- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Intertwined fluctuations and isotope effects in the Hubbard-Holstein model on the square lattice from functional renormalization

Aiman Al-Eryani1,*, Sabine Andergassen2,3,†, and Michael M. Scherer1,‡

  • *Contact author: aiman.al-eryani@rub.de
  • †Contact author: sabine.andergassen@tuwien.ac.at
  • ‡Contact author: scherer@tp3.rub.de
PDF

Phys. Rev. Research 7, 043052 – Published 15 October, 2025

DOI: https://doi.org/10.1103/6vyb-h379

Abstract

Electron-electron and electron-phonon interactions are responsible for the formation of spin, charge, and superconducting correlations in layered quantum materials. A paradigmatic model for such materials that captures both kinds of interactions is the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon. In this work, we provide a detailed analysis of the magnetic, density, and superconducting fluctuations at and away from half filling. To that end, we employ the functional renormalization group using the recently introduced extension of the single-boson exchange formulation. More precisely, we go beyond previous approaches to the model by resolving the full frequency dependence of the two-particle vertex and taking into account the feedback from the electronic self-energy. We perform broad parameter scans in the space of Hubbard repulsion, electron-phonon coupling strength, and phonon frequency to explore the leading magnetic, density, and superconducting susceptibilities from the adiabatic to the anti-adiabatic regime. Our numerical data reveal that self-energy effects lead to an enhancement of the d-wave superconducting susceptibility toward larger phonon frequencies, in contrast to earlier isotope-effect studies. At small phonon frequencies, large density contributions to the s-wave superconducting susceptibility change sign and eventually lead to a reduction of s-wave superconductivity with increasing electron-phonon coupling, signaling the breakdown of Migdal-Eliashberg theory. We analyze our findings systematically, employing detailed diagnostics of the intertwined fluctuations and pinning down the various positive and negative isotope effects of the physical susceptibilities. Our results pave the way for a more comprehensive understanding of the interplay of electron-electron and electron-phonon interactions in correlated quantum materials.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (138)

  1. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957).
  2. E. Maxwell, Isotope effect in the superconductivity of mercury, Phys. Rev. 78, 477 (1950).
  3. C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt, Superconductivity of isotopes of mercury, Phys. Rev. 78, 487 (1950).
  4. R. T. Scalettar, N. E. Bickers, and D. J. Scalapino, Competition of pairing and Peierls–Charge-density-wave correlations in a two-dimensional electron-phonon model, Phys. Rev. B 40, 197 (1989).
  5. R. M. Noack, D. J. Scalapino, and R. T. Scalettar, Charge-density-wave and pairing susceptibilities in a two-dimensional electron-phonon model, Phys. Rev. Lett. 66, 778 (1991).
  6. F. Marsiglio, Pairing and charge-density-wave correlations in the Holstein model at half-filling, Phys. Rev. B 42, 2416 (1990).
  7. M. Feldbacher, F. F. Assaad, F. Hébert, and G. G. Batrouni, Coexistence of s-Wave superconductivity and antiferromagnetism, Phys. Rev. Lett. 91, 056401 (2003).
  8. B. Xing, W.-T. Chiu, D. Poletti, R. T. Scalettar, and G. Batrouni, Quantum Monte Carlo simulations of the 2D Su-Schrieffer-Heeger model, Phys. Rev. Lett. 126, 017601 (2021).
  9. X. Cai, Z.-X. Li, and H. Yao, Antiferromagnetism induced by bond Su-Schrieffer-Heeger electron-phonon coupling: A quantum Monte Carlo study, Phys. Rev. Lett. 127, 247203 (2021).
  10. J. Hubbard, Electron correlations in narrow energy bands, Proc. R. Soc. Lond. Ser. A 276, 238 (1963).
  11. D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annu. Rev. Condens. Matter Phys. 13, 239 (2022).
  12. M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective, Annu. Rev. Condens. Matter Phys. 13, 275 (2022).
  13. B. Batlogg, R. J. Cava, A. Jayaraman, R. B. van Dover, G. A. Kourouklis, S. Sunshine, D. W. Murphy, L. W. Rupp, H. S. Chen, A. White, K. T. Short, A. M. Mujsce, and E. A. Rietman, Isotope effect in the high-Tc superconductors Ba2YCu3O7 and Ba2EuCu3O7, Phys. Rev. Lett. 58, 2333 (1987).
  14. J. P. Franck, S. Harker, and J. H. Brewer, Copper and oxygen isotope effects in La2−xSrxCuO4, Phys. Rev. Lett. 71, 283 (1993).
  15. D. Zech, H. Keller, K. Conder, E. Kaldis, E. Liarokapis, N. Poulakis, and K. A. Müller, Site-selective oxygen isotope effect in optimally doped YBa2Cu3O6+x, Nature (London) 371, 681 (1994).
  16. K. A. Müller, The unique properties of superconductivity in cuprates, J. Supercond. Novel Magn. 27, 2163 (2014).
  17. R. H. Liu, T. Wu, G. Wu, H. Chen, X. F. Wang, Y. L. Xie, J. J. Ying, Y. J. Yan, Q. J. Li, B. C. Shi, W. S. Chu, Z. Y. Wu, and X. H. Chen, A large iron isotope effect in SmFeAsO1−xFx and Ba1−xKxFe2As2, Nature (London) 459, 64 (2009).
  18. P. M. Shirage, K. Kihou, K. Miyazawa, C.-H. Lee, H. Kito, H. Eisaki, T. Yanagisawa, Y. Tanaka, and A. Iyo, Inverse iron isotope effect on the transition temperature of the (Ba,K)Fe2As2 superconductor, Phys. Rev. Lett. 103, 257003 (2009).
  19. E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  20. P. A. Pantaleón, A. Jimeno-Pozo, H. Sainz-Cruz, V. T. Phong, T. Cea, and F. Guinea, Superconductivity and correlated phases in non-twisted bilayer and trilayer graphene, Nat. Rev. Phys. 5, 304 (2023).
  21. K. F. Mak and J. Shan, Semiconductor moiré materials, Nat. Nanotechnol. 17, 686 (2022).
  22. T. Holstein, Studies of polaron motion: Part I. The molecular-crystal model, Ann. Phys. 8, 325 (1959).
  23. J. K. Freericks and M. Jarrell, Competition between electron-phonon attraction and weak Coulomb repulsion, Phys. Rev. Lett. 75, 2570 (1995).
  24. M. Capone, G. Sangiovanni, C. Castellani, C. Di Castro, and M. Grilli, Phase separation close to the density-driven Mott transition in the Hubbard-Holstein model, Phys. Rev. Lett. 92, 106401 (2004).
  25. W. Koller, D. Meyer, and A. C. Hewson, Dynamic response functions for the Holstein-Hubbard model, Phys. Rev. B 70, 155103 (2004).
  26. P. Paci, M. Capone, E. Cappelluti, S. Ciuchi, C. Grimaldi, and L. Pietronero, Polaronic and nonadiabatic phase diagram from anomalous isotope effects, Phys. Rev. Lett. 94, 036406 (2005).
  27. G. Sangiovanni, M. Capone, C. Castellani, and M. Grilli, Electron-phonon interaction close to a Mott transition, Phys. Rev. Lett. 94, 026401 (2005).
  28. C. A. Perroni, V. Cataudella, G. De Filippis, and V. M. Ramaglia, Effects of electron-phonon coupling near and within the insulating Mott phase, Phys. Rev. B 71, 113107 (2005).
  29. E. Piegari, C. A. Perroni, and V. Cataudella, Signatures of polaron formation in systems with local and non-local electron-phonon couplings, Eur. Phys. J. B 44, 415 (2005).
  30. G. Sangiovanni, O. Gunnarsson, E. Koch, C. Castellani, and M. Capone, Electron-phonon interaction and antiferromagnetic correlations, Phys. Rev. Lett. 97, 046404 (2006).
  31. P. Paci, M. Capone, E. Cappelluti, S. Ciuchi, and C. Grimaldi, Isotope effects in the Hubbard-Holstein model within dynamical mean-field theory, Phys. Rev. B 74, 205108 (2006).
  32. P. Barone, R. Raimondi, M. Capone, and C. Castellani, Effective electron-phonon coupling and polaronic transition in the presence of strong correlation, Phys. Rev. B 73, 085120 (2006).
  33. G. Sangiovanni, M. Capone, and C. Castellani, Relevance of phonon dynamics in strongly correlated systems coupled to phonons: Dynamical mean-field theory analysis, Phys. Rev. B 73, 165123 (2006).
  34. G. Aprea, C. Di Castro, M. Grilli, and J. Lorenzana, Effective electron-electron and electron-phonon interactions in the Hubbard-Holstein model, Nucl. Phys. B 744, 277 (2006).
  35. P. Werner and A. J. Millis, Efficient dynamical mean field simulation of the Holstein-Hubbard model, Phys. Rev. Lett. 99, 146404 (2007).
  36. M. Tezuka, R. Arita, and H. Aoki, Phase diagram for the one-dimensional Hubbard-Holstein model: A density-matrix renormalization group study, Phys. Rev. B 76, 155114 (2007).
  37. P. Barone, R. Raimondi, M. Capone, C. Castellani, and M. Fabrizio, Gutzwiller scheme for electrons and phonons: The half-filled Hubbard-Holstein model, Phys. Rev. B 77, 235115 (2008).
  38. A. Di Ciolo, J. Lorenzana, M. Grilli, and G. Seibold, Charge instabilities and electron-phonon interaction in the Hubbard-Holstein model, Phys. Rev. B 79, 085101 (2009).
  39. J. Bauer and A. C. Hewson, Competition between antiferromagnetic and charge order in the Hubbard-Holstein model, Phys. Rev. B 81, 235113 (2010).
  40. A. Macridin, B. Moritz, M. Jarrell, and T. Maier, Suppression of superconductivity in the Hubbard model by buckling and breathing phonons, J. Phys.: Condens. Matter 24, 475603 (2012).
  41. E. A. Nowadnick, S. Johnston, B. Moritz, R. T. Scalettar, and T. P. Devereaux, Competition between antiferromagnetic and charge-density-wave order in the half-filled Hubbard-Holstein model, Phys. Rev. Lett. 109, 246404 (2012).
  42. M. Hohenadler and F. F. Assaad, Excitation spectra and spin gap of the half-filled Holstein-Hubbard model, Phys. Rev. B 87, 075149 (2013).
  43. S. Johnston, E. A. Nowadnick, Y. F. Kung, B. Moritz, R. T. Scalettar, and T. P. Devereaux, Determinant quantum Monte Carlo study of the two-dimensional single-band Hubbard-Holstein model, Phys. Rev. B 87, 235133 (2013).
  44. Y. Murakami, P. Werner, N. Tsuji, and H. Aoki, Ordered phases in the Holstein-Hubbard model: Interplay of strong Coulomb interaction and electron-phonon coupling, Phys. Rev. B 88, 125126 (2013).
  45. E. A. Nowadnick, S. Johnston, B. Moritz, and T. P. Devereaux, Renormalization of spectra by phase competition in the half-filled Hubbard-Holstein model, Phys. Rev. B 91, 165127 (2015).
  46. D. Wang, W.-S. Wang, and Q.-H. Wang, Phonon enhancement of electronic order and negative isotope effect in the Hubbard-Holstein model on a square lattice, Phys. Rev. B 92, 195102 (2015).
  47. T. Ohgoe and M. Imada, Competition among superconducting, antiferromagnetic, and charge orders with intervention by phase separation in the 2D Holstein-Hubbard model, Phys. Rev. Lett. 119, 197001 (2017).
  48. S. Karakuzu, L. F. Tocchio, S. Sorella, and F. Becca, Superconductivity, charge-density waves, antiferromagnetism, and phase separation in the Hubbard-Holstein model, Phys. Rev. B 96, 205145 (2017).
  49. M. Weber and M. Hohenadler, Two-dimensional Holstein-Hubbard model: Critical temperature, Ising universality, and bipolaron liquid, Phys. Rev. B 98, 085405 (2018).
  50. A. Ghosh, S. Kar, and S. Yarlagadda, Study of supersolidity in the two-dimensional Hubbard-Holstein model, Eur. Phys. J. B 91, 205 (2018).
  51. Z. Han, S. A. Kivelson, and H. Yao, Strong coupling limit of the Holstein-Hubbard model, Phys. Rev. Lett. 125, 167001 (2020).
  52. Y. Wang, I. Esterlis, T. Shi, J. I. Cirac, and E. Demler, Zero-temperature phases of the 2D Hubbard-Holstein model: A non-Gaussian exact diagonalization study, Phys. Rev. Res. 2, 043258 (2020).
  53. N. C. Costa, K. Seki, and S. Sorella, Magnetism and charge order in the honeycomb lattice, Phys. Rev. Lett. 126, 107205 (2021).
  54. N. C. Costa, K. Seki, S. Yunoki, and S. Sorella, Phase diagram of the two-dimensional Hubbard-Holstein model, Commun. Phys. 3, 80 (2020).
  55. G. Issa, O. Bradley, E. Khatami, and R. Scalettar, Learning by confusion: The phase diagram of the Holstein model, Phys. Rev. B 111, 155140 (2025).
  56. B. Cohen-Stead, S. M. Costa, J. Neuhaus, A. T. Ly, Y. Zhang, R. Scalettar, K. Barros, and S. Johnston, SmoQyDQMC.Jl: A flexible implementation of determinant quantum Monte Carlo for Hubbard and electron-phonon interactions, SciPost Phys. Codebases 29 (2024).
  57. P. Kopietz, L. Bartosch, and F. Schütz, Introduction to the Functional Renormalization Group, Lecture Notes in Physics (Springer, Berlin, 2010).
  58. 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).
  59. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  60. C. Platt, W. Hanke, and R. Thomale, Functional renormalization group for multi-orbital fermi surface instabilities, Adv. Phys. 62, 453 (2013).
  61. C. Honerkamp, D. M. Kennes, V. Meden, M. M. Scherer, and R. Thomale, Recent developments in the functional renormalization group approach to correlated electron systems, Eur. Phys. J. B 95, 205 (2022).
  62. J. B. Profe, D. M. Kennes, and L. Klebl, Diverge implements various exact renormalization group examples, SciPost Phys. Codebases 26 (2024).
  63. H. C. Fu, C. Honerkamp, and D.-H. Lee, Renormalization group study of the electron-phonon interaction in high-Tc cuprates, Europhys. Lett. 75, 146 (2006).
  64. L. Classen, M. M. Scherer, and C. Honerkamp, Instabilities on graphene's honeycomb lattice with electron-phonon interactions, Phys. Rev. B 90, 035122 (2014).
  65. Q.-G. Yang, D. Wang, and Q.-H. Wang, Functional renormalization group study of the two-dimensional Su-Schrieffer-Heeger-Hubbard model, Phys. Rev. B 106, 245136 (2022).
  66. N. Cichutek, M. Hansen, and P. Kopietz, Phonon renormalization and Pomeranchuk instability in the Holstein model, Phys. Rev. B 105, 205148 (2022).
  67. G. Sun, Q.-G. Yang, D. Wang, and Q.-H. Wang, Anomalous isotope effect in d-wave superconductors on the square lattice, Phys. Rev. B 109, L180508 (2024).
  68. T. Reckling and C. Honerkamp, Approximating the frequency dependence of the effective interaction in the functional renormalization group for many-fermion systems, Phys. Rev. B 98, 085114 (2018).
  69. N. K. Yirga, K.-M. Tam, and D. K. Campbell, Phonon-induced instabilities in correlated electron Hamiltonians, Phys. Rev. B 107, 235120 (2023).
  70. F. B. Kugler and J. von Delft, Multiloop functional renormalization group that sums up all parquet diagrams, Phys. Rev. Lett. 120, 057403 (2018).
  71. F. B. Kugler and J. von Delft, Multiloop functional renormalization group for general models, Phys. Rev. B 97, 035162 (2018).
  72. A. Tagliavini, C. Hille, F. Kugler, S. Andergassen, A. Toschi, and C. Honerkamp, Multiloop functional renormalization group for the two-dimensional Hubbard model: Loop convergence of the response functions, SciPost Phys. 6, 009 (2019).
  73. C. Hille, F. B. Kugler, C. J. Eckhardt, Y.-Y. He, A. Kauch, C. Honerkamp, A. Toschi, and S. Andergassen, Quantitative functional renormalization group description of the two-dimensional Hubbard model, Phys. Rev. Res. 2, 033372 (2020).
  74. F. Krien, A. Valli, and M. Capone, Single-boson exchange decomposition of the vertex function, Phys. Rev. B 100, 155149 (2019).
  75. K. Fraboulet, S. Heinzelmann, P. M. Bonetti, A. Al-Eryani, D. Vilardi, A. Toschi, and S. Andergassen, Single-boson exchange functional renormalization group application to the two-dimensional Hubbard model at weak coupling, Eur. Phys. J. B 95, 202 (2022).
  76. P. M. Bonetti, A. Toschi, C. Hille, S. Andergassen, and D. Vilardi, Single-boson exchange representation of the functional renormalization group for strongly interacting many-electron systems, Phys. Rev. Res. 4, 013034 (2022).
  77. A. Al-Eryani, S. Heinzelmann, K. Fraboulet, F. Krien, and S. Andergassen, Screening and effective rpa-like charge susceptibility in the extended Hubbard model, arXiv:2412.07323.
  78. A. A. Katanin, Fulfillment of Ward identities in the functional renormalization group approach, Phys. Rev. B 70, 115109 (2004).
  79. A. B. Migdal, Interaction between electrons and the lattice vibrations in a normal metal, Zhur. Eksp. Teor. Fiz. 34 996 (1958).
  80. G. M. Eliashberg, Interactions between electrons and lattice vibrations in a superconductor, Sov. Phys. - JETP 11, 696 (1960).
  81. F. Marsiglio, Eliashberg theory: A short review, Ann. Phys. 417, 168102 (2020).
  82. O. Gunnarsson, T. Schäfer, J. P. F. LeBlanc, E. Gull, J. Merino, G. Sangiovanni, G. Rohringer, and A. Toschi, Fluctuation diagnostics of the electron self-energy: Origin of the pseudogap physics, Phys. Rev. Lett. 114, 236402 (2015).
  83. T. Schäfer and A. Toschi, How to read between the lines of electronic spectra: The diagnostics of fluctuations in strongly correlated electron systems, J. Phys.: Condens. Matter 33, 214001 (2021).
  84. S. Heinzelmann, A. Toschi, and S. Andergassen, Entangled magnetic, charge, and superconducting pairing correlations in the two-dimensional Hubbard model: A functional renormalization-group analysis, arXiv:2308.06497.
  85. J. Hubbard, Electron correlations in narrow energy bands. III. An improved solution, Proc. R. Soc. Lond. Ser. A 281, 401 (1964).
  86. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  87. C. Husemann and M. Salmhofer, Efficient parametrization of the vertex function, Ω scheme, and the t,t′ Hubbard model at van Hove filling, Phys. Rev. B 79, 195125 (2009).
  88. D. Vilardi, C. Taranto, and W. Metzner, Nonseparable frequency dependence of the two-particle vertex in interacting fermion systems, Phys. Rev. B 96, 235110 (2017).
  89. G. Rohringer, A. Valli, and A. Toschi, Local electronic correlation at the two-particle level, Phys. Rev. B 86, 125114 (2012).
  90. N. Wentzell, G. Li, A. Tagliavini, C. Taranto, G. Rohringer, K. Held, A. Toschi, and S. Andergassen, High-frequency asymptotics of the vertex function: Diagrammatic parametrization and algorithmic implementation, Phys. Rev. B 102, 085106 (2020).
  91. W.-S. Wang, Y.-Y. Xiang, Q.-H. Wang, F. Wang, F. Yang, and D.-H. Lee, Functional renormalization group and variational Monte Carlo studies of the electronic instabilities in graphene near 14 doping, Phys. Rev. B 85, 035414 (2012).
  92. J. Lichtenstein, D. Sánchez de la Peña, D. Rohe, E. Di Napoli, C. Honerkamp, and S. Maier, High-performance functional renormalization group calculations for interacting fermions, Comput. Phys. Commun. 213, 100 (2017).
  93. S. Heinzelmann, The single-boson exchange formalism and its application to the functional renormalization group, Ph.D. thesis, Universität Tübingen, 2023.
  94. If we had considered reducibility with respect to V0 like in the original SBE formulation instead of BX, the object wX would additionally depend on fermionic arguments k and k′, which increases the numerical complexity and challenges its interpretation as a bosonic propagator.
  95. L. Hedin, New method for calculating the one-particle Green's function with application to the electron-gas problem, Phys. Rev. 139, A796 (1965).
  96. F. Krien, A. I. Lichtenstein, and G. Rohringer, Fluctuation diagnostic of the nodal/antinodal dichotomy in the Hubbard model at weak coupling: A parquet dual fermion approach, Phys. Rev. B 102, 235133 (2020).
  97. T. Denz, M. Mitter, J. M. Pawlowski, C. Wetterich, and M. Yamada, Partial bosonization for the two-dimensional Hubbard model, Phys. Rev. B 101, 155115 (2020).
  98. F. Krien, A. Valli, P. Chalupa, M. Capone, A. I. Lichtenstein, and A. Toschi, Boson-exchange parquet solver for dual fermions, Phys. Rev. B 102, 195131 (2020).
  99. F. Krien, P. Worm, P. Chalupa, A. Toschi, and K. Held, Explaining the pseudogap through damping and antidamping on the Fermi surface by imaginary spin scattering, Commun. Phys. 5, 336 (2022).
  100. F. Krien, A. Kauch, and K. Held, Tiling with triangles: Parquet and gwγ methods unified, Phys. Rev. Res. 3, 013149 (2021).
  101. F. Krien and A. Kauch, The plain and simple parquet approximation: Single-and multi-boson exchange in the two-dimensional Hubbard model, Eur. Phys. J. B 95, 69 (2022).
  102. M. Gievers, E. Walter, A. Ge, J. von Delft, and F. B. Kugler, Multiloop flow equations for single-boson exchange fRG, Eur. Phys. J. B 95, 108 (2022).
  103. S. Adler, F. Krien, P. Chalupa-Gantner, G. Sangiovanni, and A. Toschi, Non-perturbative intertwining between spin and charge correlations: A “smoking gun” single-boson-exchange result, SciPost Phys. 16, 054 (2024).
  104. D. Kiese, N. Wentzell, I. Krivenko, O. Parcollet, K. Held, and F. Krien, Embedded multi-boson exchange: A step beyond quantum cluster theories, Phys. Rev. Res. 6, 043159 (2024).
  105. C. Husemann and W. Metzner, Incommensurate nematic fluctuations in the two-dimensional Hubbard model, Phys. Rev. B 86, 085113 (2012).
  106. C. J. Eckhardt, C. Honerkamp, K. Held, and A. Kauch, Truncated unity parquet solver, Phys. Rev. B 101, 155104 (2020).
  107. S.-W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, Renormalization-group approach to strong-coupled superconductors, Phys. Rev. B 72, 054531 (2005).
  108. C. Honerkamp and M. Salmhofer, Eliashberg equations derived from the functional renormalization group, Prog. Theor. Phys. 113, 1145 (2005).
  109. The small quantitative differences with respect to the flowing results are due to the 1ℓ truncation of the fRG [73]. This inconsistency at the two-particle level can be resolved by including multiloop corrections [70, 71, 72].
  110. H. Eßl, M. Reitner, G. Sangiovanni, and A. Toschi, General Shiba mapping for on-site four-point correlation functions, Phys. Rev. Res. 6, 033061 (2024).
  111. I. Esterlis, S. A. Kivelson, and D. J. Scalapino, A bound on the superconducting transition temperature, npj Quantum Mater. 3, 59 (2018).
  112. M. V. Sadovskii, Very strong coupling limit of Eliashberg-Mcmillan theory and the upper limit for superconducting transition temperature, arXiv:2506.19326.
  113. C. Feng and R. T. Scalettar, Interplay of flat electronic bands with Holstein phonons, Phys. Rev. B 102, 235152 (2020).
  114. F. Marsiglio, Phonon self-energy effects in Migdal-Eliashberg theory, arXiv:2101.12084.
  115. E. Berger, P. Valáëk, and W. von der Linden, Two-dimensional Hubbard-Holstein model, Phys. Rev. B 52, 4806 (1995).
  116. H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction (Oxford University Press, Oxford, 2004).
  117. Z. Wang, C. Chen, J. Mo, J. Zhou, K. P. Loh, and Y. P. Feng, Decisive role of electron-phonon coupling for phonon and electron instabilities in transition metal dichalcogenides, Phys. Rev. Res. 5, 013218 (2023).
  118. S. Blackburn, M. Côté, S. G. Louie, and M. L. Cohen, Enhanced electron-phonon coupling near the lattice instability of superconducting NbC1−xNx from density-functional calculations, Phys. Rev. B 84, 104506 (2011).
  119. O. Rösch, G. Sangiovanni, and O. Gunnarsson, Sum rules and vertex corrections for electron-phonon interactions, Phys. Rev. B 75, 035119 (2007).
  120. U. F. P. Seifert, J. Willsher, M. Drescher, F. Pollmann, and J. Knolle, Spin-Peierls instability of the U(1) Dirac spin liquid, Nat. Commun. 15 (2024).
  121. This reversal would not occur if the ordering vector was q*=(0,0) implying that ferromagnetic fluctuations suppress d-wave superconductivity and uniform charge fluctuations encourage it.
  122. This approximation applies as long as the analytically continued Σ(z=iν) does not present any features below the first Matsubara frequency. Otherwise, Gcoh has to be determined by an analytic continuation.
  123. H. Shinaoka, J. Otsuki, M. Ohzeki, and K. Yoshimi, Compressing Green's function using intermediate representation between imaginary-time and real-frequency domains, Phys. Rev. B 96, 035147 (2017).
  124. N. Chikano, J. Otsuki, and H. Shinaoka, Performance analysis of a physically constructed orthogonal representation of imaginary-time Green's function, Phys. Rev. B 98, 035104 (2018).
  125. J. Kaye, K. Chen, and O. Parcollet, Discrete Lehmann representation of imaginary time Green's functions, Phys. Rev. B 105, 235115 (2022).
  126. J. Kaye, K. Chen, and H. U. Strand, Libdlr: Efficient imaginary time calculations using the discrete Lehmann representation, Comput. Phys. Commun. 280, 108458 (2022).
  127. H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, and A. Kauch, Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains, Phys. Rev. X 13, 021015 (2023).
  128. S. Rohshap, M. K. Ritter, H. Shinaoka, J. von Delft, M. Wallerberger, and A. Kauch, Two-particle calculations with quantics tensor trains—Solving the parquet equations, Phys. Rev. Res. 7, 023087 (2025).
  129. W. Metzner and D. Vollhardt, Correlated lattice fermions in d=∞ dimensions, Phys. Rev. Lett. 62, 324 (1989).
  130. 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).
  131. C. Taranto, S. Andergassen, J. Bauer, K. Held, A. Katanin, W. Metzner, G. Rohringer, and A. Toschi, From infinite to two dimensions through the functional renormalization group, Phys. Rev. Lett. 112, 196402 (2014).
  132. D. Vilardi, C. Taranto, and W. Metzner, Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group, Phys. Rev. B 99, 104501 (2019).
  133. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  134. A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, Solitons in conducting polymers, Rev. Mod. Phys. 60, 781 (1988).
  135. A. Al-Eryani, S. Andergassen, and M. Scherer, Numerical results for “Intertwined fluctuations and isotope effects in the Hubbard-Holstein model on the square lattice from functional renormalization”, TU Wien (2025).
  136. P. Morel and P. W. Anderson, Calculation of the superconducting state parameters with retarded electron-phonon interaction, Phys. Rev. 125, 1263 (1962).
  137. C. Hille, D. Rohe, C. Honerkamp, and S. Andergassen, Pseudogap opening in the two-dimensional Hubbard model: A functional renormalization group analysis, Phys. Rev. Res. 2, 033068 (2020).
  138. In contrast to the SBEb approximation that neglects all multiboson diagrams, the SBEa neglects only the flow of MX and is less severe; multiboson insertions into the single-boson part of the vertex can still occur [75].

Outline

Information

Phys. Rev. Research 7, 043052– Published 15 October, 2025
  • Received 22 April 2025
  • Accepted 26 August 2025
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