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Explicit formula for perturbation theory at any order with infinitely many perturbations

J. M. Jones* and M. W. Long

  • *Contact author: jxj898@theory.bham.ac.uk
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Phys. Rev. B 113, 115126 – Published 12 March, 2026

DOI: https://doi.org/10.1103/2m8h-hvkj

Abstract

We provide a systematic formula, in terms of integer partitions, that generates perturbation theory explicitly at an arbitrary order. Our approach naturally includes an infinite number of perturbations and uses a single matrix equation that contains the information for both the eigenvalue and eigenvector corrections. The formula reduces to the standard case of one perturbation in the appropriate limit. This formulation streamlines the derivations that are traditionally tedious in perturbation theory, facilitating high-order calculations.

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References (37)

  1. P. O. Löwdin, Studies in perturbation theory. IV. Solution of eigenvalue problem by projection operator formalism, J. Math. Phys. 3, 969 (1962).
  2. T. Kato, On the convergence of the perturbation method. I, Prog. Theor. Phys. 4, 514 (1949).
  3. A. Messiah, Quantum Mechanics (Dover, New York, 2014), pp. 712–720.
  4. K. A. Brueckner, Many-body problem for strongly interacting particles. II. Linked cluster expansion, Phys. Rev. 100, 36 (1955).
  5. C. E. Soliverez, An effective Hamiltonian and time-independent perturbation theory, J. Phys. C: Solid State Phys. 2, 2161 (1969).
  6. L. Bracci and L. E. Picasso, A simple iterative method to write the terms of any order of perturbation theory in quantum mechanics, Eur. Phys. J. Plus 127, 119 (2012).
  7. J. C. Moodie and M. W. Long, An exact power series representation of the Baker–Campbell–Hausdorff formula, J. Phys. A: Math. Theor. 54, 015208 (2021).
  8. J. M. Jones and M. W. Long, A permutation-based power series representation of the Baker-Campbell-Hausdorff formula, arXiv:2511.16790.
  9. J. M. Jones and M. W. Long, Perturbation theory for operators related by the Baker–Campbell–Hausdorff formula (unpublished).
  10. J. R. Schrieffer and P. A. Wolff, Relation between the Anderson and Kondo Hamiltonians, Phys. Rev. 149, 491 (1966).
  11. A. H. MacDonald, S. M. Girvin, and D. Yoshioka, tU expansion for the Hubbard model, Phys. Rev. B 37, 9753 (1988).
  12. G. T. Landi, Eigenoperator approach to Schrieffer-Wolff perturbation theory and dispersive interactions, arXiv:2409.10656.
  13. X. Wang, F. P. M. Méndez-Córdoba, D. Jaksch, and F. Schlawin, Floquet Schrieffer-Wolff transform based on Sylvester equations, Phys. Rev. B 110, 245108 (2024).
  14. A. Bácsi, T. Iličin, and R. Žitko, Systematic Schrieffer-Wolff transformation approach to Josephson junctions: Quasiparticle effects and Josephson harmonics, Phys. Rev. B 113, 064501 (2026).
  15. J. W. S. Rayleigh, Theory of Sound, 2nd ed. (Macmillan, London, 1894), Vol. I, pp. 115–118.
  16. E. Schrödinger, Quantisierung als eigenwert problem, Ann. Phys. 385, 437 (1926).
  17. J. E. Lennard-Jones, Perturbation problems in quantum mechanics, Proc. R. Soc. Lond. Ser. A 129, 598 (1930).
  18. L. Brillouin, Les problèmes de perturbations et les champs self-consistents, J. Phys. Radium 3, 373 (1932).
  19. E. P. Wigner, On a modification of the Rayleigh-Schrödinger perturbation theory, in Part I: Physical Chemistry. Part II: Solid State Physics, edited by A. S. Wightman (Springer, Berlin, Heidelberg, 1997), pp. 131–136.
  20. We use integer partitions where the ordering matters, for example the ordered partitions of 4 are {(4), (3,1), (1,3), (2,2), (2,1,1), (1,2,1), (1,1,2), (1,1,1,1)}. These are sometimes referred to as compositions [34] by the mathematical community.
  21. For readers familiar with high-order expansions, the physical motivation for defining ΔH(n) can be interpreted using a disconnected-cluster argument [35, 36]. The subtraction of E(n) removes disconnected, nonextensive contributions to the final energy, ensuring that only connected components of the perturbation survive at each order.
  22. B. Blankleider and A. N. Kvinikhidze, Shortest derivation of time-independent perturbation theory, Am. J. Phys. 93, 652 (2025).
  23. P. J. Knowles, Perturbation-adapted perturbation theory, J. Chem. Phys. 156, 011101 (2022).
  24. M. D. Burke, M. Grandadam, and J. P. F. LeBlanc, Renormalized perturbation theory for fast evaluation of Feynman diagrams on the real frequency axis, Phys. Rev. B 107, 115151 (2023).
  25. M. Preuß, Simplifying higher-order perturbation theory for ring-shaped Bose–Hubbard systems, Phys. Scr. 100, 075225 (2025).
  26. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  27. K. Ding, C. Fang, and G. Ma, Non-Hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4, 745 (2022).
  28. W. M. Chen, Y. T. Lin, and C. Y. Ju, Non-Hermitian generalization of Rayleigh-Schrödinger perturbation theory, Phys. Rev. A 111, 022211 (2025).
  29. We could have equivalently used M(z)≡ΔH(z)+M(z)ΓΔH(z) and completed the analysis for this relationship which has the same solution as Eq. (3b).
  30. The inverse in the equation below exists since ΔH(z) contains no constant term, as long as Γ exists, which it does but this is subtle and explained rigorously in two of Löwdin's papers [1, 37].
  31. R. Scharf, The Campbell–Baker–Hausdorff expansion for classical and quantum kicked dynamics, J. Phys. A: Math. Gen. 21, 2007 (1988).
  32. L. D'Alessio and A. Polkovnikov, Many-body energy localization transition in periodically driven systems, Ann. Phys. 333, 19 (2013).
  33. S. Vajna, K. Klobas, T. Prosen, and A. Polkovnikov, Replica resummation of the Baker–Campbell–Hausdorff series, Phys. Rev. Lett. 120, 200607 (2018).
  34. R. P. Stanley, Enumerative Combinatorics, 2nd ed., Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2011), Vol. 1, pp. 17–18.
  35. F. J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75, 486 (1949).
  36. R. Kubo, Generalized cumulant expansion method, J. Phys. Soc. Jpn. 17, 1100 (1962).
  37. P. O. Löwdin, Studies in perturbation theory. V. Some aspects on the exact self-consistent field theory, J. Math. Phys. 3, 1171 (1962).

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Phys. Rev. B 113, 115126– Published 12 March, 2026
  • Received 20 October 2025
  • Revised 19 February 2026
  • Accepted 24 February 2026
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