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Quantum behavior of multiple monomeric arrays of spin 12

Vinicius T. Santana1,*, Rosana P. Sartoris2, and Rafael Calvo2,3,†

  • *Contact author: santana@vutbr.cz
  • †Contact author: calvo.rafael@gmail.com
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Phys. Rev. B 112, 014445 – Published 25 July, 2025

DOI: https://doi.org/10.1103/71h7-vhj7

Abstract

Spin arrays provide a framework for studying cooperative spin phenomena and correlated states. Advances in high-frequency electron paramagnetic resonance (HFEPR) above 90 GHz offer new opportunities to investigate these systems, determine the effect of weak spin-spin interactions, and evaluate their magnitudes. Here, we investigate the cupric coordination compound Cu(L−met)2, where anisotropic S=12 spins occupy two rotated structural sites, forming weakly coupled spin arrays via supramolecular interactions. Previous EPR studies at 9.7 GHz and 33.6 GHz identified pseudo-one-dimensional magnetic behavior with a single, orientation-independent spectral peak, allowing estimates of exchange couplings. In this work, HFEPR spectra up to 450 GHz and magnetic fields (B0) up to 16 T display a more complex spectral landscape: the number of resonances switches between one and two as a function of the orientation of B0, revealing different dynamical regimes. This behavior indicates a crossover between a collective quantum-correlated phase and a regime where spins behave independently, allowing us to construct quantum dynamical phase diagrams across frequency and field space. These findings reinforce molecular compounds as promising platforms for exploring tunable spin dynamics, with implications for quantum information science. The observation of spin-spin interactions persisting at room temperature highlights their potential for robust quantum coherence in molecular systems.

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References (104)

  1. A. Bencini and D. Gatteschi, Electron Paramagnetic Resonance of Exchange Coupled Systems (Springer, Berlin, 1990).
  2. O. Kahn, Molecular Magnetism (Wiley VCH, New York, 1993).
  3. A. Gaita-Ariño, F. Luis, S. Hill, and E. Coronado, Molecular spins for quantum computation, Nat. Chem. 11, 301 (2019).
  4. A. Chiesa, P. Santini, E. Garlatti, F. Luis, and S. Carretta, Molecular nanomagnets: A viable path toward quantum information processing? Rep. Prog. Phys. 87, 034501 (2024).
  5. D. Gatteschi, A. L. Barra, A. Caneschi, A. Cornia, R. Sessoli, and L. Sorace, EPR of molecular nanomagnets, Coord. Chem. Rev. 250, 1514 (2006).
  6. P. W. Anderson, A mathematical model for the narrowing of spectral lines by exchange or motion, J. Phys. Soc. Jpn. 9, 316 (1954).
  7. M. Kubo and K. Tomita, A general theory of magnetic resonance absorption, J. Phys. Soc. Jpn. 9, 888 (1954).
  8. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford, 1970), p. 911.
  9. J. A. Weil and J. R. Bolton, Electron Paramagnetic Resonance. Elementary Theory and Practical Applications, 2nd ed. (Wiley-Interscience, Hoboken, NJ, 2007), p. 664.
  10. C. J. Gorter and J. H. Van Vleck, The role of exchange interaction in paramagnetic absorption, Phys. Rev. 72, 1128 (1947).
  11. J. H. Van Vleck, The dipolar broadening of magnetic resonance lines in crystals, Phys. Rev. 74, 1168 (1948).
  12. D. M. S. Bagguley and J. H. E. Griffiths, Paramagnetic resonance and magnetic energy levels in chrome alum, Nature (London) 160, 532 (1947).
  13. D. M. S. Bagguley and J. H. E. Griffiths, Paramagnetic resonance in copper sulphate, Nature (London) 162, 538 (1948).
  14. M. H. L. Pryce, Theory, Nature (London) 162, 539 (1948).
  15. D. Halliday and J. Wheatley, Paramagnetic resonance absorption in chrome Alum, Phys. Rev. 74, 1712 (1948).
  16. R. D. Arnold and A. F. Kip, Paramagnetic resonance absorption in two sulfates of copper, Phys. Rev. 75, 1199 (1949).
  17. J. Wheatley and D. Halliday, Paramagnetic absorption in single crystals of copper sulfate pentahydrate, Phys. Rev. 75, 1412 (1949).
  18. D. M. S. Bagguley and J. H. E. Griffiths, Paramagnetic resonance in copper sulphate, Proc. R. Soc. London A 201, 366 (1950).
  19. H. Abe and K. Ôno, Paramagnetic resonance spectra of some cupric salts (I), J. Phys. Soc. Jpn. 11, 947 (1956).
  20. D. M. S. Bagguley and J. Owen, Microwave properties of solids, Rep. Prog. Phys. 20, 304 (1957).
  21. P. W. Anderson and P. R. Weiss, Exchange narrowing in paramagnetic resonance, Rev. Mod. Phys. 25, 269 (1953).
  22. M. Yokota and S. Koide, The exchange effects on the absorption spectrum by the paramagnetic crystal with anisotropic g-factors, J. Phys. Soc. Jpn. 9, 953 (1954).
  23. R. A. Sack, A contribution to the theory of the exchange narrowing of spectral lines, Mol. Phys. 1, 163 (1958).
  24. C. J. Ballhausen, Introduction to Ligand Field Theory (McGraw-Hill, New York, 1962), pp. 268.
  25. B. N. Figgis and M. A. Hitchman, Ligand Field Theory and its Applications (Wiley-VCH, New York, 2000), Chap. 10.
  26. R. E. Dietz, F. R. Merritt, R. Dingle, D. Hone, B. G. Silbernagel, and P. M. Richards, Exchange narrowing in one-dimensional systems, Phys. Rev. Lett. 26, 1186 (1971).
  27. P. M. Richards and M. B. Salamon, Exchange narrowing of electron spin resonance in a two-dimensional system, Phys. Rev. B 9, 32 (1974).
  28. Y. Servant, J.-C. Bissey, and M. Maini, An X-band and Q-band single crystal electron paramagnetic resonance study of CuCl2·2H2O, Phys. B: Condens. Matter 106, 343 (1981).
  29. R. Calvo and M. A. Mesa, EPR study of electronic and magnetic properties of bis(DL−α-amino-n-butyrato)copper(II). A layered magnetic system, Phys. Rev. B 28, 1244 (1983).
  30. R. Calvo, M. A. Mesa, G. Oliva, J. Zukerman-Schpector, O. R. Nascimento, M. Tovar, and R. Arce, Structural and magnetic properties of a copper amino acid salt. Copper(II)bis(α-amino-isobutyrato), J. Chem. Phys. 81, 4584 (1984).
  31. R. Calvo, H. Isern, and M. A. Mesa, EPR study of Cu(L-ILE)2, a copper amino acid salt, Chem. Phys. 100, 89 (1985).
  32. A. M. Gennaro, P. R. Levstein, C. A. Steren, and R. Calvo, EPR of layered magnetic metal—amino acid salts. I. Cu(L-phe)2, Chem. Phys. 111, 431 (1987).
  33. P. R. Levstein, C. A. Steren, A. M. Gennaro, and R. Calvo, EPR of layered magnetic metal-amino acid salts. II. Cu(L-Met)2, Chem. Phys. 120, 449 (1988).
  34. C. A. Steren, A. M. Gennaro, P. R. Levstein, and R. Calvo, Exchange interactions in bis(L-leucinato)copper(II), J. Phys.: Condens. Matter 1, 637 (1989).
  35. P. R. Levstein, R. Calvo, E. E. Castellano, O. E. Piro, and B. E. Rivero, Molecular structure and exchange interactions in trans-bis(L-2-aminobutyrato)copper(II) and trans-bis(DL-2-aminobutyrato)copper(II), Inorg. Chem. 29, 3918 (1990).
  36. P. R. Levstein and R. Calvo, Superexchange coupling mediated by carboxylate and hydrogen bridges in copper amino acid complexes, Inorg. Chem. 29, 1581 (1990).
  37. P. R. Levstein, H. M. Pastawski, and R. Calvo, Spin diffusion in low-dimensional copper-amino-acid complexes, J. Phys.: Condens. Matter 3, 1877 (1991).
  38. R. Calvo, M. C. G. Passeggi, M. A. Novak, O. G. Symko, S. B. Oseroff, O. R. Nascimento, and M. C. Terrile, Exchange interactions and magnetic dimension in Cu(L-alanine)2, Phys. Rev. B 43, 1074 (1991).
  39. R. Calvo, P. R. Levstein, E. E. Castellano, S. M. Fabiane, O. E. Piro, and S. B. Oseroff, Crystal structure and magnetic interactions in bis(D,L-alaninato)copper(II)hydrate, Inorg. Chem. 30, 216 (1991).
  40. C. D. Brondino, N. M. C. Casado, M. C. G. Passeggi, and R. Calvo, Exchange interaction between copper(II) ions through glutamic acid molecules, Inorg. Chem. 32, 2078 (1993).
  41. R. Calvo, C. A. Steren, O. E. Piro, T. Rojo, F. J. Zuniga, and E. E. Castellano, Crystal structure and magnetic properties of diaqua(L-aspartato)copper(II), Inorg. Chem. 32, 6016 (1993).
  42. M. C. G. Passeggi and R. Calvo, Evaluation by EPR of the exchange interactions coupling anisotropic spins at symmetry-related sites in paramagnetic crystals, J. Magn. Reson. Ser. A 114, 1 (1995).
  43. D. M. Martino, M. C. G. Passeggi, and R. Calvo, Magnetic interactions in Cu(L-isoleucine)2·H2O: An EPR measurement, Phys. Rev. B 52, 9466 (1995).
  44. R. P. Sartoris, L. Ortigoza, N. M. C. Casado, R. Calvo, E. E. Castellano, and O. E. Piro, Structure, single crystal EPR spectra, and exchange interactions in [Cu(L-proline)2]2·5H2O and Cu(D,L-proline)2·2H2O, Inorg. Chem. 38, 3598 (1999).
  45. A. J. Costa-Filho, C. E. Munte, C. Barberato, E. E. Castellano, M. P. D. Mattioli, R. Calvo, and O. R. Nascimento, Crystal structures and magnetic properties of CuX2(pdmp)2 complexes (X = Br, Cl), Inorg. Chem. 38, 4413 (1999).
  46. S. K. Hoffmann and L. S. Szczepaniak, Dynamical averaging of two symmetry-related magnetic resonance tensors in single crystals, J. Magn. Reson. 52, 182 (1983).
  47. S. K. Hoffmann, Temperature dependence of a superexchange interaction determined from merging effect of copper(II) EPR lines, Chem. Phys. Lett. 98, 329 (1983).
  48. S. K. Hoffmann, J. Goslar, and L. S. Szczepaniak, Weak exchange interaction in cis-Cu(NH2CH2COO)2·H2O) single crystal: EPR studies, Phys. Rev. B 37, 7331 (1988).
  49. J. Goslar, W. Hilczer, S. K. Hoffmann, and M. Krupski, Temperature and pressure EPR studies of superexchange coupling in Cu(1-phenylpyrazole)2Cl2 crystals, Phys. Status Solidi (b) 167, 291 (1991).
  50. J. Goslar, W. Hilczer, and S. K. Hoffmann, Weak exchange effects in EPR anisotropy of dichlorobis(1-phenyl-3,5-dimethylpyrazole) Copper(II) Single Crystals, Phys. Status Solidi (b) 175, 465 (1993).
  51. S. K. Hoffmann, W. Hilczer, and J. Goslar, Weak long-distance superexchange interaction and its temperature variations in copper(II) compounds studied by single crystal EPR, Appl. Magn. Reson. 7, 289 (1994).
  52. G. E. Pake, Paramagnetic Resonance. An Introductory Monograph, 1st ed. (Benjamin, Inc., New York, 1962).
  53. D. M. Martino, M. C. G. Passeggi, R. Calvo, and O. R. Nascimento, EPR spectroscopy and exchange interaction parameters in Cu(glycine)2H2O, Phys. B: Condens. Matter 225, 63 (1996).
  54. R. Calvo, M. C. G. Passeggi, N. O. Moreno, G. E. Barberis, A. B. Chaves, B. C. M. Torres, L. Lezama, and T. Rojo, Magnetic properties of Cu(L-aspartato)(H2O): A linear chain antiferromagnet, Phys. Rev. B 60, 1197 (1999).
  55. E. D. Vieira, N. M. C. Casado, G. Facchin, M. H. Torre, A. J. Costa-Filho, and R. Calvo, Weak exchange interaction supported by a biologically relevant long chemical bridge in a Cu-peptide model compound, Inorg. Chem. 45, 2942 (2006).
  56. D. Venegas-Yazigi, K. A. Brown, A. Vega, R. Calvo, C. Aliaga, R. C. Santana, R. Cardoso-Gil, R. Kniep, W. Schnelle, and E. Spodine, Exchange interactions through π–π stacking in the lamellar compound [{Cu(bipy)(en)}{Cu(bipy)(H2O)}{VO3}4]n, Inorg. Chem. 50, 11461 (2011).
  57. V. Paredes–García, R. C. Santana, R. Madrid, B. Baldo, A. Vega, and M. Spohn, Single crystal electron paramagnetic resonance spectra of Cu(II) ions in Cu(tyrosine)2. A study of weak exchange interactions mediated by resonance assisted hydrogen bonds (RAHB), J. Inorg. Biochem. 114, 75 (2012).
  58. V. T. Santana, B. N. Cunha, A. M. Plutín, R. G. Silveira, E. E. Castellano, A. A. Batista, R. Calvo, and O. R. Nascimento, Magnetic-field-tuned phase transition of a molecular material from the isolated-spin to the coupled-spin regime, Phys. Chem. Chem. Phys. 21, 4394 (2019).
  59. F. Troiani, V. Bellini, A. Candini, G. Lorusso, and M. Affronte, Spin entanglement in supramolecular structures, Nanotechnology 21, 274009 (2010).
  60. M. R. Wasielewski, M. D. E. Forbes, N. L. Frank, K. Kowalski, G. D. Scholes, J. Yuen-Zhou, M. A. Baldo, D. E. Freedman, R. H. Goldsmith, T. Goodson III et al., Exploiting chemistry and molecular systems for quantum information science, Nat. Rev. Chem. 4, 490 (2020).
  61. Y. Chen, Y. Bae, and A. J. Heinrich, Harnessing the quantum behavior of spins on surfaces, Adv. Mater. 35, 2107534 (2023).
  62. A. Sojka, M. Šedivý, O. Laguta, A. Marko, V. T. Santana, and P. Neugebauer, High-frequency EPR: current state and perspectives, in Electron Paramagnetic Resonance: Volume 27, edited by V. Chechik, D. M. Murphy, and B. E. Bode (Royal Society of Chemistry, 2020), pp. 214–252.
  63. L. M. B. Napolitano, O. R. Nascimento, S. Cabaleiro, J. Castro, and R. Calvo, Isotropic and anisotropic spin-spin interactions and a quantum phase transition in a dinuclear Cu(II) compound, Phys. Rev. B 77, 214423 (2008).
  64. M. Perec, R. Baggio, R. P. Sartoris, R. C. Santana, O. Peña, and R. Calvo, Magnetism and structure in chains of copper dinuclear paddlewheel units, Inorg. Chem. 49, 695 (2010).
  65. R. Calvo, J. E. Abud, R. P. Sartoris, and R. C. Santana, Collapse of the EPR fine structure of a one-dimensional array of weakly interacting binuclear units: A dimensional quantum phase transition, Phys. Rev. B 84, 104433 (2011).
  66. R. Calvo, V. T. Santana, and O. R. Nascimento, Temperature dependence of the effective interdimer exchange interaction in a weakly coupled antiferromagnetic dimer copper compound, Phys. Rev. B 96, 064424 (2017).
  67. R. P. Sartoris, V. T. Santana, E. Freire, R. F. Baggio, O. R. Nascimento, and R. Calvo, Exchange couplings and quantum phases in two dissimilar arrays of similar copper dinuclear units, Dalton Trans. 49, 5228 (2020).
  68. R. Calvo, R. P. Sartoris, O. R. Nascimento, M. Šedivý, A. Sojka, P. Neugebauer, and V. T. Santana, Quantum phase transitions probed by EPR spectra in dimeric spin arrays with supramolecular couplings, Coord. Chem. Rev. 480, 215007 (2023).
  69. Č. Brukner, V. Vedral, and A. Zeilinger, Crucial role of quantum entanglement in bulk properties of solids, Phys. Rev. A 73, 012110 (2006).
  70. T. Chakraborty, H. Singh, D. Das, T. K. Sen, and C. Mitra, Quantification of entanglement from magnetic susceptibility for a Heisenberg spin 1/2 system, Phys. Lett. A 376, 2967 (2012).
  71. S. Sahling, G. Remenyi, C. Paulsen, P. Monceau, V. Saligrama, C. Marin, A. Revcolevschi, L. P. Regnault, S. Raymond, and J. E. Lorenzo, Experimental realization of long-distance entanglement between spins in antiferromagnetic quantum spin chains, Nat. Phys. 11, 255 (2015).
  72. P. Laurell, A. Scheie, E. Dagotto, and D. A. Tennant, Witnessing entanglement and quantum correlations in condensed matter: A review, Adv. Quantum Technol. 8, 2400196 (2025).
  73. J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Coherent manipulation of coupled electron spins in semiconductor quantum dots, Science 309, 2180 (2005).
  74. G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Rev. Mod. Phys. 95, 025003 (2023).
  75. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, 2011), p. 501.
  76. V. Zapf, M. Jaime, and C. D. Batista, Bose-Einstein condensation in quantum magnets, Rev. Mod. Phys. 86, 563 (2014).
  77. E. Fogh, M. Nayak, O. Prokhnenko, M. Bartkowiak, K. Munakata, J.-R. Soh, A. A. Turrini, M. E. Zayed, E. Pomjakushina, H. Kageyama et al., Field-induced bound-state condensation and spin-nematic phase in SrCu2(BO3)2 revealed by neutron scattering up to 25.9 T, Nat. Commun. 15, 442 (2024).
  78. R. Grimaudo, A. Messina, A. Sergi, N. Vitanov, and S. Filippov, Two-qubit entanglement generation through non-Hermitian Hamiltonians induced by repeated measurements on an ancilla, Entropy 22, 1184 (2020).
  79. R. Grimaudo, A. S. Magalhães de Castro, A. Messina, and D. Valenti, Spin-chain-star systems: Entangling multiple chains of spin qubits, Fortschr. Phys. 70, 2200042 (2022).
  80. R. Grimaudo, SU(2)-Symmetric exactly solvable models of two interacting qubits, Physics 6, 1111 (2024).
  81. C.-C. Ou, D. A. Powers, J. A. Thich, T. R. Felthouse, D. N. Hendrickson, J. A. Potenza, and H. J. Schugar, Molecular structure and magnetic properties of trans-bis(L-methioninato)copper(II), Cu(C5H10NO2S)2, Inorg. Chem. 17, 34 (1978).
  82. H. M. Pastawski, Revisiting the Fermi golden rule: Quantum dynamical phase transition as a paradigm shift, Phys. B: Condens. Matter 398, 278 (2007).
  83. R. Calvo, EPR measurements of weak exchange interactions coupling unpaired spins in model compounds, Appl. Magn. Reson. 31, 271 (2007).
  84. S. Stoll and A. Schweiger, EasySpin, a comprehensive software package for spectral simulation and analysis in EPR, J. Magn. Reson. 178, 42 (2006).
  85. Note that a closed ring of NS=12 was used as a representation of an infinite array since this was the highest number of spins we could add in order to have a successful simulation given our available computational resources. In any case, it was enough to demonstrate the expected merging of peaks expected for an infinite array.
  86. A. Abragam, Principles of Nuclear Magnetism (Clarendon Press, Oxford, 1961).
  87. See Supplemental Material at http://link.aps.org/supplemental/10.1103/71h7-vhj7 for additional details on low-frequency EPR data used to calculate the g2 matrix given in Table 1 and the value of J for the same set of data.
  88. C. F. Macrae, I. J. Bruno, J. A. Chisholm, P. R. Edgington, F. Mazzi, E. Pidcock, L. Rodriguez-Monge, R. Taylor, J. Van de Streek, and P. A. Wood, Mercury CSD 2.0 – New features for the visualization and investigation of crystal structures, J. Appl. Crystallogr. 41, 466 (2008).
  89. Crystal structure of TBLMCU, Available from The Cambridge Crystallographic Data Centre. CCDC: 1267555 (1978).
  90. D. Van der Helm, M. B. Lawson, and E. L. Enwall, The crystal structure of bis-(L-phenylalaninato)copper(II), Acta Crystallogr. B 27, 2411 (1971).
  91. T. G. Fawcett, M. Ushay, J. P. Rose, R. A. Lalancette, J. A. Potenza, and H. J. Schugar, Molecular structures of the copper-amino acid complexes Bis(L-leucinato)copper(II) and Bis(D,L-2-aminobutyrato)copper(II), Inorg. Chem. 18, 327 (1979).
  92. J. M. Schveigkardt, A. C. Rizzi, O. E. Piro, E. E. Castellano, R. C. Santana, R. Calvo, and C. D. Brondino, Structural and single crystal EPR studies of the complex copper L-glutamine: A weakly exchange-coupled system with syn-anti carboxylate bridges, Eur. J. Inorg. Chem. 2002, 2913 (2002).
  93. S. Hill, R. S. Edwards, N. Aliaga-Alcalde, and G. Christou, Quantum coherence in an exchange-coupled dimer of single-molecule magnets, Science 302, 1015 (2003).
  94. S. Hill and A. Wilson, Calculation of the EPR spectrum for an entangled dimer of S = 9/2 Mn4 single-molecule magnets, J Low Temp Phys 142, 271 (2007).
  95. P. M. Richards, Correlations in magnetic exchange-lattice relaxation, Phys. Rev. 137, A1327 (1965).
  96. S. Mankovsky, S. Polesya, and H. Ebert, Exchange coupling constants at finite temperature, Phys. Rev. B 102, 134434 (2020).
  97. M. L. Siqueira, R. E. Rapp, and R. Calvo, Specific heat and exchange interactions of two isomer and isostructural copper–amino-acid complexes, Phys. Rev. B 48, 3257 (1993).
  98. A. J. Costa-Filho, O. R. Nascimento, L. Ghivelder, and R. Calvo, Magnetic properties of carboxylate-bridged ferromagnetic copper(II) chains coupled by cation-π interactions, J. Phys. Chem. B 105, 5039 (2001).
  99. R. Calvo and M. C. G. Passeggi, Shifts with temperature of the EPR signal in Cu(L-alanine)2: A low-dimensional paramagnet, Phys. Rev. B 44, 5111 (1991).
  100. A. Fainstein, Dipolar fields in a low-dimensional paramagnet: Cu(L-leucine)2, J. Phys.: Condens. Matter 4, 4989 (1992).
  101. A. Sojka, M. Šedivý, A. Lagiň, A. Gabriš, T. Láznička, V. T. Santana, O. Laguta, and P. Neugebauer, Sample holders for sub-THz electron spin resonance spectroscopy, IEEE Trans. Instrum. Meas. 71, 1 (2022).
  102. M. Šedivý, V. Santana, A. Sojka, O. Laguta, and P. Neugebauer, MEPROS – Modular electron paramagnetic resonance operating software for multifunctional high-frequency EPR spectrometer, J. Magn. Reson. 355, 107556 (2023).
  103. M. Bak and N. C. Nielsen, Repulsion, A novel approach to efficient powder averaging in solid-state NMR, J. Magn. Reson. 125, 132 (1997).
  104. V. T. Santana, R. P. Sartoris, and R. Calvo, Quantum behavior of multiple monomeric arrays of spins 1/2, Zenodo (2024), doi: 10.5281/zenodo.13869324.

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Phys. Rev. B 112, 014445– Published 25 July, 2025
  • Received 12 February 2025
  • Revised 1 July 2025
  • Accepted 8 July 2025
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