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Superdiffusion and anomalous fluctuations in chiral integrable dynamics

Cristiano Muzzi1,2,*,†, Devendra Singh Bhakuni3,*,‡, Marcello Dalmonte3,§, Lenart Zadnik4,∥, and Hernan B. Xavier1,3,¶

  • *These authors contributed equally to this work.
  • †Contact author: cmuzzi@sissa.it
  • ‡Contact author: dbhakuni@ictp.it
  • §Contact author: mdalmont@ictp.it
  • ∥Contact author: lenart.zadnik@fmf.uni-lj.si
  • Contact author: hxavier@ictp.it
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Phys. Rev. B 113, 184305 – Published 6 May, 2026

DOI: https://doi.org/10.1103/q549-whw6

Abstract

Symmetries strongly influence transport properties of quantum many-body systems and can lead to deviations from the generic case of diffusion. In this work, we study the impact of time-reversal symmetry breaking on the transport and its universal aspects in integrable chiral spin ladders. We observe that the infinite-temperature spin transport is superdiffusive with a dynamical critical exponent z=3/2 matching the one of the Kardar-Parisi-Zhang (KPZ) universality class, which also lacks the time reversal symmetry. However, we find that fluctuations of the net magnetization transfer deviate from the KPZ predictions. Moreover, the full probability distribution of the associated spin current obeys fluctuation symmetry despite broken time-reversal and space-reflection symmetries. To further investigate the role of conserved quantities, we introduce an integrable quantum circuit that shares the essential symmetries with the chiral ladder and which exhibits analogous dynamical behavior in the absence of energy conservation. Our work shows that time-reversal symmetry breaking is compatible with superdiffusion but insufficient to stabilize the KPZ universality in integrable systems. This suggests that additional fundamental features are missing in order to identify the emergence of such dynamics in quantum matter.

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References (81)

  1. P. Romatschke and U. Romatschke, Viscosity information from relativistic nuclear collisions: How perfect is the fluid observed at RHIC? Phys. Rev. Lett. 99, 172301 (2007).
  2. D. A. Teaney, Viscous hydrodynamics and the quark gluon plasma, in Quark-Gluon Plasma 4 (World Scientific, 2010), pp. 207–266.
  3. R. Snellings, Elliptic flow: A brief review, New J. Phys. 13, 055008 (2011).
  4. W. Busza, K. Rajagopal, and W. van der Schee, Heavy ion collisions: The big picture and the big questions, Annu. Rev. Nucl. Part. Sci. 68, 339 (2018).
  5. U. Schneider, L. Hackermüller, J. P. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch, and A. Rosch, Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, Nat. Phys. 8, 213 (2012).
  6. S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas, Nat. Phys. 8, 325 (2012).
  7. S. Hild, T. Fukuhara, P. Schauß, J. Zeiher, M. Knap, E. Demler, I. Bloch, and C. Gross, Far-from-equilibrium spin transport in Heisenberg quantum magnets, Phys. Rev. Lett. 113, 147205 (2014).
  8. I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  9. R. C. Brown, R. Wyllie, S. B. Koller, E. A. Goldschmidt, M. Foss-Feig, and J. V. Porto, Two-dimensional superexchange-mediated magnetization dynamics in an optical lattice, Science 348, 540 (2015).
  10. C.-C. Chien, S. Peotta, and M. Di Ventra, Quantum transport in ultracold atoms, Nat. Phys. 11, 998 (2015).
  11. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  12. M. A. Nichols, L. W. Cheuk, M. Okan, T. R. Hartke, E. Mendez, T. Senthil, E. Khatami, H. Zhang, and M. W. Zwierlein, Spin transport in a Mott insulator of ultracold fermions, Science 363, 383 (2019).
  13. P. T. Brown, D. Mitra, E. Guardado-Sanchez, R. Nourafkan, A. Reymbaut, C.-D. Hébert, S. Bergeron, A.-M. S. Tremblay, J. Kokalj, D. A. Huse, P. Schauß, and W. S. Bakr, Bad metallic transport in a cold atom Fermi-Hubbard system, Science 363, 379 (2019).
  14. M. Žnidarič, Spin transport in a one-dimensional anisotropic Heisenberg model, Phys. Rev. Lett. 106, 220601 (2011).
  15. M. Ljubotina, M. Žnidarič, and T. Prosen, Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet, Phys. Rev. Lett. 122, 210602 (2019).
  16. M. Dupont and J. E. Moore, Universal spin dynamics in infinite-temperature one-dimensional quantum magnets, Phys. Rev. B 101, 121106(R) (2020).
  17. Y. Bar Lev, G. Cohen, and D. R. Reichman, Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice, Phys. Rev. Lett. 114, 100601 (2015).
  18. K. Agarwal, S. Gopalakrishnan, M. Knap, M. Müller, and E. Demler, Anomalous diffusion and Griffiths effects near the many-body localization transition, Phys. Rev. Lett. 114, 160401 (2015).
  19. Y. Bar Lev, D. M. Kennes, C. Klöckner, D. R. Reichman, and C. Karrasch, Transport in quasiperiodic interacting systems: From superdiffusion to subdiffusion, Europhys. Lett. 119, 37003 (2017).
  20. M. Žnidarič, Comment on “Nonequilibrium steady state phases of the interacting Aubry-André-Harper model,” Phys. Rev. B 103, 237101 (2021).
  21. J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, and M. Knap, Anomalous diffusion in dipole- and higher-moment-conserving systems, Phys. Rev. Lett. 125, 245303 (2020).
  22. A. Morningstar, V. Khemani, and D. A. Huse, Kinetically constrained freezing transition in a dipole-conserving system, Phys. Rev. B 101, 214205 (2020).
  23. S. Moudgalya, A. Prem, D. A. Huse, and A. Chan, Spectral statistics in constrained many-body quantum chaotic systems, Phys. Rev. Res. 3, 023176 (2021).
  24. A. Gromov, A. Lucas, and R. M. Nandkishore, Fracton hydrodynamics, Phys. Rev. Res. 2, 033124 (2020).
  25. J. Iaconis, A. Lucas, and R. Nandkishore, Multipole conservation laws and subdiffusion in any dimension, Phys. Rev. E 103, 022142 (2021).
  26. M. Žnidarič, Superdiffusive magnetization transport in the XX spin chain with nonlocal dephasing, Phys. Rev. B 109, 075105 (2024).
  27. M. Ljubotina, J.-Y. Desaules, M. Serbyn, and Z. Papić, Superdiffusive energy transport in kinetically constrained models, Phys. Rev. X 13, 011033 (2023).
  28. Y.-P. Wang, C. Fang, and J. Ren, Superdiffusive transport in quasi-particle dephasing models, SciPost Phys. 17, 150 (2024).
  29. D. S. Bhakuni, R. Verdel, J.-Y. Desaules, M. Serbyn, M. Ljubotina, and M. Dalmonte, Anomalously fast transport in non-integrable lattice gauge theories, arXiv:2509.08889.
  30. C. P. Moca, B. Dóra, D. Sticlet, A. Valli, T. Prosen, and G. Zaránd, Dynamic scaling and Family-Vicsek universality in SU(N) quantum spin chains, Phys. Rev. B 113, L020405 (2026).
  31. A. Scheie, N. Sherman, M. Dupont, S. Nagler, M. Stone, G. Granroth, J. Moore, and D. Tennant, Detection of Kardar–Parisi–Zhang hydrodynamics in a quantum Heisenberg spin-1/2 chain, Nat. Phys. 17, 726 (2021).
  32. D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, et al., Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science 376, 716 (2022).
  33. M. K. Joshi, F. Kranzl, A. Schuckert, I. Lovas, C. Maier, R. Blatt, M. Knap, and C. F. Roos, Observing emergent hydrodynamics in a long-range quantum magnet, Science 376, 720 (2022).
  34. E. Rosenberg, T. Andersen, R. Samajdar, A. Petukhov, J. Hoke, D. Abanin, A. Bengtsson, I. Drozdov, C. Erickson, P. Klimov, et al., Dynamics of magnetization at infinite temperature in a Heisenberg spin chain, Science 384, 48 (2024).
  35. V. B. Bulchandani, S. Gopalakrishnan, and E. Ilievski, Superdiffusion in spin chains, J. Stat. Mech. (2021) 084001.
  36. S. Gopalakrishnan and R. Vasseur, Anomalous transport from hot quasiparticles in interacting spin chains, Rep. Prog. Phys. 86, 036502 (2023).
  37. E. Ilievski, J. De Nardis, M. Medenjak, and T. Prosen, Superdiffusion in one-dimensional quantum lattice models, Phys. Rev. Lett. 121, 230602 (2018).
  38. E. Ilievski, J. De Nardis, S. Gopalakrishnan, R. Vasseur, and B. Ware, Superuniversality of superdiffusion, Phys. Rev. X 11, 031023 (2021).
  39. S. Gopalakrishnan and R. Vasseur, Kinetic theory of spin diffusion and superdiffusion in xxz spin chains, Phys. Rev. Lett. 122, 127202 (2019).
  40. K. A. Takeuchi, K. Takasan, O. Busani, P. L. Ferrari, R. Vasseur, and J. De Nardis, Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains, Phys. Rev. Lett. 134, 097104 (2025).
  41. A. Das, M. Kulkarni, H. Spohn, and A. Dhar, Kardar-Parisi-Zhang scaling for an integrable lattice Landau-Lifshitz spin chain, Phys. Rev. E 100, 042116 (2019).
  42. Ž. Krajnik, E. Ilievski, T. Prosen, and V. Pasquier, Anisotropic Landau-Lifshitz model in discrete space-time, SciPost Phys. 11, 051 (2021).
  43. J. De Nardis, S. Gopalakrishnan, and R. Vasseur, Nonlinear fluctuating hydrodynamics for Kardar-Parisi-Zhang scaling in isotropic spin chains, Phys. Rev. Lett. 131, 197102 (2023).
  44. B. Doyon, G. Perfetto, T. Sasamoto, and T. Yoshimura, Ballistic macroscopic fluctuation theory, SciPost Phys. 15, 136 (2023).
  45. L. Zadnik, M. Ljubotina, Ž. Krajnik, E. Ilievski, and T. Prosen, Quantum many-body spin ratchets, PRX Quantum 5, 030356 (2024).
  46. A. Valli, C. P. Moca, M. A. Werner, M. Kormos, Ž. Krajnik, T. Prosen, and G. Zaránd, Efficient computation of cumulant evolution and full counting statistics: Application to infinite temperature quantum spin chains, Phys. Rev. Lett. 135, 100401 (2025).
  47. V. Yu. Popkov and A. A. Zvyagin, “Antichiral” exactly solvable effectively two-dimensional quantum spin model, Phys. Lett. A 175, 295 (1993).
  48. H. Frahm and C. Rödenbeck, Integrable models of coupled Heisenberg chains, Europhys. Lett. 33, 47 (1996).
  49. H. Frahm and C. Rödenbeck, Properties of the chiral spin liquid state in generalized spin ladders, J. Phys. A: Math. Gen. 30, 4467 (1997).
  50. B. Doyon and H. Spohn, Drude weight for the Lieb-Liniger Bose gas, SciPost Phys. 3, 039 (2017).
  51. B. Doyon, Lecture notes on generalised hydrodynamics, SciPost Phys. Lect. Notes 18 (2020).
  52. B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, and R. Vasseur, Generalized hydrodynamics: A perspective, Phys. Rev. X 15, 010501 (2025).
  53. Note that our definition of Hχ differs by a factor of 1/2 from Ref. [49], but is consistent with the model Hamiltonian reported in Ref. [48], and Bethe ansatz equations found in Refs. [47, 48, 49].
  54. X. G. Wen, F. Wilczek, and A. Zee, Chiral spin states and superconductivity, Phys. Rev. B 39, 11413 (1989).
  55. B. Bauer, L. Cincio, B. Keller, M. Dolfi, G. Vidal, S. Trebst, and A. Ludwig, Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator, Nat. Commun. 5, 5137 (2014).
  56. G. Gorohovsky, R. G. Pereira, and E. Sela, Chiral spin liquids in arrays of spin chains, Phys. Rev. B 91, 245139 (2015).
  57. F. Oliviero, J. A. Sobral, E. C. Andrade, and R. G. Pereira, Noncoplanar magnetic orders and gapless chiral spin liquid on the Kagome lattice with staggered scalar spin chirality, SciPost Phys. 13, 050 (2022).
  58. P. Lecheminant and A. M. Tsvelik, Lattice spin models for non-Abelian chiral spin liquids, Phys. Rev. B 95, 140406(R) (2017).
  59. H. B. Xavier, C. Chamon, and R. G. Pereira, Network construction of non-Abelian chiral spin liquids, SciPost Phys. 14, 105 (2023).
  60. J. Willsher and J. Knolle, Dynamics and stability of U(1) spin liquids beyond mean-field theory, arXiv:2503.13831.
  61. D. Sen and R. Chitra, Large-U limit of a Hubbard model in a magnetic field: Chiral spin interactions and paramagnetism, Phys. Rev. B 51, 1922 (1995).
  62. O. I. Motrunich, Orbital magnetic field effects in spin liquid with spinon Fermi sea: Possible application to κ–(ET)2cu2(CN)3, Phys. Rev. B 73, 155115 (2006).
  63. S. Kitamura, T. Oka, and H. Aoki, Probing and controlling spin chirality in Mott insulators by circularly polarized laser, Phys. Rev. B 96, 014406 (2017).
  64. M. Claassen, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Dynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated Mott insulators, Nat. Commun. 8, 1192 (2017).
  65. M. Vanicat, L. Zadnik, and T. Prosen, Integrable Trotterization: Local conservation laws and boundary driving, Phys. Rev. Lett. 121, 030606 (2018).
  66. C. Paletta, U. Duh, B. Pozsgay, and L. Zadnik, Integrability and charge transport in asymmetric quantum-circuit geometries, J. Phys. A: Math. Theor. 58, 275001 (2025).
  67. M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, New York, 1999).
  68.  A quasiparticle traversing the distance ℓ∼|vst| encounters ℓ spins that can be thought of as randomly chosen from some probability distribution with a finite variance σ2 and zero mean. The magnetic field felt by the quasiparticle corresponds to the mean field h of the sequence of ℓ traversed spins. For large ℓ (or t), by the central limit theorem, ℓh is normally distributed with zero mean and variance σ2. Therefore, 〈h2〉∼|vst|−1.
  69. As an additional check of self-consistency, we note that for slow quasiparticles with s∼s* the background cannot be treated as frozen. These quasiparticles move with the background, i.e., they diffuse with a time-dependent diffusion constant s*(t), since m2(t)∼s*(t)t. Relating the length scales as |vs*t|2∼s*t, where |vs*|∼s*−1, we again find s*(t)∼t1/3.
  70. To enhance numerical convergence, we adopt a symmetric evolution scheme in which R(λ,0) is propagated forward by +t/2 and R†(λ*,0) is propagated backward by −t/2.
  71. M. Prähofer and H. Spohn, Universal distributions for growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett. 84, 4882 (2000).
  72. J. Baik and E. M. Rains, Limiting distributions for a polynuclear growth model with external sources, J. Stat. Phys. 100, 523 (2000).
  73. C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159, 151 (1994).
  74. M. Kalinowski, N. Maskara, and M. D. Lukin, Non-Abelian Floquet spin liquids in a digital Rydberg simulator, Phys. Rev. X 13, 031008 (2023).
  75. S. J. Evered, M. Kalinowski, A. A. Geim, T. Manovitz, D. Bluvstein, S. H. Li, N. Maskara, H. Zhou, S. Ebadi, M. Xu, et al., Probing the Kitaev honeycomb model on a neutral-atom quantum computer, Nature (London) 645, 341 (2025).
  76. We implement open boundary conditions by removing one of the four-site unitary gate, such that UOBC=∏j=1L4−1U[4j−1,4j+2]∏j=1L4U[4j−3,4j].
  77. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  78. M. Fishman, S. R. White, and E. M. Stoudenmire, Codebase release 0.3 for ITensor, SciPost Phys. Codebases 4 (2022).
  79. P. Weinberg and M. Bukov, QuSpin: A Python package for dynamics and exact diagonalisation of quantum many body systems part I: Spin chains, SciPost Phys. 2, 003 (2017).
  80. P. Weinberg and M. Bukov, QuSpin: A Python package for dynamics and exact diagonalisation of quantum many body systems. Part II: Bosons, fermions and higher spins, SciPost Phys. 7, 020 (2019).
  81. L. D. Faddeev, How Algebraic Bethe ansatz works for integrable model, arXiv:hep-th/9605187.

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Phys. Rev. B 113, 184305– Published 6 May, 2026
  • Received 8 January 2026
  • Revised 6 March 2026
  • Accepted 13 March 2026
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