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Fluctuation corrections to the free energy of strongly correlated electron systems

David Riegler1,2,*, Jannis Seufert2,3, Ronny Thomale2,3, and Peter Wölfle1,4,†

  • *Contact author: riegler.physics@gmail.com
  • †Contact author: peter.woelfle@kit.edu
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Phys. Rev. B 113, 155131 – Published 15 April, 2026

DOI: https://doi.org/10.1103/yc1z-kr21

Abstract

We determine the free energy of strongly correlated electron systems in the example of the Hubbard model by calculating the contribution of spin and charge fluctuations to the Gutzwiller approximation mean-field result. We employ the slave-boson formulation of Kotliar and Ruckenstein in its spin-rotation-invariant form in the usual continuous-time approximation of the functional integral representation, corrected by “high-frequency contributions” (SRIKR+). In contrast to all previous attempts to calculate the fluctuation contribution to the free energy within the spin-rotation-invariant Kotliar-Ruckenstein slave-boson representation (SRIKR) scheme our result preserves the noninteracting limit. We show that the correct form of the kinetic energy renormalization is essential. The results for the ground-state energy in the paramagnetic phase are in very good agreement with state-of-the-art results obtained by methods such as density matrix embedding theory, quantum Monte Carlo, and others. The leading low-temperature behavior of the free energy allows us to extract the quasiparticle effective mass, in particular its enhancement near a continuous phase transition into an ordered state. Our work demonstrates that the SRIKR+ method is competitive with the best available alternative methods and equips the slave-boson approach with an improved synoptic power to explore strongly correlated electron systems.

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References (61)

  1. J. Hubbard, Electron correlations in narrow energy bands, Proc. R. Soc. Lond. A 276, 238 (1963).
  2. J. Hubbard, Electron correlations in narrow energy bands III. An improved solution, Proc. R. Soc. Lond. A 281, 401 (1964).
  3. J. Hubbard, Electron correlations in narrow energy bands-IV. The atomic representation, Proc. R. Soc. Lond. A 285, 542 (1965).
  4. J. Kanamori, Electron correlation and ferromagnetism of transition metals, Prog. Theor. Phys. 30, 275 (1963).
  5. P. A. Lee, T. M. Rice, J. W. Serene, L. J. Sham, and J. W. Wilkins, Theories of heavy-electron systems, Comments Condens. Matter Phys. 12, 99 (1986).
  6. P. Anderson, Basic notions of condensed matter physics, Front. Phys. 55, 49 (1984).
  7. W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and K. Schönhammer, Functional renormalization group approach to correlated fermion systems, Rev. Mod. Phys. 84, 299 (2012).
  8. J. Beyer, J. B. Hauck, and L. Klebl, Reference results for the momentum space functional renormalization group, Eur. Phys. J. B 95, 65 (2022).
  9. W. Metzner and D. Vollhardt, Correlated Lattice fermions in d=∞ dimensions, Phys. Rev. Lett. 62, 324 (1989).
  10. 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).
  11. J. Gubernatis, N. Kawashima, and P. Werner, Quantum Monte Carlo Methods (Cambridge University Press, Cambridge, UK, 2016).
  12. F. Verstraete, T. Nishino, U. Schollwöck, M. C. Bañuls, G. K. Chan, and M. E. Stoudenmire, Density matrix renormalization group, 30 years on, Nat. Rev. Phys. 5, 273 (2023).
  13. P. R. C. Kent and G. Kotliar, Toward a predictive theory of correlated materials, Science 361, 348 (2018).
  14. Q. Sun and G. K.-L. Chan, Quantum embedding theories, Acc. Chem. Res. 49, 2705 (2016).
  15. S. Wouters, C. A. Jiménez-Hoyos, Q. Sun, and G. K.-L. Chan, A practical guide to density matrix embedding theory in quantum chemistry, J. Chem. Theory Comput. 12, 2706 (2016).
  16. M. C. Gutzwiller, Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett. 10, 159 (1963).
  17. W. F. Brinkman and T. M. Rice, Application of Gutzwiller's variational method to the metal-insulator transition, Phys. Rev. B 2, 4302 (1970).
  18. D. Vollhardt, Normal He3: An almost localized Fermi liquid, Rev. Mod. Phys. 56, 99 (1984).
  19. D. Vollhardt, P. Wölfle, and P. W. Anderson, Gutzwiller-Hubbard lattice-gas model with variable density: Application to normal liquid He3, Phys. Rev. B 35, 6703 (1987).
  20. N. Lanatà, T.-H. Lee, Y.-X. Yao, and V. Dobrosavljevic, Emergent Bloch excitations in Mott matter, Phys. Rev. B 96, 195126 (2017).
  21. M. S. Frank, T.-H. Lee, G. Bhattacharyya, P. K. H. Tsang, V. L. Quito, V. Dobrosavljevic, O. Christiansen, and N. Lanatà, Quantum embedding description of the Anderson lattice model with the ghost Gutzwiller approximation, Phys. Rev. B 104, L081103 (2021).
  22. C. Mejuto-Zaera and M. Fabrizio, Towards efficient computational screening of strongly correlated materials—Multiorbital phenomenology within the ghost Gutzwiller approximation, Phys. Rev. B 107, 235150 (2023).
  23. T.-H. Lee, C. Melnick, R. Adler, N. Lanatà, and G. Kotliar, Accuracy of ghost-rotationally-invariant slave-boson theory for multiorbital Hubbard models and realistic materials, Phys. Rev. B 108, 245147 (2023).
  24. T.-H. Lee, N. Lanatà, and G. Kotliar, Accuracy of ghost rotationally invariant slave-boson and dynamical mean field theory as a function of the impurity-model bath size, Phys. Rev. B 107, L121104 (2023).
  25. R. Frésard, J. Kroha, and P. Wölfle, The pseudoparticle approach to strongly correlated electron systems, in Strongly Correlated Systems: Theoretical Methods, edited by A. Avella and F. Mancini (Springer, Berlin, 2012), pp. 65–101.
  26. R. Frésard, The slave-boson approach to correlated fermions, in Many-Body Physics: From Kondo to Hubbard, edited by E. K. E. Pavarini and P. Coleman (Forschungszentrum Jülich, Jülich, 2015),Vol.5, pp. 1–30.
  27. S. E. Barnes, New method for the Anderson model, J. Phys. F 6, 1375 (1976).
  28. P. Coleman, New approach to the mixed-valence problem, Phys. Rev. B 29, 3035 (1984).
  29. G. Kotliar and A. E. Ruckenstein, New functional integral approach to strongly correlated Fermi systems: The Gutzwiller approximation as a saddle point, Phys. Rev. Lett. 57, 1362 (1986).
  30. T. Li, P. Wölfle, and P. J. Hirschfeld, Spin-rotation-invariant slave-boson approach to the Hubbard model, Phys. Rev. B 40, 6817 (1989).
  31. R. Frésard and P. Wölfle, Unified slave boson representation of spin and charge degrees of freedom for strongly correlated Fermi systems, Int. J. Mod. Phys. B 06, 685 (1992).
  32. V. H. Dao and R. Frésard, Exact functional integration of radial and complex slave-boson fields: Thermodynamics and dynamics of the two-site extended Hubbard model, Ann. Phys. 536, 2400029 (2024).
  33. J. W. Rasul and T. Li, One-loop corrections to the Gutzwiller approach to strongly interacting fermions, with application to liquid He3, J. Phys. C 21, 5119 (1988).
  34. M. Lavagna, Functional-integral approach to strongly correlated Fermi systems: Quantum fluctuations beyond the Gutzwiller approximation, Phys. Rev. B 41, 142 (1990).
  35. T. Li, Y. S. Sun, and P. Wölfle, Dynamic response functions of Hubbard model in Gutzwiller approximation, Z. Phys. B: Condens. Matter 82, 369 (1991).
  36. W. Zimmermann, R. Frésard, and P. Wölfle, Spin and charge structure factor of the two-dimensional Hubbard model, Phys. Rev. B 56, 10097 (1997).
  37. T. Jolicoeur and J. C. Le Guillou, Fluctuations beyond the Gutzwiller approximation in the slave-boson approach, Phys. Rev. B 44, 2403 (1991).
  38. E. Arrigoni, C. Castellani, M. Grilli, R. Raimondi, and G. C. Strinati, Functional-integral formulation of the slave-boson approach: Beyond the mean-field treatment with the correct continuum limit, Phys. Rep. 241, 291 (1994).
  39. E. Arrigoni and G. C. Strinati, Correct continuum limit of the functional-integral representation for the four-slave-boson approach to the Hubbard model: Paramagnetic phase, Phys. Rev. B 52, 2428 (1995).
  40. D. Riegler, M. Klett, T. Neupert, R. Thomale, and P. Wölfle, Slave-boson analysis of the two-dimensional Hubbard model, Phys. Rev. B 101, 235137 (2020).
  41. See Supplemental Material at http://link.aps.org/supplemental/10.1103/yc1z-kr21 for a discussion of the path integral for the partition function of free bosons, showcasing the validity of the high frequency contributions in an analytically controlled example. Moreover, we derive the SRIKR+ fluctuation contributions for the extended Hubbard model including non-local interactions.
  42. J. W. Negele and H. Orland, Quantum Many-Particle Systems, Frontiers in Physics Vol. 68 (Westview Press, Boulder, CO, 1988).
  43. R. Frésard and T. Kopp, Slave bosons in radial gauge: The correct functional integral representation and inclusion of non-local interactions, Nucl. Phys. B 594, 769 (2001).
  44. R. Frésard, H. Ouerdane, and T. Kopp, Slave bosons in radial gauge: A bridge between path integral and Hamiltonian language, Nucl. Phys. B 785, 286 (2007).
  45. R. Frésard and P. Wölfle, Spiral magnetic states in the large-U Hubbard model: A slave boson approach, J. Phys.: Condens. Matter 4, 3625 (1992).
  46. E. Arrigoni and G. C. Strinati, Doping-induced incommensurate antiferromagnetism in a Mott-Hubbard insulator, Phys. Rev. B 44, 7455 (1991).
  47. R. Frésard, M. Dzierzawa, and P. Wölfle, Slave-boson approach to spiral magnetic order in the Hubbard model, Europhys. Lett. 15, 325 (1991).
  48. P. Igoshev, M. Timirgazin, A. Arzhnikov, and V. Irkhin, Effect of electron correlations on the formation of spiral magnetic states in the two-dimensional t- t' Hubbard model, J. Exp. Theor. Phys. 98, 150 (2013).
  49. P. A. Igoshev, M. A. Timirgazin, V. F. Gilmutdinov, A. K. Arzhnikov, and V. Y. Irkhin, Spiral magnetism in the single-band Hubbard model: The Hartree–Fock and slave-boson approaches, J. Condens. Matter Phys. 27, 446002 (2015).
  50. J. Seufert, D. Riegler, M. Klett, R. Thomale, and P. Wölfle, Breakdown of charge homogeneity in the two-dimensional Hubbard model: Slave-boson study of magnetic order, Phys. Rev. B 103, 165117 (2021).
  51. M. Klett, J. Beyer, D. Riegler, J. Seufert, P. Wölfle, S. Rachel, and R. Thomale, Incommensurate magnetic order: A fingerprint for electronic correlations in hole-doped cuprates, Phys. Rev. B 110, 085104 (2024).
  52. D. Riegler, J. Seufert, E. H. da Silva Neto, P. Wölfle, R. Thomale, and M. Klett, Interplay of spin and charge order in the electron-doped cuprates, Phys. Rev. B 108, 195141 (2023).
  53. V. H. Dao and R. Frésard, Collective modes in the paramagnetic phase of the Hubbard model, Phys. Rev. B 95, 165127 (2017).
  54. J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, C. A. Jiménez-Hoyos, E. Kozik, X.-W. Liu, A. J. Millis, N. V. Prokof'ev, M. Qin, G. E. Scuseria, H. Shi, B. V. Svistunov, L. F. Tocchio, I. S. Tupitsyn, and Simons Collaboration on the Many-Electron Problem, Solutions of the two-dimensional Hubbard model: Benchmarks and results from a wide range of numerical algorithms, Phys. Rev. X 5, 041041 (2015).
  55. B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155 (2017).
  56. D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, F. Becca, X. Cao, J. Carrasquilla, F. Ferrari, A. Georges, M. Hibat-Allah, M. Imada, A. M. Läuchli, G. Mazzola, A. Mezzacapo, A. Millis, J. R. Moreno, T. Neupert, Y. Nomura, J. Nys, O. Parcollet, et al., Variational benchmarks for quantum many-body problems, Science 386, 296 (2024).
  57. H. J. Schulz, Incommensurate antiferromagnetism in the two-dimensional Hubbard model, Phys. Rev. Lett. 64, 1445 (1990).
  58. B.-X. Zheng and G. K.-L. Chan, Ground-state phase diagram of the square lattice Hubbard model from density matrix embedding theory, Phys. Rev. B 93, 035126 (2016).
  59. W. F. Brinkman and S. Engelsberg, Spin-fluctuation contributions to the specific heat, Phys. Rev. 169, 417 (1968).
  60. P. Wölfle and T. Li, Spin fluctuation contribution to the specific heat of strongly correlated fermions, Z. Phys. B: Condens. Matter 78, 45 (1990).
  61. W. Brenig, H. Mikeska, and E. Riedel, On the specific heat of Fermi liquids, Z. Phys. 206, 439 (1967).

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Phys. Rev. B 113, 155131– Published 15 April, 2026
  • Received 22 April 2025
  • Revised 31 October 2025
  • Accepted 29 January 2026
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