1958, 4, No. 4, 241–255.
Moriya T., Phys. Rev., 1960, 120, No. 1, 91.
Nagaosa N., Sinova J., Onoda S., MacDonald A. H., Ong N. P., Rev. Mod. Phys., 2010, 82, 1539–1592.
Xiao D., Chang M.-C., Niu Q., Rev. Mod. Phys., 2010, 82, 1959–2007.
Ohgushi K., Murakami S., Nagaosa N., Phys. Rev. B, 2000, 62, R6065–R6068.
Taguchi Y., Oohara Y., Yoshizawa H., Nagaosa N., Tokura Y., Science, 2001, 291, No. 5513, 2573–2576.
Tatara G., Kawamura H., J. Phys. Soc. Jpn., 2002, 71, No. 11, 2613–2616.
Machida Y., Nakatsuji S., Maeno Y., Tayama T., Sakakibara T., Onoda S., Phys. Rev. Lett., 2007, 98, No. 5, 057203.
Shindou R., Nagaosa N., Phys. Rev. Lett., 2001, 87, 116801.
Martin I., Batista C. D., Phys. Rev. Lett., 2008, 101, 156402.
Neubauer A., Pfleiderer C., Binz B., Rosch A., Ritz R., Niklowitz P. G., Böni P., Phys. Rev. Lett., 2009, 102, 186602.
Takatsu H., Yonezawa S., Fujimoto S., Maeno Y., Phys. Rev. Lett., 2010, 105, No. 13, 137201.
Ueland B. G., Miclea C. F., Kato Y., Ayala-Valenzuela O., McDonald R. D., Okazaki R., Tobash P. H., Torrez M. A., Ronning F., Movshovich R., Fisk Z., Martin E. D. B. I., Thompson J. D., Nat. Commun., 2012, 3, 1067.
Hamamoto K., Ezawa M., Nagaosa N., Phys. Rev. B, 2015, 92, 115417.
Nakazawa K., Bibes M., Kohno H., J. Phys. Soc. Jpn., 2018, 87, No. 3, 033705.
Tokura Y., Nagaosa N., Nat. Commun., 2018, 9, No. 1, 3740.
Hayami S., Yambe R., Phys. Rev. Res., 2021, 3, 043158.
Hayami S., Yatsushiro M., Phys. Rev. B, 2022, 106, 014420.
Eto R., Pohle R., Mochizuki M., Phys. Rev. Lett., 2022, 129, 017201.
Akagi Y., Motome Y., J. Phys. Soc. Jpn., 2010, 79, No. 8, 083711.
Barros K., Kato Y., Phys. Rev. B, 2013, 88, 235101.
Venderbos J. W. F., Kourtis S., van den Brink J., Daghofer M., Phys. Rev. Lett., 2012, 108, 126405.
Jiang K., Zhang Y., Zhou S., Wang Z., Phys. Rev. Lett., 2015, 114, 216402.
Venderbos J. W. F., Phys. Rev. B, 2016, 93, 115108.
Barros K., Venderbos J. W. F., Chern G.-W., Batista C. D., Phys. Rev. B, 2014, 90, 245119.
Ghosh S., O’Brien P., Henley C. L., Lawler M. J., Phys. Rev. B, 2016, 93, 024401.
Takagi R., White J., Hayami S., Arita R., Honecker D., Rønnow H., Tokura Y., Seki S., Sci. Adv., 2018, 4, No. 11, eaau3402.
Wang Z., Su Y., Lin S.-Z., Batista C. D., Phys. Rev. Lett., 2020, 124, 207201.
Saito H., Kon F., Hidaka H., Amitsuka H., Kwanghee C., Hagihala M., Kamiyama T., Itoh S., Nakajima T., Phys. Rev. B, 2023, 108, 094440.
Ruderman M. A., Kittel C., Phys. Rev., 1954, 96, 99–102.
Kasuya T., Prog. Theor. Phys., 1956, 16, No. 1, 45–57.
Yosida K., Phys. Rev., 1957, 106, 893–898.
Saha S. R., Sugawara H., Matsuda T. D., Sato H., Mallik R., Sampathkumaran E. V., Phys. Rev. B, 1999, 60, 12162–12165.
Kurumaji T., Nakajima T., Hirschberger M., Kikkawa A., Yamasaki Y., Sagayama H., Nakao H., Taguchi Y., Arima T.-h., Tokura Y., Science, 2019, 365, No. 6456, 914–918.
Hirschberger M., Spitz L., Nomoto T., Kurumaji T., Gao S., Masell J., Nakajima T., Kikkawa A., Yamasaki Y., Sagayama H., Nakao H., Taguchi Y., Arita R., Arima T.-h., Tokura Y., Phys. Rev. Lett., 2020, 125, 076602.
Nomoto T., Koretsune T., Arita R., Phys. Rev. Lett., 2020, 125, 117204.
Kumar R., Iyer K. K., Paulose P. L., Sampathkumaran E. V., Phys. Rev. B, 2020, 101, 144440.
Spachmann S., Elghandour A., Frontzek M., Löser W., Klingeler R., Phys. Rev. B, 2021, 103, 184424.
Khanh N. D., Nakajima T., Yu X., Gao S., Shibata K., Hirschberger M., Yamasaki Y., Sagayama H., Nakao H., Peng L., Nakajima K., Takagi R., Arima T.-h., Tokura Y., Seki S., Nat. Nanotechnol., 2020, 15, 444.
Matsuyama N., Nomura T., Imajo S., Nomoto T., Arita R., Sudo K., Kimata M., Khanh N. D., Takagi R., Tokura Y., Seki S., Kindo K., Kohama Y., Phys. Rev. B, 2023, 107, 104421.
Wood G. D. A., Khalyavin D. D., Mayoh D. A., Bouaziz J., Hall A. E., Holt S. J. R., Orlandi F., Manuel P., Blügel S., Staunton J. B., Petrenko O. A., Lees M. R., Balakrishnan G., Phys. Rev. B, 2023, 107, L180402.
Eremeev S. V., Glazkova D., Poelchen G., Kraiker A., Ali K., Tarasov A. V., Schulz S., Kliemt K., Chulkov E. V., Stolyarov V. S., Ernst A., Krellner C., Usachov D. Yu., Vyalikh D. V., Nanoscale Adv., 2023, 5, No. 23, 6678–6687.
Spethmann J., Khanh N. D., Yoshimochi H., Takagi R., Hayami S., Motome Y., Wiesendanger R., Seki S., von Bergmann K., Phys. Rev. Mater., 2024, 8, 064404.
Dong Y., Kinoshita Y., Ochi M., Nakachi R., Higashinaka R., Hayami S., Wan Y., Arai Y., Huh S., Hashimoto M., Lu D., Tokunaga M., Aoki Y., Matsuda T. D., Kondo T., Science, 2025, 388, No. 6747, 624–630.
Nakamura S., Kabeya N., Kobayashi M., Araki K., Katoh K., Ochiai A., Phys. Rev. B, 2018, 98, 054410.
Hirschberger M., Nakajima T., Gao S., Peng L., Kikkawa A., Kurumaji T., Kriener M., Yamasaki Y., Sagayama H., Nakao H., Ohishi K., Kakurai K., Taguchi Y., Yu X., Arima T.-H., Tokura Y., Nat. Commun., 2019, 10, No. 1, 5831.
Hirschberger M., Hayami S., Tokura Y., New J. Phys., 2021, 23, 023039.
Nakamura S., Kabeya N., Kobayashi M., Araki K., Katoh K., Ochiai A., Phys. Rev. B, 2023, 107, 014422.
Ogunbunmi M. O., Nair H. S., Strydom A. M., Crit. Rev. Solid State Mater. Sci., 2023, 48, No. 4, 480–501.
Gao S., Zaharko O., Tsurkan V., Su Y., White J. S., Tucker G. S., Roessli B., Bourdarot F., Sibille R., Chernyshov D., Fennell T., Loidl A., Rüegg C., Nat. Phys., 2017, 13, No. 2, 157–161.
Gao S., Rosales H. D., Gómez Albarracín F. A., Tsurkan V., Kaur G., Fennell T., Steffens P., Boehm M., Čermák P., Schneidewind A., Ressouche E., Cabra D. C., Rüegg C., Zaharko O., Nature, 2020, 586, No. 7828, 37–41.
Rosales H. D., Albarracín F. A. G., Guratinder K., Tsurkan V., Prodan L., Ressouche E., Zaharko O., Phys. Rev. B, 2022, 105, 224402.
Takeda H., Kawano M., Tamura K., Akazawa M., Yan J., Waki T., Nakamura H., Sato K., Narumi Y., Hagiwara M., Yamashita M., Hotta C., Nat. Commun., 2024, 15, No. 1, 566.
Ishiwata S., Tokunaga M., Kaneko Y., Okuyama D., Tokunaga Y., Wakimoto S., Kakurai K., Arima T., Taguchi Y., Tokura Y., Phys. Rev. B, 2011, 84, 054427.
Ishiwata S., Nakajima T., Kim J.-H., Inosov D. S., Kanazawa N., White J. S., Gavilano J. L., Georgii R., Seemann K. M., Brandl G., et al., Phys. Rev. B, 2020, 101, 134406.
Rogge P. C., Green R. J., Sutarto R., May S. J., Phys. Rev. Mater., 2019, 3, 084404.
Onose M., Takahashi H., Sagayama H., Yamasaki Y., Ishiwata S., Phys. Rev. Mater., 2020, 4, 114420.
Venderbos J. W. F., Daghofer M., van den Brink J., Kumar S., Phys. Rev. Lett., 2012, 109, 166405.
Solenov D., Mozyrsky D., Martin I., Phys. Rev. Lett., 2012, 108, 096403.
Shahzad M., Sengupta P., Phys. Rev. B, 2017, 96, 224402.
Shahzad M., Sengupta P., J. Phys.: Condens. Matter, 2017, 29, No. 30, 305802.
Marcus G. G., Kim D.-J., Tutmaher J. A., Rodriguez-Rivera J. A., Birk J. O., Niedermeyer C., Lee H., Fisk Z., Broholm C. L., Phys. Rev. Lett., 2018, 120, 097201.
Seo S., Wang X., Thomas S. M., Rahn M. C., Carmo D., Ronning F., Bauer E. D., dos Reis R. D., Janoschek M., Thompson J. D., Fernandes R. M., Rosa P. F. S., Phys. Rev. X, 2020, 10, 011035.
Singh D., Fujishiro Y., Hayami S., Moody S. H., Nomoto T., Baral P. R., Ukleev V., Cubitt R., Steinke N.-J., Gawryluk D. J., Pomjakushina E., Ōnuki Y., Arita R., Tokura Y., Kanazawa N., White J. S., Nat. Commun., 2023, 14, 8050.
Akagi Y., Udagawa M., Motome Y., Phys. Rev. Lett., 2012, 108, 096401.
Hayami S., Yambe R., Mater. Today Quantum, 2024, 3, 100010.
Leonov A. O., Mostovoy M., Nat. Commun., 2015, 6, 8275.
Hayami S., Motome Y., Phys. Rev. B, 2019, 99, 094420.
Hayami S., J. Magn. Magn. Mater., 2020, 513, 167181.
Hayami S., Phys. Rev. B, 2021, 103, 224418.
Becker M., Hermanns M., Bauer B., Garst M., Trebst S., Phys. Rev. B, 2015, 91, 155135.
Lee E. K.-H., Kim Y. B., Phys. Rev. B, 2015, 91, 064407.
Janssen L., Andrade E. C., Vojta M., Phys. Rev. Lett., 2016, 117, 277202.
Rousochatzakis I., Rössler U. K., van den Brink J., Daghofer M., Phys. Rev. B, 2016, 93, 104417.
Yao X., Dong S., Sci. Rep., 2016, 6, 26750.
Amoroso D., Barone P., Picozzi S., Nat. Commun., 2020, 11, No. 1, 5784.
Hayami S., Yambe R., J. Phys. Soc. Jpn., 2020, 89, No. 10, 103702.
Hayami S., Yambe R., Phys. Rev. B, 2022, 105, 104428.
Goetsch R. J., Anand V. K., Johnston D. C., Phys. Rev. B, 2013, 87, 064406.
Fabrèges X., Gukasov A., Bonville P., Maurya A., Thamizhavel A., Dhar S. K., Phys. Rev. B, 2016, 93, 214414.
Matsumura T., Kurauchi K., Tsukagoshi M., Higa N., Nakao H., Kakihana M., Hedo M., Nakama T., Ōnuki Y., J. Phys. Soc. Jpn., 2024, 93, No. 7, 074705.
Kakihana M., Aoki D., Nakamura A., Honda F., Nakashima M., Amako Y., Nakamura S., Sakakibara T., Hedo M., Nakama T., Ōnuki Y., J. Phys. Soc. Jpn., 2018, 87, No. 2, 023701.
Kaneko K., Frontzek M. D., Matsuda M., Nakao A., Munakata K., Ohhara T., Kakihana M., Haga Y., Hedo M., Nakama T., Ōnuki Y., J. Phys. Soc. Jpn., 2019, 88, No. 1, 013702.
Tabata C., Matsumura T., Nakao H., Michimura S., Kakihana M., Inami T., Kaneko K., Hedo M., Nakama T., Ōnuki Y., J. Phys. Soc. Jpn., 2019, 88, No. 9, 093704.
Kakihana M., Aoki D., Nakamura A., Honda F., Nakashima M., Amako Y., Takeuchi T., Harima H., Hedo M., Nakama T., Ōnuki Y., J. Phys. Soc. Jpn., 2019, 88, No. 9, 094705.
Mishra A. K., Ganesan V., Phys. Rev. B, 2019, 100, 125113.
Takeuchi T., Kakihana M., Hedo M., Nakama T., Ōnuki Y., J. Phys. Soc. Jpn., 2019, 88, No. 5, 053703.
Hayami S., Yambe R., J. Phys. Soc. Jpn., 2021, 90, No. 7, 073705.
Matsumura T., Tabata C., Kaneko K., Nakao H., Kakihana M., Hedo M., Nakama T., Ōnuki Y., Phys. Rev. B, 2024, 109, 174437.
Hayami S., J. Phys. Soc. Jpn., 2022, 91, No. 2, 023705.
Hayami S., Phys. Rev. B, 2022, 105, 174437.
Hayami S., Lin S.-Z., Kamiya Y., Batista C. D., Phys. Rev. B, 2016, 94, 174420.
Brey L., Fertig H., Côté R., MacDonald A., Phys. Scr., 1996, 1996, No. T66, 154.
Ezawa M., Phys. Rev. B, 2011, 83, 100408(R).
Lin S.-Z., Saxena A., Batista C. D., Phys. Rev. B, 2015, 91, 224407.
Tan A., Li J., Scholl A., Arenholz E., Young A. T., Li Q., Hwang C., Qiu Z. Q., Phys. Rev. B, 2016, 94, 014433.
Yu X. Z., Koshibae W., Tokunaga Y., Shibata K., Taguchi Y., Nagaosa N., Tokura Y., Nature, 2018, 564, No. 7734, 95–98.
Hayami S., Yambe R., Phys. Rev. B, 2021, 104, 094425.
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