- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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

Beyond fragmented dopant dynamics in quantum spin lattices: Robust localization and non-Gaussian diffusion

Mingru Yang1,2, Sajant Anand3,4, and Kristian Knakkergaard Nielsen5,1

PDF

Phys. Rev. B 112, 165129 – Published 20 October, 2025

DOI: https://doi.org/10.1103/clhr-4h26

Abstract

The motion of dopants in magnetic spin lattices has received tremendous attention for at least four decades due to its connection to high-temperature superconductivity. Despite these efforts, we lack a complete understanding of their behavior, especially out of the equilibrium and at nonzero temperatures. In this paper, we take a significant step towards a much deeper understanding based on state-of-the-art matrix-product-state calculations. In particular, we investigate the nonequilibrium dynamics of a dopant in two-leg t−J ladders with antiferromagnetic XXZ spin interactions. In the Ising limit, we find that the dopant is localized for all investigated nonzero temperatures due to an emergent disordered potential, with a localization length controlled by the underlying correlation length of the spin lattice, which increases exponentially with decreasing temperature. The dopant, hereby, only delocalizes asymptotically in the zero temperature limit. This greatly generalizes the localization effect discovered recently in Hilbert space fragmented models [Phys. Rev. Res. 6, 023325 (2024), SciPost Phys. Core 7, 054 (2024)]. In the presence of spin-exchange processes at rate α, the dopant diffuses with a diffusion coefficient, Dh, depending nonmonotonically on α. It initially increases linearly as Dh∝α for α≪1 before dropping off as α−1 for α>1. Moreover, we show that the underlying spin dynamics at infinite temperature behaves qualitatively the same, albeit with important quantitative differences. We substantiate these findings by showing that the dynamics features self-similar scaling behavior, which strongly deviates from the Gaussian behavior of regular diffusion, especially for weak spin exchange. Finally, we show that the diffusion coefficient Dh follows an Arrhenius relation at high temperatures, whereby it is exponentially suppressed upon cooling.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (120)

  1. M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. (2007) P08024.
  2. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  3. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  4. P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, Direct observation of dynamical quantum phase transitions in an interacting many-body system, Phys. Rev. Lett. 119, 080501 (2017).
  5. S. Karch, S. Bandyopadhyay, Z.-H. Sun, A. Impertro, S. Huh, I. P. Rodríguez, J. F. Wienand, W. Ketterle, M. Heyl, A. Polkovnikov, I. Bloch, and M. Aidelsburger, Probing quantum many-body dynamics using subsystem Loschmidt echos, arXiv:2501.16995.
  6. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  7. S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: A review of exact results, Rep. Prog. Phys. 85, 086501 (2022).
  8. S. Moudgalya and O. I. Motrunich, Hilbert space fragmentation and commutant algebras, Phys. Rev. X 12, 011050 (2022).
  9. D. Adler, D. Wei, M. Will, K. Srakaew, S. Agrawal, P. Weckesser, R. Moessner, F. Pollmann, I. Bloch, and J. Zeiher, Observation of Hilbert space fragmentation and fractonic excitations in 2d, Nature (London) 636, 80 (2024).
  10. W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice, Nature (London) 462, 74 (2009).
  11. J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature (London) 467, 68 (2010).
  12. L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe, and M. W. Zwierlein, Quantum-gas microscope for fermionic atoms, Phys. Rev. Lett. 114, 193001 (2015).
  13. E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G. D. Bruce, and S. Kuhr, Single-atom imaging of fermions in a quantum-gas microscope, Nat. Phys. 11, 738 (2015).
  14. C. Gross and W. S. Bakr, Quantum gas microscopy for single atom and spin detection, Nat. Phys. 17, 1316 (2021).
  15. D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, and J. Zeiher, Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science 376, 716 (2022).
  16. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  17. L. Tarruell and L. Sanchez-Palencia, Quantum simulation of the Hubbard model with ultracold fermions in optical lattices, Comptes Rendus. Physique 19, 365 (2018).
  18. F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices, Nat. Rev. Phys. 2, 411 (2020).
  19. D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Short-range quantum magnetism of ultracold fermions in an optical lattice, Science 340, 1307 (2013).
  20. R. A. Hart, P. M. Duarte, T.-L. Yang, X. Liu, T. Paiva, E. Khatami, R. T. Scalettar, N. Trivedi, D. A. Huse, and R. G. Hulet, Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms, Nature (London) 519, 211 (2015).
  21. M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains, Science 353, 1257 (2016).
  22. M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and M. Greiner, Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model, Science 353, 1253 (2016).
  23. L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan, H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol, and M. W. Zwierlein, Observation of spatial charge and spin correlations in the 2D Fermi-Hubbard model, Science 353, 1260 (2016).
  24. A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom Fermi–Hubbard antiferromagnet, Nature (London) 545, 462 (2017).
  25. D. Mitra, P. T. Brown, E. Guardado-Sanchez, S. S. Kondov, T. Devakul, D. A. Huse, P. Schauß, and W. S. Bakr, Quantum gas microscopy of an attractive Fermi–Hubbard system, Nat. Phys. 14, 173 (2018).
  26. T. Hartke, B. Oreg, C. Turnbaugh, N. Jia, and M. Zwierlein, Direct observation of nonlocal fermion pairing in an attractive Fermi-Hubbard gas, Science 381, 82 (2023).
  27. J. R. Schrieffer and J. S. Brooks, Handbook of High-Temperature Superconductivity (Springer, New York, NY, 2007).
  28. J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker, E. Demler, G. Salomon, I. Bloch, and C. Gross, Imaging magnetic polarons in the doped Fermi–Hubbard model, Nature (London) 572, 358 (2019).
  29. T. Chalopin, P. Bojović, S. Wang, T. Franz, A. Sinha, Z. Wang, D. Bourgund, J. Obermeyer, F. Grusdt, A. Bohrdt, L. Pollet, A. Wietek, A. Georges, T. Hilker, and I. Bloch, Probing the magnetic origin of the pseudogap using a Fermi-Hubbard quantum simulator, arXiv:2412.17801.
  30. G. Ji, M. Xu, L. H. Kendrick, C. S. Chiu, J. C. Brüggenjürgen, D. Greif, A. Bohrdt, F. Grusdt, E. Demler, M. Lebrat, and M. Greiner, Coupling a mobile hole to an antiferromagnetic spin background: Transient dynamics of a magnetic polaron, Phys. Rev. X 11, 021022 (2021).
  31. S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Spectral function of holes in a quantum antiferromagnet, Phys. Rev. Lett. 60, 2793 (1988).
  32. C. L. Kane, P. A. Lee, and N. Read, Motion of a single hole in a quantum antiferromagnet, Phys. Rev. B 39, 6880 (1989).
  33. K. K. Nielsen, T. Pohl, and G. M. Bruun, Nonequilibrium hole dynamics in antiferromagnets: Damped strings and polarons, Phys. Rev. Lett. 129, 246601 (2022).
  34. K. K. Nielsen, M. Zwierlein, and G. M. Bruun, Dual spectroscopy of quantum simulated Fermi-Hubbard systems, Phys. Rev. Lett. 135, 043401 (2025).
  35. D. N. Sheng, Y. C. Chen, and Z. Y. Weng, Phase string effect in a doped antiferromagnet, Phys. Rev. Lett. 77, 5102 (1996).
  36. K. Wu, Z. Y. Weng, and J. Zaanen, Sign structure of the t−J model, Phys. Rev. B 77, 155102 (2008).
  37. Z. Zhu and Z.-Y. Weng, Quasiparticle collapsing in an anisotropic t−J ladder, Phys. Rev. B 92, 235156 (2015).
  38. S. R. White, D. J. Scalapino, and S. A. Kivelson, One hole in the two-leg t−J ladder and adiabatic continuity to the noninteracting limit, Phys. Rev. Lett. 115, 056401 (2015).
  39. R.-Y. Sun, Z. Zhu, and Z.-Y. Weng, Localization in a t−J-type model with translational symmetry, Phys. Rev. Lett. 123, 016601 (2019).
  40. J.-Y. Zhao, S. A. Chen, H.-K. Zhang, and Z.-Y. Weng, Two-hole ground state: Dichotomy in pairing symmetry, Phys. Rev. X 12, 011062 (2022).
  41. A. Bohrdt, F. Grusdt, and M. Knap, Dynamical formation of a magnetic polaron in a two-dimensional quantum antiferromagnet, New J. Phys. 22, 123023 (2020).
  42. J. Carlström, N. Prokof'ev, and B. Svistunov, Quantum walk in degenerate spin environments, Phys. Rev. Lett. 116, 247202 (2016).
  43. M. Kanász-Nagy, I. Lovas, F. Grusdt, D. Greif, M. Greiner, and E. A. Demler, Quantum correlations at infinite temperature: The dynamical Nagaoka effect, Phys. Rev. B 96, 014303 (2017).
  44. K. K. Nielsen, Thermally induced localization of dopants in a magnetic spin ladder, Phys. Rev. Res. 6, 023325 (2024).
  45. J. Crank, Methods of solution when the diffusion coefficient is constant, in The Mathematics of Diffusion (Oxford University Press, 1975), pp. 11–13.
  46. S. Arrhenius, Über die dissociationswärme und den einfluss der temperatur auf den dissociationsgrad der elektrolyte, Zeitschrift für Physikalische Chemie 4U, 96 (1889).
  47. K. K. Nielsen, Localized dopant motion across the 2D ising phase transition, SciPost Phys. Core 7, 054 (2024).
  48. K. K. Nielsen, Dopant pairing in a disordered magnetic spin ladder, Phys. Rev. Res. 7, L012033 (2025).
  49. A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, Disorder-free localization, Phys. Rev. Lett. 118, 266601 (2017).
  50. G. Gyawali, S. Kumar, Y. D. Lensky, E. Rosenberg, A. Szasz, T. Cochran, R. Chen, A. H. Karamlou, K. Kechedzhi, J. Berndtsson, T. Westerhout, A. Asfaw, D. Abanin, R. Acharya, L. A. Beni et al., Observation of disorder-free localization using a (2+1)d lattice gauge theory on a quantum processor, arXiv:2410.06557.
  51. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  52. In disorder-free localization, the local gauge symmetries commute and are responsible for the Hilbert space fragmentation.
  53. J. Zeiher, J.-Y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen, I. Bloch, and C. Gross, Coherent many-body spin dynamics in a long-range interacting ising chain, Phys. Rev. X 7, 041063 (2017).
  54. P. Weckesser, K. Srakaew, T. Blatz, D. Wei, D. Adler, S. Agrawal, A. Bohrdt, I. Bloch, and J. Zeiher, Realization of a rydberg-dressed extended bose Hubbard model, arXiv:2405.20128.
  55. L. Homeier, T. J. Harris, T. Blatz, S. Geier, S. Hollerith, U. Schollwöck, F. Grusdt, and A. Bohrdt, Antiferromagnetic bosonic t−J models and their quantum simulation in tweezer arrays, Phys. Rev. Lett. 132, 230401 (2024).
  56. M. Qiao, G. Emperauger, C. Chen, L. Homeier, S. Hollerith, G. Bornet, R. Martin, B. Gély, L. Klein, D. Barredo, S. Geier, N.-C. Chiu, F. Grusdt, A. Bohrdt, T. Lahaye, and A. Browaeys, Realization of a doped quantum antiferromagnet with dipolar tunnelings in a rydberg tweezer array, Nature (London) 644, 889 (2025).
  57. E. Bobrow, K. Stubis, and Y. Li, Exact results on itinerant ferromagnetism and the 15-puzzle problem, Phys. Rev. B 98, 180101(R) (2018).
  58. S. R. White, Minimally entangled typical quantum states at finite temperature, Phys. Rev. Lett. 102, 190601 (2009).
  59. M. Yang and S. R. White, Time-dependent variational principle with ancillary krylov subspace, Phys. Rev. B 102, 094315 (2020).
  60. 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).
  61. 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).
  62. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  63. S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
  64. E. M. Stoudenmire and S. R. White, Minimally entangled typical thermal state algorithms, New J. Phys. 12, 055026 (2010).
  65. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21, 1087 (1953).
  66. W. K. Hastings, Monte carlo sampling methods using markov chains and their applications, Biometrika 57, 97 (1970).
  67. K. K. Nielsen, Exact dynamics of two holes in two-leg antiferromagnetic ladders, Phys. Rev. B 108, 085125 (2023).
  68. S. A. Trugman, Interaction of holes in a Hubbard antiferromagnet and high-temperature superconductivity, Phys. Rev. B 37, 1597 (1988).
  69. C. D. White, M. Zaletel, R. S. K. Mong, and G. Refael, Quantum dynamics of thermalizing systems, Phys. Rev. B 97, 035127 (2018).
  70. S. Anand, C. D. White, J. Kemp, J. Wei, N. Y. Yao, and M. P. Zaletel, Efficient classical simulation of infinite-temperature operator dynamics beyond one-dimensional, local spin chains (unpublished).
  71. M. Gazo, A. Karailiev, T. Satoor, C. Eigen, M. Gałka, and Z. Hadzibabic, Universal coarsening in a homogeneous two-dimensional bose gas, Science 389, 802 (2025).
  72. R. Steinigeweg, F. Heidrich-Meisner, J. Gemmer, K. Michielsen, and H. De Raedt, Scaling of diffusion constants in the spin-12 XX ladder, Phys. Rev. B 90, 094417 (2014).
  73. B. Kloss, Y. B. Lev, and D. Reichman, Time-dependent variational principle in matrix-product state manifolds: Pitfalls and potential, Phys. Rev. B 97, 024307 (2018).
  74. T. Rakovszky, C. W. von Keyserlingk, and F. Pollmann, Dissipation-assisted operator evolution method for capturing hydrodynamic transport, Phys. Rev. B 105, 075131 (2022).
  75. S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15, 1 (1943).
  76. M. A. Islam, Einstein–smoluchowski diffusion equation: A discussion, Phys. Scr. 70, 120 (2004).
  77. P. Calabrese and A. Gambassi, Ageing properties of critical systems, J. Phys. A: Math. Gen. 38, R133 (2005).
  78. A. Chandran, A. Nanduri, S. S. Gubser, and S. L. Sondhi, Equilibration and coarsening in the quantum o(n) model at infinite n, Phys. Rev. B 88, 024306 (2013).
  79. A. Maraga, A. Chiocchetta, A. Mitra, and A. Gambassi, Aging and coarsening in isolated quantum systems after a quench: Exact results for the quantum O(N) model with N→∞, Phys. Rev. E 92, 042151 (2015).
  80. 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).
  81. J. F. Rodriguez-Nieva, A. P. Orioli, and J. Marino, Far-from-equilibrium universality in the two-dimensional Heisenberg model, Proc. Natl. Acad. Sci. 119, e2122599119 (2022).
  82. J. Berges, A. Rothkopf, and J. Schmidt, Nonthermal fixed points: Effective weak coupling for strongly correlated systems far from equilibrium, Phys. Rev. Lett. 101, 041603 (2008).
  83. S. Bhattacharyya, J. F. Rodriguez-Nieva, and E. Demler, Universal prethermal dynamics in Heisenberg ferromagnets, Phys. Rev. Lett. 125, 230601 (2020).
  84. A. N. Mikheev, I. Siovitz, and T. Gasenzer, Universal dynamics and non-thermal fixed points in quantum fluids far from equilibrium, Eur. Phys. J.: Spec. Top. 232, 3393 (2023).
  85. E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, Three-dimensional direct imaging of structural relaxation near the colloidal glass transition, Science 287, 627 (2000).
  86. S. Hapca, J. W. Crawford, and I. M. Young, Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level, J. R. Soc., Interface 6, 111 (2009).
  87. B. Wang, S. M. Anthony, S. C. Bae, and S. Granick, Anomalous yet Brownian, Proc. Natl. Acad. Sci. USA 106, 15160 (2009).
  88. T. Toyota, D. A. Head, C. F. Schmidt, and D. Mizuno, Non-Gaussian athermal fluctuations in active gels, Soft Matter 7, 3234 (2011).
  89. B. Parry, I. Surovtsev, M. Cabeen, C. O'Hern, E. Dufresne, and C. Jacobs-Wagner, The bacterial cytoplasm has glass-like properties and is fluidized by metabolic activity, Cell 156, 183 (2014).
  90. M. C. Munder, D. Midtvedt, T. Franzmann, E. Nüske, O. Otto, M. Herbig, E. Ulbricht, P. Müller, A. Taubenberger, S. Maharana, L. Malinovska, D. Richter, J. Guck, V. Zaburdaev, S. Alberti, and R. H. Singer, A ph-driven transition of the cytoplasm from a fluid- to a solid-like state promotes entry into dormancy, eLife 5, e09347 (2016).
  91. A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov, Brownian yet non-Gaussian diffusion: From superstatistics to subordination of diffusing diffusivities, Phys. Rev. X 7, 021002 (2017).
  92. R. Metzler, Superstatistics and non-Gaussian diffusion, Eur. Phys. J.: Spec. Top. 229, 711 (2020).
  93. In Refs. [91, 92], they give an experimentally relevant example of the exponential distribution p(D), which gives rise to an exponential scaling function F(u)∼exp[−|u|].
  94. M. V. Chubynsky and G. W. Slater, Diffusing diffusivity: A model for anomalous, yet Brownian, diffusion, Phys. Rev. Lett. 113, 098302 (2014).
  95. L. Katz, M. Guinan, and R. J. Borg, Diffusion of H2, D2, and T2 in single-crystal Ni and Cu, Phys. Rev. B 4, 330 (1971).
  96. N. Peterson, Self-diffusion in pure metals, J. Nucl. Mater. 69-70, 3 (1978).
  97. T. Ishikawa and R. B. McLellan, The diffusivity of hydrogen in copper at low temperatures, J. Phys. Chem. Solids 46, 445 (1985).
  98. H. Bonzel, A surface diffusion mechanism at high temperature, Surf. Sci. 21, 45 (1970).
  99. S. Fujita, Statistical mechanical theory of fast interstitial diffusion in metals, J. Phys. Chem. Solids 49, 41 (1988).
  100. B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, and R. Vasseur, Generalized hydrodynamics: A perspective, Phys. Rev. X 15, 010501 (2025).
  101. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  102. E. Tarquini, G. Biroli, and M. Tarzia, Critical properties of the anderson localization transition and the high-dimensional limit, Phys. Rev. B 95, 094204 (2017).
  103. B. V. Tiggelen, Diffuse Waves in Complex Media, edited by J.-P. Fouque (Springer, Dordrecht, 1999) Chap. Localization of waves.
  104. U. Wolff, Collective Monte Carlo updating for spin systems, Phys. Rev. Lett. 62, 361 (1989).
  105. S. A. Trugman, Spectral function of a hole in a Hubbard antiferromagnet, Phys. Rev. B 41, 892 (1990).
  106. G. J. Dolan and D. D. Osheroff, Nonmetallic conduction in thin metal films at low temperatures, Phys. Rev. Lett. 43, 721 (1979).
  107. B. L. Altshuler, A. G. Aronov, and P. A. Lee, Interaction effects in disordered fermi systems in two dimensions, Phys. Rev. Lett. 44, 1288 (1980).
  108. B. L. Altshuler, D. Khmel'nitzkii, A. I. Larkin, and P. A. Lee, Magnetoresistance and Hall effect in a disordered two-dimensional electron gas, Phys. Rev. B 22, 5142 (1980).
  109. H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics—An Introduction (Oxford University Press, 2016).
  110. Ny=2 is a strip with open boundary conditions in the short direction, while Ny>2 are cylinders with periodic boundary conditions in the short direction.
  111. T. Guthardt, M. Scheb, J. von Delft, F. Grusdt, and A. Bohrdt, Finite-temperature real-time properties of magnetic polarons in two-dimensional quantum antiferromagnets, arXiv:2504.07715.
  112. There are two minor differences between the investigated models, both of which should not lead to qualitative differences. (1) Ref. [111] includes a nearest-neighbor density-density term which arises in the large U/t limit of the Fermi-Hubbard model, which, however, leads to a fixed energy shift for a single dopant and thereby no differences in the dynamics. (2) The investigations in Ref. [111] is at lower spin couplings, J/t<1.
  113. A. E. Feiguin and S. R. White, Finite-temperature density matrix renormalization using an enlarged Hilbert space, Phys. Rev. B 72, 220401(R) (2005).
  114. L. P. Kadanoff and P. C. Martin, Hydrodynamic equations and correlation functions, Ann. Phys. 24, 419 (1963).
  115. P. M. Chaikin and T. C. Lubensky, Hydrodynamics, in Principles of Condensed Matter Physics (Cambridge University Press, 1995), pp. 417–494.
  116. M. Fishman, S. R. White, and E. M. Stoudenmire, The itensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  117. J. Hauschild and F. Pollmann, Efficient numerical simulations with tensor networks: Tensor network python (tenpy), SciPost Phys. Lect. Notes 5 (2018).
  118. J. Hauschild, J. Unfried, S. Anand, B. Andrews, M. Bintz, U. Borla, S. Divic, M. Drescher, J. Geiger, M. Hefel, K. Hémery, W. Kadow, J. Kemp, N. Kirchner, V. S. Liu, G. Möller, D. Parker, M. Rader, A. Romen, S. Scalet et al., Tensor network python (tenpy) version 1, SciPost Phys. Codebases 41 (2024).
  119. K. K. Nielsen, Data for “beyond fragmented dopant dynamics in quantum spin lattices: Robust localization and non-Gaussian diffusion” Zenodo (2025), 10.5281/zenodo.16606487.
  120. A. Wietek, Y.-Y. He, S. R. White, A. Georges, and E. M. Stoudenmire, Stripes, antiferromagnetism, and the pseudogap in the doped Hubbard model at finite temperature, Phys. Rev. X 11, 031007 (2021).

Outline

Information

Phys. Rev. B 112, 165129– Published 20 October, 2025
  • Received 30 July 2025
  • Revised 2 October 2025
  • Accepted 7 October 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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation