- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Spectral Decomposition and High-Accuracy Green’s Functions: Overcoming the Nyquist-Shannon Limit via Complex-time Krylov Expansion

S. Paeckel

PDF

Phys. Rev. Lett. 136, 160401 – Published 21 April, 2026

DOI: https://doi.org/10.1103/bx76-hps4

Abstract

The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green’s functions, is a long-standing problem posing enormous challenges to numerical methods. When the spectral decomposition is obtained from Fourier transforming a time series, the Nyquist-Shannon theorem limits the frequency resolution Δω according to the numerically accessible time domain size T via Δω=2π/T. In tensor network methods, increasing the domain size is exponentially hard due to the ubiquitous spread of correlations, limiting the frequency resolution and thereby restricting this ansatz class mostly to one-dimensional systems with small quasiparticle velocities. Here, we show how this limitation can be overcome by augmenting the time series with complex-time Krylov states. With the example of the critical S−1/2 Heisenberg model and light bipolarons in the two-dimensional Su-Schrieffer-Heeger model, we demonstrate the enormous improvements in accuracy, which can be achieved using this method.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (81)

  1. H. Benthien, F. Gebhard, and E. Jeckelmann, Spectral function of the one-dimensional hubbard model away from half filling, Phys. Rev. Lett. 92, 256401 (2004).
  2. S. R. White and I. Affleck, Spectral function for the s=1 Heisenberg antiferromagetic chain, Phys. Rev. B 77, 134437 (2008).
  3. R. G. Pereira, S. R. White, and I. Affleck, Spectral function of spinless fermions on a one-dimensional lattice, Phys. Rev. B 79, 165113 (2009).
  4. M. Mourigal, M. Enderle, A. Klöpperpieper, J.-S. Caux, A. Stunault, and H. M. Rønnow, Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain, Nat. Phys. 9, 435 (2013).
  5. M. Weber, F. F. Assaad, and M. Hohenadler, Phonon spectral function of the one-dimensional Holstein-Hubbard model, Phys. Rev. B 91, 235150 (2015).
  6. M. Gohlke, R. Verresen, R. Moessner, and F. Pollmann, Dynamics of the Kitaev-Heisenberg model, Phys. Rev. Lett. 119, 157203 (2017).
  7. A. Keselman, C. Murthy, B. van Heck, and B. Bauer, Spectral response of Josephson junctions with low-energy quasiparticles, SciPost Phys. 7, 050 (2019).
  8. F. Grusdt, E. Demler, and A. Bohrdt, Pairing of holes by confining strings in antiferromagnets, SciPost Phys. 14, 090 (2023).
  9. M. c. v. Kebrič, J. C. Halimeh, U. Schollwöck, and F. Grusdt, Confinement in (1+1)-dimensional z2 lattice gauge theories at finite temperature, Phys. Rev. B 109, 245110 (2024).
  10. H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Renormalization-group approach to the anderson model of dilute magnetic alloys (part i+part ii), Phys. Rev. B 21, 1003 (1980).
  11. 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).
  12. R. Bulla, A. C. Hewson, and T. Pruschke, Numerical renormalization group calculations for the self-energy of the impurity Anderson model, J. Phys. Condens. Matter 10, 8365 (1998).
  13. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Electronic structure calculations with dynamical mean-field theory, Rev. Mod. Phys. 78, 865 (2006).
  14. M. Potthoff, M. Aichhorn, and C. Dahnken, Variational cluster approach to correlated electron systems in low dimensions, Phys. Rev. Lett. 91, 206402 (2003).
  15. M. Potthoff, Self-energy-functional approach to systems of correlated electrons, Eur. Phys. J. B 32, 429 (2003).
  16. M. Potthoff, Self-energy-functional approach: Analytical results and the Mott-Hubbard transition, Eur. Phys. J. B 36, 335 (2003).
  17. C. Dahnken, M. Aichhorn, W. Hanke, E. Arrigoni, and M. Potthoff, Variational cluster approach to spontaneous symmetry breaking: The itinerant antiferromagnet in two dimensions, Phys. Rev. B 70, 245110 (2004).
  18. P. Werner and A. J. Millis, Efficient dynamical mean field simulation of the Holstein-Hubbard model, Phys. Rev. Lett. 99, 146404 (2007).
  19. T. E. Reinhard, U. Mordovina, C. Hubig, J. S. Kretchmer, U. Schollwöck, H. Appel, M. A. Sentef, and A. Rubio, Density-matrix embedding theory study of the one-dimensional hubbard–holstein model, J. Chem. Theory Comput. 15, 2221 (2019).
  20. G. Bollmark, T. Köhler, L. Pizzino, Y. Yang, J. S. Hofmann, H. Shi, S. Zhang, T. Giamarchi, and A. Kantian, Solving 2d and 3d lattice models of correlated fermions—Combining matrix product states with mean-field theory, Phys. Rev. X 13, 011039 (2023).
  21. B. Lenz, C. Martins, and S. Biermann, Spectral functions of Sr2IrO4: Theory versus experiment, J. Phys. Condens. Matter 31, 293001 (2019).
  22. J. Karp, A. Hampel, and A. J. Millis, Superconductivity and antiferromagnetism in NdNiO2 and CaCuO2: A cluster DMFT study, Phys. Rev. B 105, 205131 (2022).
  23. M. Bramberger, J. Mravlje, M. Grundner, U. Schollwöck, and M. Zingl, BaOSo3: A Hund’s metal in the presence of strong spin-orbit coupling, Phys. Rev. B 103, 165133 (2021).
  24. M. Bramberger, B. Bacq-Labreuil, M. Grundner, S. Biermann, U. Schollwöck, S. Paeckel, and B. Lenz, Formation of CuO2 sublattices by suppression of interlattice correlations in tetragonal CuO, SciPost Phys. 14, 010 (2023).
  25. S. Paeckel, T. Köhler, S. R. Manmana, and B. Lenz, Matrix-product-state-based Band-Lanczos solver for quantum cluster approaches, Phys. Rev. B 112, 235108 (2025).
  26. M. Grundner, F. B. Kugler, O. Parcollet, U. Schollwöck, A. Georges, and A. Hampel, LiV2O4: Hund-assisted orbital-selective mottness, Phys. Rev. B 112, L041106 (2025).
  27. E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, Cambridge, England, 2013).
  28. F. Aryasetiawan and O. Gunnarsson, The GW method, Rep. Prog. Phys. 61, 237 (1998).
  29. G. Onida, L. Reining, and A. Rubio, Electronic excitations: Density-functional versus many-body Green’s-function approaches, Rev. Mod. Phys. 74, 601 (2002).
  30. J. Reuther and R. Thomale, Functional renormalization group for the anisotropic triangular antiferromagnet, Phys. Rev. B 83, 024402 (2011).
  31. 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).
  32. 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).
  33. T. D. Kühner and S. R. White, Dynamical correlation functions using the density matrix renormalization group, Phys. Rev. B 60, 335 (1999).
  34. R. Bulla, The numerical renormalization group method for correlated electrons, in Advances in Solid State Physics 40, edited by B. Kramer (Springer, Berlin, Heidelberg, 2000), pp. 169–182.
  35. M. Aichhorn, M. Daghofer, H. G. Evertz, and W. von der Linden, Low-temperature Lanczos method for strongly correlated systems, Phys. Rev. B 67, 161103(R) (2003).
  36. A. Weichselbaum and J. von Delft, Sum-rule conserving spectral functions from the numerical renormalization group, Phys. Rev. Lett. 99, 076402 (2007).
  37. R. Bulla, T. A. Costi, and T. Pruschke, Numerical renormalization group method for quantum impurity systems, Rev. Mod. Phys. 80, 395 (2008).
  38. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  39. U. Schollwoeck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (Amsterdam) 326, 96 (2010).
  40. N. V. Prokof’ev and B. V. Svistunov, Polaron problem by diagrammatic quantum monte carlo, Phys. Rev. Lett. 81, 2514 (1998).
  41. K. Van Houcke, E. Kozik, N. Prokof’ev, and B. Svistunov, Diagrammatic monte carlo, Phys. Procedia 6, 95 (2010).
  42. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  43. J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nat. Phys. 11, 124 (2015).
  44. E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
  45. H. Nyquist, Certain topics in telegraph transmission theory, Trans. Am. Inst. Electr. Eng. 47, 617 (1928).
  46. C. E. Shannon, Communication in the presence of noise, Proc. IRE 37, 10 (1949).
  47. E. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory 51, 4203 (2005).
  48. E. J. Candès, J. K. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math. 59, 1207 (2006).
  49. C. Lánczos, An iterative method for the solution of the eigenvalue problem of linear differential and integral, J. Res. Natl. Bur. Stand. 45, 225 (1950).
  50. J. K. Cullum and R. A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1 (Progress in Scientific Computing, Basel, Switzerland, 1985).
  51. R. W. Freund, Band lanczos method, in Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, edited by Z. Bai, J. D. Demmel, A. Ruhe, J. Dongarra, and H. van der Vorst (SIAM, Philadelphia, 2000), Chap. 4.6, pp. 80–88.
  52. M. Grundner, P. Westhoff, F. B. Kugler, O. Parcollet, and U. Schollwöck, Complex time evolution in tensor networks and time-dependent Green’s functions, Phys. Rev. B 109, 155124 (2024).
  53. X. Cao, Y. Lu, E. M. Stoudenmire, and O. Parcollet, Dynamical correlation functions from complex time evolution, Phys. Rev. B 109, 235110 (2024).
  54. S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. (Amsterdam) 411, 167998 (2019).
  55. See Supplemental Material at http://link.aps.org/supplemental/10.1103/bx76-hps4 for a detailed error analysis, which includes Refs. [56–59].
  56. M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 34, 1911 (1997).
  57. M. Hochbruck and C. Lubich, Exponential integrators for quantum-classical molecular dynamics, BIT 39, 620 (1999).
  58. E. M. Stoudenmire and S. R. White, Real-space parallel density matrix renormalization group, Phys. Rev. B 87, 155137 (2013).
  59. Y. Shen, K. Klymko, J. Sud, D. B. Williams-Young, W. A. d. Jong, and N. M. Tubman, Real-time Krylov theory for quantum computing algorithms, Quantum 7, 1066 (2023).

  60. Bringing Heff into the upper Hessenberg form directly allows for reading of the matrix element hr,r−1.

  61. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  62. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  63. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (Amsterdam) 326, 96 (2011).
  64. S. Barišić, Rigid-atom electron-phonon coupling in the tight-binding approximation. I, Phys. Rev. B 5, 932 (1972).
  65. S. Barišić, Self-consistent electron-phonon coupling in the tight-binding approximation. II, Phys. Rev. B 5, 941 (1972).
  66. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  67. J. Sous, M. Chakraborty, R. V. Krems, and M. Berciu, Light bipolarons stabilized by Peierls electron-phonon coupling, Phys. Rev. Lett. 121, 247001 (2018).
  68. C. Zhang, J. Sous, D. R. Reichman, M. Berciu, A. J. Millis, N. V. Prokof’ev, and B. V. Svistunov, Bipolaronic high-temperature superconductivity, Phys. Rev. X 13, 011010 (2023).
  69. M. Grundner, T. Blatz, J. Sous, U. Schollwöck, and S. Paeckel, Cooper-paired bipolaronic superconductors, arXiv:2308.13427.
  70. K.-S. Kim, Z. Han, and J. Sous, Semiclassical theory of bipolaronic superconductivity in a bond-modulated electron-phonon model, Phys. Rev. B 109, L220502 (2024).
  71. M. Berciu, Few-particle Green’s functions for strongly correlated systems on infinite lattices, Phys. Rev. Lett. 107, 246403 (2011).
  72. T. Köhler, J. Stolpp, and S. Paeckel, Efficient and flexible approach to simulate low-dimensional quantum lattice models with large local Hilbert spaces, SciPost Phys. 10, 058 (2021).
  73. J. Stolpp, T. Köhler, S. R. Manmana, E. Jeckelmann, F. Heidrich-Meisner, and S. Paeckel, Comparative study of state-of-the-art matrix-product-state methods for lattice models with large local Hilbert spaces without U(1) symmetry, Comput. Phys. Commun. 269, 108106 (2021).
  74. J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011).
  75. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116 (2016).
  76. M. Yang and S. R. White, Time-dependent variational principle with ancillary Krylov subspace, Phys. Rev. B 102, 094315 (2020).

  77. In fact, the maximum time T=8 could be reached only for a few initial states due to the rapid growth of memory requirements.

  78. Because of the finite cluster size there are strong finite size effects. We checked this by computing the ground state of the non interacting model yielding a ground -state energy, which is always larger than the one obtained when incorporating finite λ.

  79. C. Hubig, F. Lachenmaier, N.-O. Linden, T. Reinhard, L. Stenzel, A. Swoboda, M. Grundner, S. Mardazad, and S. Paeckel, The syten toolkit, https://syten.eu/.
  80. C. Hubig, Symmetry-protected tensor networks, Ph.D. thesis, LMU München, 2017).
  81. S. Paeckel and T. Köhler, symmps, https://www.symmps.eu, accessed: 2019-12-29.

Outline

Information

Phys. Rev. Lett. 136, 160401– Published 21 April, 2026
  • Received 18 February 2025
  • Revised 14 December 2025
  • Accepted 16 March 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