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Superspin renormalization and slow relaxation in random spin systems

Yi J. Zhao1,*, Samuel J. Garratt1, and Joel E. Moore1,2

  • *Contact author: yi_zhao@berkeley.edu
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Phys. Rev. B 112, 054436 – Published 14 August, 2025

DOI: https://doi.org/10.1103/ppkd-dff1

Abstract

We develop an excited-state real-space renormalization group (RSRG-X) formalism to describe the dynamics of conserved densities in randomly interacting spin-12 systems. Our formalism is suitable for systems with U(1) and Z2 symmetries, and we apply it to chains of randomly positioned spins with dipolar XX+YY interactions, as arise in Rydberg quantum simulators and other platforms. The formalism generates a sequence of effective Hamiltonians that provide approximate descriptions for dynamics on successively smaller energy scales. These effective Hamiltonians involve “superspins”: two-level collective degrees of freedom constructed from (anti)aligned microscopic spins. Conserved densities can then be understood as relaxing via coherent collective spin flips. For the well-studied simpler case of randomly interacting nearest-neighbor XX+YY chains, the superspins reduce to single spins. Our formalism also leads to a numerical method capable of simulating the dynamics up to an otherwise inaccessible combination of large system size and late time. Focusing on disorder-averaged infinite-temperature autocorrelation functions, in particular the spin survival probability Sp¯(t), we demonstrate quantitative agreement between our algorithm and exact diagonalization (ED) at low but nonzero frequencies. Such agreement holds for chains with nearest-neighbor, next-nearest-neighbor, and long-range dipolar interactions. Our results indicate decay of Sp¯(t) slower than any power law and feature no significant deviation from the ∼1/ln2(t) asymptote expected from the infinite-randomness fixed-point of the nearest-neighbor model. We also apply the RSRG-X formalism to two-dimensional long-range systems of moderate size and find slow late-time decay of Sp¯(t).

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References (62)

  1. J. Choi, S. Choi, G. Kucsko, P. C. Maurer, B. J. Shields, H. Sumiya, S. Onoda, J. Isoya, E. Demler, F. Jelezko, N. Y. Yao, and M. D. Lukin, Depolarization dynamics in a strongly interacting solid-state spin ensemble, Phys. Rev. Lett. 118, 093601 (2017).
  2. G. Kucsko, S. Choi, J. Choi, P. C. Maurer, H. Zhou, R. Landig, H. Sumiya, S. Onoda, J. Isoya, F. Jelezko, E. Demler, N. Y. Yao, and M. D. Lukin, Critical thermalization of a disordered dipolar spin system in diamond, Phys. Rev. Lett. 121, 023601 (2018).
  3. C. Zu, F. Machado, B. Ye, S. Choi, B. Kobrin, T. Mittiga, S. Hsieh, P. Bhattacharyya, M. Markham, D. Twitchen et al., Emergent hydrodynamics in a strongly interacting dipolar spin ensemble, Nature (London) 597, 45 (2021).
  4. E. J. Davis, B. Ye, F. Machado, S. A. Meynell, W. Wu, T. Mittiga, W. Schenken, M. Joos, B. Kobrin, Y. Lyu et al., Probing many-body dynamics in a two-dimensional dipolar spin ensemble, Nat. Phys. 19, 836 (2023).
  5. A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  6. L. Su, A. Douglas, M. Szurek, R. Groth, S. F. Ozturk, A. Krahn, A. H. Hébert, G. A. Phelps, S. Ebadi, S. Dickerson et al., Dipolar quantum solids emerging in a Hubbard quantum simulator, Nature (London) 622, 724 (2023).
  7. D. Bloch, B. Hofer, S. R. Cohen, A. Browaeys, and I. Ferrier-Barbut, Trapping and imaging single dysprosium atoms in optical tweezer arrays, Phys. Rev. Lett. 131, 203401 (2023).
  8. D. S. Grün, S. J. M. White, A. Ortu, A. Di Carli, H. Edri, M. Lepers, M. J. Mark, and F. Ferlaino, Optical tweezer arrays of erbium atoms, Phys. Rev. Lett. 133, 223402 (2024).
  9. S. L. Cornish, M. R. Tarbutt, and K. R. Hazzard, Quantum computation and quantum simulation with ultracold molecules, Nat. Phys. 20, 730 (2024).
  10. P. Peng, B. Ye, N. Y. Yao, and P. Cappellaro, Exploiting disorder to probe spin and energy hydrodynamics, Nat. Phys. 19, 1027 (2023).
  11. T. Franz, S. Geier, C. Hainaut, A. Braemer, N. Thaicharoen, M. Hornung, E. Braun, M. Gärttner, G. Zürn, and M. Weidemüller, Observation of anisotropy-independent magnetization dynamics in spatially disordered Heisenberg spin systems, Phys. Rev. Res. 6, 033131 (2024).
  12. D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams, Coherent excitation transfer in a spin chain of three Rydberg atoms, Phys. Rev. Lett. 114, 113002 (2015).
  13. F. Iglói and C. Monthus, Strong disorder RG approach of random systems, Phys. Rep. 412, 277 (2005).
  14. G. Refael and E. Altman, Strong disorder renormalization group primer and the superfluid–insulator transition, C. R. Phys. 14, 725 (2013).
  15. F. Iglói and C. Monthus, Strong disorder RG approach—A short review of recent developments, Eur. Phys. J. B 91, 290 (2018).
  16. D. A. Huse, Strong-randomness renormalization groups, in 50 Years of the Renormalization Group: Dedicated to the Memory of Michael E. Fisher, edited by A. Aharony, O. Entin-Wohlman, D. A. Huse, and L. Radzihovsky (World Scientific, Singapore, 2024), pp. 363–372.
  17. S.-K. Ma, C. Dasgupta, and C.-K. Hu, Random antiferromagnetic chain, Phys. Rev. Lett. 43, 1434 (1979).
  18. C. Dasgupta and S.-K. Ma, Low-temperature properties of the random Heisenberg antiferromagnetic chain, Phys. Rev. B 22, 1305 (1980).
  19. R. N. Bhatt and P. A. Lee, Scaling studies of highly disordered spin-12 antiferromagnetic systems, Phys. Rev. Lett. 48, 344 (1982).
  20. D. S. Fisher, Random transverse field Ising spin chains, Phys. Rev. Lett. 69, 534 (1992).
  21. D. S. Fisher, Random antiferromagnetic quantum spin chains, Phys. Rev. B 50, 3799 (1994).
  22. E. Westerberg, A. Furusaki, M. Sigrist, and P. A. Lee, Random quantum spin chains: A real-space renormalization group study, Phys. Rev. Lett. 75, 4302 (1995).
  23. D. S. Fisher, Critical behavior of random transverse-field Ising spin chains, Phys. Rev. B 51, 6411 (1995).
  24. E. Westerberg, A. Furusaki, M. Sigrist, and P. A. Lee, Low-energy fixed points of random quantum spin chains, Phys. Rev. B 55, 12578 (1997).
  25. K. Damle, O. Motrunich, and D. A. Huse, Dynamics and transport in random antiferromagnetic spin chains, Phys. Rev. Lett. 84, 3434 (2000).
  26. O. Motrunich, K. Damle, and D. A. Huse, Dynamics and transport in random quantum systems governed by strong-randomness fixed points, Phys. Rev. B 63, 134424 (2001).
  27. G. Refael, S. Kehrein, and D. S. Fisher, Spin reduction transition in spin-32 random Heisenberg chains, Phys. Rev. B 66, 060402(R) (2002).
  28. K. Damle and D. A. Huse, Permutation-symmetric multicritical points in random antiferromagnetic spin chains, Phys. Rev. Lett. 89, 277203 (2002).
  29. G. Refael and J. E. Moore, Entanglement entropy of random quantum critical points in one dimension, Phys. Rev. Lett. 93, 260602 (2004).
  30. Y. Mohdeb, J. Vahedi, N. Moure, A. Roshani, H.-Y. Lee, R. N. Bhatt, S. Kettemann, and S. Haas, Entanglement properties of disordered quantum spin chains with long-range antiferromagnetic interactions, Phys. Rev. B 102, 214201 (2020).
  31. B. Roberts and O. I. Motrunich, Infinite randomness with continuously varying critical exponents in the random XYZ spin chain, Phys. Rev. B 104, 214208 (2021).
  32. L. F. C. Faria, V. L. Quito, J. C. Getelina, J. A. Hoyos, and E. Miranda, Signatures of infinite randomness in transport properties of disordered spin chains, arXiv:2312.07474.
  33. R. Vosk and E. Altman, Many-body localization in one dimension as a dynamical renormalization group fixed point, Phys. Rev. Lett. 110, 067204 (2013).
  34. D. Pekker, G. Refael, E. Altman, E. Demler, and V. Oganesyan, Hilbert-glass transition: New universality of temperature-tuned many-body dynamical quantum criticality, Phys. Rev. X 4, 011052 (2014).
  35. R. Vosk and E. Altman, Dynamical quantum phase transitions in random spin chains, Phys. Rev. Lett. 112, 217204 (2014).
  36. Y. Huang and J. E. Moore, Excited-state entanglement and thermal mutual information in random spin chains, Phys. Rev. B 90, 220202(R) (2014).
  37. R. Vasseur, A. C. Potter, and S. A. Parameswaran, Quantum criticality of hot random spin chains, Phys. Rev. Lett. 114, 217201 (2015).
  38. Y.-Z. You, X.-L. Qi, and C. Xu, Entanglement holographic mapping of many-body localized system by spectrum bifurcation renormalization group, Phys. Rev. B 93, 104205 (2016).
  39. R. Vasseur, A. J. Friedman, S. A. Parameswaran, and A. C. Potter, Particle-hole symmetry, many-body localization, and topological edge modes, Phys. Rev. B 93, 134207 (2016).
  40. C. Monthus, Strong disorder real-space renormalization for the many-body-localized phase of random Majorana models, J. Phys. A: Math. Theor. 51, 115304 (2018).
  41. I. V. Protopopov, R. K. Panda, T. Parolini, A. Scardicchio, E. Demler, and D. A. Abanin, Non-Abelian symmetries and disorder: A broad nonergodic regime and anomalous thermalization, Phys. Rev. X 10, 011025 (2020).
  42. Y. Mohdeb, J. Vahedi, and S. Kettemann, Excited-eigenstate entanglement properties of XX spin chains with random long-range interactions, Phys. Rev. B 106, 104201 (2022).
  43. A. Braemer, T. Franz, M. Weidemüller, and M. Gärttner, Pair localization in dipolar systems with tunable positional disorder, Phys. Rev. B 106, 134212 (2022).
  44. Y. Mohdeb, J. Vahedi, R. N. Bhatt, S. Haas, and S. Kettemann, Global quench dynamics and the growth of entanglement entropy in disordered spin chains with tunable range interactions, Phys. Rev. B 108, L140203 (2023).
  45. A. Braemer, J. Vahedi, and M. Gärttner, Cluster truncated Wigner approximation for bond-disordered Heisenberg spin models, Phys. Rev. B 110, 054204 (2024).
  46. A. S. Aramthottil, P. Sierant, M. Lewenstein, and J. Zakrzewski, Phenomenology of many-body localization in bond-disordered spin chains, Phys. Rev. Lett. 133, 196302 (2024).
  47. S. Gopalakrishnan, M. Müller, V. Khemani, M. Knap, E. Demler, and D. A. Huse, Low-frequency conductivity in many-body localized systems, Phys. Rev. B 92, 104202 (2015).
  48. P. J. D. Crowley and A. Chandran, A constructive theory of the numerically accessible many-body localized to thermal crossover, SciPost Phys. 12, 201 (2022).
  49. S. J. Garratt, S. Roy, and J. T. Chalker, Local resonances and parametric level dynamics in the many-body localized phase, Phys. Rev. B 104, 184203 (2021).
  50. S. J. Garratt and S. Roy, Resonant energy scales and local observables in the many-body localized phase, Phys. Rev. B 106, 054309 (2022).
  51. D. M. Long, P. J. D. Crowley, V. Khemani, and A. Chandran, Phenomenology of the prethermal many-body localized regime, Phys. Rev. Lett. 131, 106301 (2023).
  52. F. Iglói, R. Juhász, and H. Rieger, Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions, Phys. Rev. B 61, 11552 (2000).
  53. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, 2011).
  54. One can also formulate the RSRG-X and derive the same update rule by directly working with the single particle Hamiltonian in the fermion language.
  55. O. Motrunich, S.-C. Mau, D. A. Huse, and D. S. Fisher, Infinite-randomness quantum Ising critical fixed points, Phys. Rev. B 61, 1160 (2000).
  56. C. R. Laumann, D. A. Huse, A. W. W. Ludwig, G. Refael, S. Trebst, and M. Troyer, Strong-disorder renormalization for interacting non-Abelian anyon systems in two dimensions, Phys. Rev. B 85, 224201 (2012).
  57. C. L. Baldwin, A. Ehrenberg, A. Y. Guo, and A. V. Gorshkov, Disordered Lieb-Robinson bounds in one dimension, PRX Quantum 4, 020349 (2023).
  58. C. L. Baldwin, Sub-ballistic operator growth in spin chains with heavy-tailed random fields, Phys. Rev. B 111, 184204 (2025).
  59. W. D. Roeck, L. Giacomin, F. Huveneers, and O. Prosniak, Absence of normal heat conduction in strongly disordered interacting quantum chains, arXiv:2408.04338.
  60. C.-F. Chen and A. Lucas, Finite speed of quantum scrambling with long range interactions, Phys. Rev. Lett. 123, 250605 (2019).
  61. M. C. Tran, A. Y. Guo, C. L. Baldwin, A. Ehrenberg, A. V. Gorshkov, and A. Lucas, Lieb-Robinson light cone for power-law interactions, Phys. Rev. Lett. 127, 160401 (2021).
  62. F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A: Math. Gen. 15, 3241 (1982).

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Information

Phys. Rev. B 112, 054436– Published 14 August, 2025
  • Received 21 February 2025
  • Revised 1 July 2025
  • Accepted 2 July 2025
Article part of CHORUS
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