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Equation of motion method for composite field operators


The Green’s function formalism in Condensed Matter Physics is reviewed within the equation of motion approach. Composite operators and their Green’s functions naturally appear as building blocks of generalized perturbative approaches and require fully self-consistent treatments in order to be properly handled. It is shown how to unambiguously set the representation of the Hilbert space by fixing both the unknown parameters, which appear in the linearized equations of motion and in the spectral weights of non-canonical operators, and the zero-frequency components of Green’s functions in a way that algebra and symmetries are preserved. To illustrate this procedure some examples are given: the complete solution of the two-site Hubbard model, the evaluation of spin and charge correlators for a narrow-band Bloch system, the complete solution of the three-site Heisenberg model, and a study of the spin dynamics in the Double-Exchange model.

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  1. It is also possible to have fermionic systems with occupation numbers per site equal to either 0 or 1 independently from the spin (e.g., the t-J model) or just fixed to 1 (e.g., the spin-\(\frac12\) Heisenberg model)

  2. N. Bogoliubov, S. Tyablikov, Dokl. Akad. Nauk. USSR 126, 53 (1959)

    Google Scholar 

  3. H. Mori, Progr. Theor. Phys. 33, 423 (1965)

    MATH  Google Scholar 

  4. D.J. Rowe, Rev. Mod. Phys. 40, 153 (1968)

    Article  Google Scholar 

  5. L.M. Roth, Phys. Rev. 184, 451 (1969)

    Article  Google Scholar 

  6. W. Nolting, Z. Phys. 255, 25 (1972)

    Google Scholar 

  7. Y.A. Tserkovnikov, Teor. Mat. Fiz. 49, 219 (1981)

    MathSciNet  Google Scholar 

  8. W. Nolting, W. Borgiel, Phys. Rev. B 39, 6962 (1989)

    Article  Google Scholar 

  9. N.M. Plakida, V.Y. Yushankhai, I.V. Stasyuk, Physica C 162-164, 787 (1989)

  10. A.J. Fedro, Y. Zhou, T.C. Leung, B.N. Harmon, S.K. Sinha, Phys. Rev. B 46, 14785 (1992)

    Google Scholar 

  11. P. Fulde, Electron Correlations in Molecules and Solids, 3rd edn. (Springer-Verlag, 1995),

  12. F. Mancini, S. Marra, H. Matsumoto, Physica C 244, 49 (1995)

    Google Scholar 

  13. A. Avella, F. Mancini, D. Villani, L. Siurakshina, V.Y. Yushankhai, Int. J. Mod. Phys. B 12, 81 (1998)

    Google Scholar 

  14. S. Ishihara, H. Matsumoto, S. Odashima, M. Tachiki, F. Mancini, Phys. Rev. B 49, 1350 (1994)

    Article  Google Scholar 

  15. V. Vaks, A. Larkin, S. Pikin, Sov. Phys. JETP 26, 188 (1962)

    Google Scholar 

  16. R. Zaitsev, Sov. Phys. JETP 43, 574 (1976)

    Google Scholar 

  17. Y. Izyumov, B. Letfulov, J. Phys.: Condens. Matter 2, 8905 (1990)

    Google Scholar 

  18. H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Physics (A.I.P., New York, 1993), and references therein

  19. J.C. Ward, Phys. Rev. 78, 182 (1950)

    MATH  Google Scholar 

  20. We can choose: the higher order fields emerging from the equations of motion (i.e., the conservation of some spectral moments is assured) [37], the eigenoperators of some relevant interacting terms (i.e., the relevant interactions are correctly treated) [38], the eigenoperators of the problem reduced to a small cluster [39], the composite field describing the Kondo-like singlet emerging at low-energy in any electronic spin system [40], ...

  21. N. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1133 (1966)

    Article  Google Scholar 

  22. We choose fermionic operators to study the local, thermodynamic and single-particle properties and bosonic ones to analyze the response functions

  23. D.N. Zubarev, Sov. Phys. Uspekhi 3, 320 (1960)

    Google Scholar 

  24. D. Zubarev, Non Equilibrium Statistical Thermodynamics (Consultants Bureau, New York, 1974)

  25. R. Kubo, J. Phys. Soc. Jpn 12, 570 (1957)

    MATH  Google Scholar 

  26. H. Callen, R. Swendsen, R. Tahir-Kheli, Phys. Lett. A 25, 505 (1967)

    Google Scholar 

  27. M. Suzuki, Physica 51, 277 (1971)

    Article  Google Scholar 

  28. D.L. Huber, Physica A 87, 199 (1977)

    Google Scholar 

  29. V. Aksenov, H. Konvent, J. Schreiber, Phys. Status Solids (b) 88, K43 (1978)

  30. V. Aksenov, J. Schreiber, Phys. Lett. A 69, 56 (1978)

    Article  MathSciNet  Google Scholar 

  31. V. Aksenov, M. Bobeth, N. Plakida, J. Schreiber, J. Phys. C. 20, 375 (1987)

    Google Scholar 

  32. F. Mancini, A. Avella, Condens. Matter Phys. 1, 11 (1998)

    Google Scholar 

  33. A. Avella, F. Mancini, T. Saikawa (2001), cond-mat/0103610

  34. C. Zener, Phys. Rev. B 82, 403 (1951)

    Article  Google Scholar 

  35. F. Mancini, N. Perkins, N. Plakida, Phys. Lett. A 284, 286 (2001)

    Google Scholar 

  36. S. Tyablikov, Methods in Quantum Theory of Magnetism (Plenum Press, New York, 1967)

  37. F. Mancini, Phys. Lett. A 249, 231 (1998)

    Article  Google Scholar 

  38. A. Avella, F. Mancini, S. Odashima, Physica C 388, 76 (2003)

    Google Scholar 

  39. A. Avella, F. Mancini, S. Odashima, preprint of the University of Salerno, to be published in Magn. Magn. Mater.

  40. D. Villani, E. Lange, A. Avella, G. Kotliar, Phys. Rev. Lett. 85, 804 (2000)

    Google Scholar 

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Correspondence to A. Avella.

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Received: 9 June 2003, Published online: 19 November 2003


71.10.-w Theories and models of many-electron systems - 71.27. + a Strongly correlated electron systems; heavy fermions - 71.10.Fd Lattice fermion models (Hubbard model, etc.)

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Mancini, F., Avella, A. Equation of motion method for composite field operators. Eur. Phys. J. B 36, 37–56 (2003).

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  • Hilbert Space
  • Function Formalism
  • Complete Solution
  • Hubbard Model
  • Spin Dynamic