- 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

Correlation-driven phonon renormalization and the equation of state of γ-cerium

Yao Wei1,2,3,*, Siyu Chen4,5,6, Evgeny Plekhanov1, Ivan Štich3, Cedric Weber7,†, and Jan M. Tomczak1,8,‡

  • *Contact author: yao.wei@kcl.ac.uk
  • †Contact author: cedric.weber.phd@gmail.com
  • ‡Contact author: jan.tomczak@kcl.ac.uk
PDF

Phys. Rev. B 114, 055110 – Published 10 July, 2026

DOI: https://doi.org/10.1103/zm4r-vs8j

Abstract

We investigate the thermodynamic properties of elemental cerium by assessing the crucial role of phonon free energy within the framework of dynamical mean-field theory (DMFT). While conventional density functional theory (DFT) often fails to capture the intricate energy landscape of f-electron materials, our approach integrates many-body electronic correlations with lattice dynamics to achieve a more rigorous description of the equation of state. We calculate the total energy as a function of the lattice constant at both the DFT and DFT+DMFT levels, subsequently incorporating the vibrational free energy derived from the phonon density of states. Our findings reveal that electronic renormalization of the force constants significantly alters the phonon spectra, particularly in the strongly correlated γ phase. By applying these phonon corrections to the energy profiles, we observe a substantial refinement in the predicted equilibrium volumes. Using principal component-based machine learning, we interpolate phonon dispersions continuously from a finite set of first-principles calculations and compare them with experiment, finding significantly closer agreement than conventional DFT and DFT+U calculations that neglect dynamical many-body correlations. This study underlines the necessity of accounting for both electronic and vibrational entropy when evaluating the phase stability and structural transitions of lanthanide systems under varying pressures and temperatures.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (64)

  1. D. C. Koskenmaki and J. K. A. Gschneidner, Cerium, Handb. Phys. Chem. Rare Earths 1, 337 (1978).
  2. A. W. Lawson and T.-Y. Tang, Concerning the high pressure allotropic modification of cerium, Phys. Rev. 76, 301 (1949).
  3. D. Wieliczka, J. H. Weaver, D. W. Lynch, and C. G. Olson, Photoemission studies of the γ- α phase transition in Ce: Changes in 4f character, Phys. Rev. B 26, 7056 (1982).
  4. C. Laubschat, E. Weschke, C. Holtz, M. Domke, O. Strebel, and G. Kaindl, Surface-electronic structure of α-like Ce compounds, Phys. Rev. Lett. 65, 1639 (1990).
  5. B. Johansson, The α−γ transition in cerium is a Mott transition, Philos. Mag. 30, 469 (1974).
  6. J. W. Allen and R. M. Martin, Kondo volume collapse and the γ→α transition in cerium, Phys. Rev. Lett. 49, 1106 (1982).
  7. M. Lavagna, C. Lacroix, and M. Cyrot, Volume collapse in the Kondo lattice, Phys. Lett. A 90, 210 (1982).
  8. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  9. G. Kotliar and D. Vollhardt, Strongly correlated materials: Insights from dynamical mean-field theory, Phys. Today 57(3), 53 (2004).
  10. K. Held, A. K. McMahan, and R. T. Scalettar, Cerium volume collapse: Results from the merger of dynamical mean-field theory and local density approximation, Phys. Rev. Lett. 87, 276404 (2001).
  11. A. K. McMahan, K. Held, and R. T. Scalettar, Thermodynamic and spectral properties of compressed Ce calculated using a combined local-density approximation and dynamical mean-field theory, Phys. Rev. B 67, 075108 (2003).
  12. K. Haule, V. Oudovenko, S. Y. Savrasov, and G. Kotliar, The α→γ transition in Ce: A theoretical view from optical spectroscopy, Phys. Rev. Lett. 94, 036401 (2005).
  13. B. Amadon, S. Biermann, A. Georges, and F. Aryasetiawan, The α−γ transition of cerium is entropy driven, Phys. Rev. Lett. 96, 066402 (2006).
  14. B. Chakrabarti, M. E. Pezzoli, G. Sordi, K. Haule, and G. Kotliar, α−γ transition in cerium: Magnetic form factor and dynamic magnetic susceptibility in dynamical mean-field theory, Phys. Rev. B 89, 125113 (2014).
  15. I. Leonov, V. I. Anisimov, and D. Vollhardt, First-principles calculation of atomic forces and structural distortions in strongly correlated materials, Phys. Rev. Lett. 112, 146401 (2014).
  16. K. Haule and G. L. Pascut, Forces for structural optimizations in correlated materials within a DFT+ embedded DMFT functional approach, Phys. Rev. B 94, 195146 (2016).
  17. E. Plekhanov, P. Hasnip, V. Sacksteder, M. Probert, S. J. Clark, K. Refson, and C. Weber, Many-body renormalization of forces in f-electron materials, Phys. Rev. B 98, 075129 (2018).
  18. J. Bieder and B. Amadon, Thermodynamics of the α−γ transition in cerium from first principles, Phys. Rev. B 89, 195132 (2014).
  19. M. E. Manley, R. J. McQueeney, B. Fultz, R. Osborn, G. H. Kwei, and P. D. Bogdanoff, Vibrational and electronic entropy of β-cerium and γ-cerium measured by inelastic neutron scattering, Phys. Rev. B 65, 144111 (2002).
  20. I.-K. Jeong, T. W. Darling, M. J. Graf, T. Proffen, R. H. Heffner, Y. Lee, T. Vogt, and J. D. Jorgensen, Role of the lattice in the γ→α phase transition of Ce: A high-pressure neutron and x-ray diffraction study, Phys. Rev. Lett. 92, 105702 (2004).
  21. M. Krisch, D. L. Farber, R. Xu, D. Antonangeli, C. M. Aracne, A. Beraud, T.-C. Chiang, J. Zarestky, D. Y. Kim, and E. I. Isaev, et al., Phonons of the anomalous element cerium, Proc. Natl. Acad. Sci. USA 108, 9342 (2011).
  22. I. Leonov, A. I. Poteryaev, V. I. Anisimov, and D. Vollhardt, Calculated phonon spectra of paramagnetic iron at the α−γ phase transition, Phys. Rev. B 85, 020401(R) (2012).
  23. I. Leonov, A. I. Poteryaev, Y. N. Gornostyrev, A. I. Lichtenstein, M. I. Katsnelson, V. I. Anisimov, and D. Vollhardt, Electronic correlations determine the phase stability of iron up to the melting temperature, Sci. Rep. 4, 5585 (2014).
  24. Q. Han, T. Birol, and K. Haule, Phonon softening due to melting of the ferromagnetic order in elemental iron, Phys. Rev. Lett. 120, 187203 (2018).
  25. C. P. Koçer, K. Haule, G. L. Pascut, and B. Monserrat, Efficient lattice dynamics calculations for correlated materials with DFT+DMFT, Phys. Rev. B 102, 245104 (2020).
  26. G. Khanal and K. Haule, Correlation driven phonon anomalies in bulk FeSe, Phys. Rev. B 102, 241108(R) (2020).
  27. P. Pandey, V. Pandey, and S. K. Pandey, Investigating the effect of electronic correlation on transport properties and phononic states of vanadium, Physica B 669, 415301 (2023).
  28. K. Haule and T. Birol, Free energy from stationary implementation of the DFT+DMFT functional, Phys. Rev. Lett. 115, 256402 (2015).
  29. P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H. Madsen, and L. D. Marks, WIEN2K: An APW+ lo program for calculating the properties of solids, J. Chem. Phys. 152, 074101 (2020).
  30. K. Haule, Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base, Phys. Rev. B 75, 155113 (2007).
  31. E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous-time Monte Carlo methods for quantum impurity models, Rev. Mod. Phys. 83, 349 (2011).
  32. K. Haule, Exact double counting in combining the dynamical mean field theory and the density functional theory, Phys. Rev. Lett. 115, 196403 (2015).
  33. K. Haule, C.-H. Yee, and K. Kim, Dynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn5, CeCoIn5, and CeRhIn5, Phys. Rev. B 81, 195107 (2010).
  34. See Supplemental Material at http://link.aps.org/supplemental/10.1103/zm4r-vs8j for details on the 4f occupancy, the effect of Hubbard U and spin-orbit coupling, validation of the fixed self-energy approximation, the machine-learning-based phonon interpolation scheme, and temperature-dependent free-energy analyses.
  35. J. Kaufmann and K. Held, ana_cont: Python package for analytic continuation, Comput. Phys. Commun. 282, 108519 (2023).
  36. L. Huang and H. Lu, Electronic structure of cerium: A comprehensive first-principles study, Phys. Rev. B 99, 045122 (2019).
  37. J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B 45, 13244 (1992).
  38. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  39. J. M. Tomczak, Thermoelectricity in correlated narrow-gap semiconductors, J. Phys.: Condens. Matter 30, 183001 (2018).
  40. K. Haule, DFT+DMFT tutorial 3: Generating input files and CTQMC setup, http://hauleweb.rutgers.edu/tutorials/Tutorial3.html, accessed 27 February 2026.
  41. S. Chen, Y. Wei, B. Monserrat, J. M. Tomczak, and S. Poncé, Impact of electronic correlations on the superconductivity of high-pressure CeH9, npj Comput. Mater. 12, 32 (2026).
  42. J. H. Lloyd-Williams and B. Monserrat, Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells, Phys. Rev. B 92, 184301 (2015).
  43. N. Lanatà, Y.-X. Yao, C.-Z. Wang, K.-M. Ho, J. Schmalian, K. Haule, and G. Kotliar, γ−α isostructural transition in cerium, Phys. Rev. Lett. 111, 196801 (2013).
  44. D. D. Koelling and B. N. Harmon, A technique for relativistic spin-polarised calculations, J. Phys. C: Solid State Phys. 10, 3107 (1977).
  45. B. Herzog, P. Thunström, and O. Eriksson, A configuration interaction approach to solve the Anderson impurity model; applications to elemental Ce, npj Comput. Mater. 11, 373 (2025).
  46. D. C. Wallace and H. Callen, Thermodynamics of crystals, Am. J. Phys. 40, 1718 (1972).
  47. J. S. Olsen, L. Gerward, U. Benedict, and J.-P. Itié, The crystal structure and the equation of state of cerium metal in the pressure range 0–46 GPa, Physica B+C 133, 129 (1985).
  48. M. Casadei, X. Ren, P. Rinke, A. Rubio, and M. Scheffler, Density-functional theory for f-electron systems: The α−γ phase transition in cerium, Phys. Rev. Lett. 109, 146402 (2012).
  49. M. Casadei, X. Ren, P. Rinke, A. Rubio, and M. Scheffler, Density functional theory study of the α−γ phase transition in cerium: Role of electron correlation and f-orbital localization, Phys. Rev. B 93, 075153 (2016).
  50. A. E. Mattsson, R. Armiento, J. Paier, G. Kresse, J. M. Wills, and T. R. Mattsson, The AM05 density functional applied to solids, J. Chem. Phys. 128, 084714 (2008).
  51. L. V. Pourovskii, B. Amadon, S. Biermann, and A. Georges, Self-consistency over the charge density in dynamical mean-field theory: A linear muffin-tin implementation and some physical implications, Phys. Rev. B 76, 235101 (2007).
  52. B. Amadon, First-principles DFT+ DMFT calculations of structural properties of actinides: Role of Hund's exchange, spin-orbit coupling, and crystal structure, Phys. Rev. B 94, 115148 (2016).
  53. S. Biermann, F. Aryasetiawan, and A. Georges, First-principles approach to the electronic structure of strongly correlated systems: Combining the GW approximation and dynamical mean-field theory, Phys. Rev. Lett. 90, 086402 (2003).
  54. J. M. Tomczak, P. Liu, A. Toschi, G. Kresse, and K. Held, Merging GW with DMFT and non-local correlations beyond, Eur. Phys. J. Spec. Top. 226, 2565 (2017).
  55. B. Fultz, Vibrational thermodynamics of materials, Prog. Mater. Sci. 55, 247 (2010).
  56. I. Jolliffe, Principal component analysis, in International Encyclopedia of Statistical Science (Springer, New York, 2011), pp. 1094–1096.
  57. M. Greenacre, P. J. F. Groenen, T. Hastie, A. I. d'Enza, A. Markos, and E. Tuzhilina, Principal component analysis, Nat. Rev. Methods Primers 2, 100 (2022).
  58. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning representations by back-propagating errors, Nature (London) 323, 533 (1986).
  59. P. Söderlind, O. Eriksson, B. Johansson, and J. M. Wills, Electronic properties of f-electron metals using the generalized gradient approximation, Phys. Rev. B 50, 7291 (1994).
  60. B. Amadon, F. Jollet, and M. Torrent, γ and β cerium: LDA + U calculations of ground-state parameters, Phys. Rev. B 77, 155104 (2008).
  61. B. Amadon, A self-consistent DFT + DMFT scheme in the projector augmented wave method: Applications to cerium, Ce2O3 and Pu2O3 with the Hubbard I solver and comparison to DFT + U, J. Phys.: Condens. Matter 24, 075604 (2012).
  62. A. Giri, C. Shahi, and A. Ruzsinszky, Isostructural α−γ phase transition in cerium from the perspective of metageneralized gradient approximations, Phys. Rev. B 112, 125115 (2025).
  63. A. Svane, Electronic structure of cerium in the self-interaction-corrected local-spin-density approximation, Phys. Rev. B 53, 4275 (1996).
  64. C. Ma, Z.-Y. Dou, H.-Y. Zhu, G.-Y. Fu, X. Tan, B. Bai, P.-C. Zhang, and Q.-L. Cui, Structure phase transformation and equation of state of cerium metal under pressures up to 51 GPa, Chin. Phys. B 25, 046401 (2016).

Outline

Information

Phys. Rev. B 114, 055110– Published 10 July, 2026
  • Received 31 March 2026
  • Revised 11 June 2026
  • Accepted 23 June 2026
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