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Spectral decimation of quantum many-body Hamiltonians

Feng He, Arthur Hutsalyuk, Giuseppe Mussardo, and Andrea Stampiggi*

  • *Contact author: astampig@sissa.it
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Phys. Rev. B 113, 245121 – Published 9 June, 2026

DOI: https://doi.org/10.1103/4dct-rs4q

Abstract

We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization. In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased, and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.

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References (147)

  1. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  2. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  3. R. Gezzi, T. Pruschke, and V. Meden, Functional renormalization group for nonequilibrium quantum many-body problems, Phys. Rev. B 75, 045324 (2007).
  4. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  5. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  6. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  7. S.-J. Ran, E. Tirrito, C. Peng, X. Chen, L. Tagliacozzo, G. Su, and M. Lewenstein, Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems (Springer Nature, Berlin, 2020).
  8. R. Orús, Tensor networks for complex quantum systems, Nat. Rev. Phys. 1, 538 (2019).
  9. P. Silvi, F. Tschirsich, M. Gerster, J. Jünemann, D. Jaschke, M. Rizzi, and S. Montangero, The tensor networks anthology: Simulation techniques for many-body quantum lattice systems, SciPost Phys. Lect. Notes 8 (2019).
  10. X. Gao and L.-M. Duan, Efficient representation of quantum many-body states with deep neural networks, Nat. Commun. 8, 662 (2017).
  11. G. Carleo, Y. Nomura, and M. Imada, Constructing exact representations of quantum many-body systems with deep neural networks, Nat. Commun. 9, 5322 (2018).
  12. A. Szabó and C. Castelnovo, Neural network wave functions and the sign problem, Phys. Rev. Res. 2, 033075 (2020).
  13. Y. Nomura and M. Imada, Dirac-type nodal spin liquid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy, Phys. Rev. X 11, 031034 (2021).
  14. D.-L. Deng, X. Li, and S. Das Sarma, Quantum entanglement in neural network states, Phys. Rev. X 7, 021021 (2017).
  15. K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4, 251 (1991).
  16. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  17. M. Medvidović and J. R. Moreno, Neural-network quantum states for many-body physics, Eur. Phys. J. Plus 139, 1 (2024).
  18. Y. Du, Y. Zhu, Y.-H. Zhang, M.-H. Hsieh, P. Rebentrost, W. Gao, Y.-D. Wu, J. Eisert, G. Chiribella, D. Tao, et al., Artificial intelligence for representing and characterizing quantum systems, arXiv:2509.04923.
  19. R. Rende, L. L. Viteritti, F. Becca, A. Scardicchio, A. Laio, and G. Carleo, Foundation neural-networks quantum states as a unified ansatz for multiple Hamiltonians, Nat. Commun. 16, 7213 (2025).
  20. F. He, A. Hutsalyuk, G. Mussardo, and A. Stampiggi, Statistical signatures of integrable and non-integrable quantum Hamiltonians, J. Stat. Mech. (2026) 023101.
  21. C. E. Porter, Statistical Theory of Spectra: Fluctuations (Elsevier Science, Amsterdam, 1965).
  22. N. Rosenzweig and C. E. Porter, “Repulsion of energy levels” in complex atomic spectra, Phys. Rev. 120, 1698 (1960).
  23. F. J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3, 140 (1962).
  24. F. J. Dyson, Statistical theory of the energy levels of complex systems. II, J. Math. Phys. 3, 157 (1962).
  25. M. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. A 356, 375 (1977).
  26. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  27. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  28. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  29. M. Rigol, Breakdown of thermalization in finite one-dimensional systems, Phys. Rev. Lett. 103, 100403 (2009).
  30. L. F. Santos and M. Rigol, Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Phys. Rev. E 81, 036206 (2010).
  31. S. Moudgalya, A. Prem, R. Nandkishore, N. Regnault, and B. A. Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, in Memorial Volume for Shoucheng Zhang (World Scientific, Singapore, 2021), p. 147.
  32. P. Calabrese, F. H. L. Essler, and G. Mussardo, Introduction to ‘quantum integrability in out of equilibrium systems’, J. Stat. Mech. (2016) 064001.
  33. P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020).
  34. V. Khemani, M. Hermele, and R. Nandkishore, Localization from Hilbert space shattering: From theory to physical realizations, Phys. Rev. B 101, 174204 (2020).
  35. S. Pai, M. Pretko, and R. M. Nandkishore, Localization in fractonic random circuits, Phys. Rev. X 9, 021003 (2019).
  36. P. Łydżba, P. Prelovšek, and M. Mierzejewski, Local integrals of motion in dipole-conserving models with Hilbert space fragmentation, Phys. Rev. Lett. 132, 220405 (2024).
  37. J. Classen-Howes, R. Senese, and A. Prakash, Universal freezing transitions of dipole-conserving chains, Phys. Rev. B 112, 125148 (2025).
  38. S. Moudgalya and O. I. Motrunich, Hilbert space fragmentation and commutant algebras, Phys. Rev. X 12, 011050 (2022).
  39. L. Caha and D. Nagaj, The pair-flip model: A very entangled translationally invariant spin chain, arXiv:1805.07168.
  40. O. Hart, Exact Mazur bounds in the pair-flip model and beyond, SciPost Phys. Core 7, 040 (2024).
  41. C. Stahl, R. Nandkishore, and O. Hart, Topologically stable ergodicity breaking from emergent higher-form symmetries in generalized quantum loop models, SciPost Phys. 16, 068 (2024).
  42. Y. Li, P. Sala, and F. Pollmann, Hilbert space fragmentation in open quantum systems, Phys. Rev. Res. 5, 043239 (2023).
  43. 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).
  44. M. Serbyn, Z. Papić, and D. A. Abanin, Quantum quenches in the many-body localized phase, Phys. Rev. B 90, 174302 (2014).
  45. A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science 364, 256 (2019).
  46. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  47. M. Serbyn, Z. Papić, and D. A. Abanin, Universal slow growth of entanglement in interacting strongly disordered systems, Phys. Rev. Lett. 110, 260601 (2013).
  48. M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Evidence for unbounded growth of the number entropy in many-body localized phases, Phys. Rev. Lett. 124, 243601 (2020).
  49. P. T. Dumitrescu, R. Vasseur, and A. C. Potter, Scaling theory of entanglement at the many-body localization transition, Phys. Rev. Lett. 119, 110604 (2017).
  50. F. Pietracaprina, G. Parisi, A. Mariano, S. Pascazio, and A. Scardicchio, Entanglement critical length at the many-body localization transition, J. Stat. Mech. (2017) 113102.
  51. J. H. Bardarson, F. Pollmann, and J. E. Moore, Unbounded growth of entanglement in models of many-body localization, Phys. Rev. Lett. 109, 017202 (2012).
  52. A. Morningstar and D. A. Huse, Renormalization-group study of the many-body localization transition in one dimension, Phys. Rev. B 99, 224205 (2019).
  53. P. T. Dumitrescu, A. Goremykina, S. A. Parameswaran, M. Serbyn, and R. Vasseur, Kosterlitz-Thouless scaling at many-body localization phase transitions, Phys. Rev. B 99, 094205 (2019).
  54. J. Niedda, G. B. Testasecca, G. Magnifico, F. Balducci, C. Vanoni, and A. Scardicchio, Renormalization group analysis of the many-body localization transition in the random-field XXZ chain, Phys. Rev. B 112, 144201 (2025).
  55. D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103(R) (2015).
  56. V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  57. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  58. F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Phys. 19, 498 (2018).
  59. P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, Rep. Prog. Phys. 88, 026502 (2025).
  60. M. L. Mehta, Random Matrices, 3rd ed., Pure and Applied Mathematics Vol. 142 (Elsevier/Academic Press, Amsterdam, 2004).
  61. O. Giraud, N. Macé, E. Vernier, and F. Alet, Probing symmetries of quantum many-body systems through gap ratio statistics, Phys. Rev. X 12, 011006 (2022).
  62. M. V. Berry and M. Robnik, Semiclassical level spacings when regular and chaotic orbits coexist, J. Phys. A: Math. Gen. 17, 2413 (1984).
  63. G. De Tomasi, D. Hetterich, P. Sala, and F. Pollmann, Dynamics of strongly interacting systems: From Fock-space fragmentation to many-body localization, Phys. Rev. B 100, 214313 (2019).
  64. T. Rakovszky, P. Sala, R. Verresen, M. Knap, and F. Pollmann, Statistical localization: From strong fragmentation to strong edge modes, Phys. Rev. B 101, 125126 (2020).
  65. I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interacting electrons in disordered wires: Anderson localization and low-T transport, Phys. Rev. Lett. 95, 206603 (2005).
  66. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, On the problem of many-body localization, arXiv:cond-mat/0602510.
  67. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys. 321, 1126 (2006).
  68. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Possible experimental manifestations of the many-body localization, Phys. Rev. B 76, 052203 (2007).
  69. A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  70. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E 102, 062144 (2020).
  71. M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Slow delocalization of particles in many-body localized phases, Phys. Rev. B 103, 024203 (2021).
  72. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Ergodicity breaking transition in finite disordered spin chains, Phys. Rev. B 102, 064207 (2020).
  73. S. Bera, G. De Tomasi, F. Weiner, and F. Evers, Density propagator for many-body localization: Finite-size effects, transient subdiffusion, and exponential decay, Phys. Rev. Lett. 118, 196801 (2017).
  74. R. K. Panda, A. Scardicchio, M. Schulz, S. R. Taylor, and M. Žnidarič, Can we study the many-body localisation transition? Europhys. Lett. 128, 67003 (2020).
  75. A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Avalanches and many-body resonances in many-body localized systems, Phys. Rev. B 105, 174205 (2022).
  76. D. Sels and A. Polkovnikov, Dynamical obstruction to localization in a disordered spin chain, Phys. Rev. E 104, 054105 (2021).
  77. T. Devakul and R. R. P. Singh, Early breakdown of area-law entanglement at the many-body delocalization transition, Phys. Rev. Lett. 115, 187201 (2015).
  78. T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Random-matrix physics: Spectrum and strength fluctuations, Rev. Mod. Phys. 53, 385 (1981).
  79. D. Freedman and P. Diaconis, On the histogram as a density estimator:L2 theory, Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 453 (1981).
  80. S. Aditya, D. Dhar, and D. Sen, Subspace-restricted thermalization in a correlated-hopping model with strong Hilbert space fragmentation characterized by irreducible strings, Phys. Rev. B 110, 045418 (2024).
  81. L. Zhao, P. R. Datla, W. Tian, M. M. Aliyu, and H. Loh, Observation of quantum thermalization restricted to Hilbert space fragments and Z2k scars, Phys. Rev. X 15, 011035 (2025).
  82. S. Balasubramanian, S. Gopalakrishnan, A. Khudorozhkov, and E. Lake, Glassy word problems: Ultraslow relaxation, Hilbert space jamming, and computational complexity, Phys. Rev. X 14, 021034 (2024).
  83. Y. H. Kwan, P. H. Wilhelm, S. Biswas, and S. A. Parameswaran, Minimal Hubbard models of maximal Hilbert space fragmentation, Phys. Rev. Lett. 134, 010411 (2025).
  84. C. M. Langlett and S. Xu, Hilbert space fragmentation and exact scars of generalized Fredkin spin chains, Phys. Rev. B 103, L220304 (2021).
  85. A. Yoshinaga, H. Hakoshima, T. Imoto, Y. Matsuzaki, and R. Hamazaki, Emergence of Hilbert space fragmentation in Ising models with a weak transverse field, Phys. Rev. Lett. 129, 090602 (2022).
  86. O. Hart and R. Nandkishore, Hilbert space shattering and dynamical freezing in the quantum Ising model, Phys. Rev. B 106, 214426 (2022).
  87. M. Ganguli, S. Aditya, and D. Sen, Aspects of Hilbert space fragmentation in the quantum East model: Fragmentation, subspace-restricted quantum scars, and effects of density-density interactions, Phys. Rev. B 111, 045411 (2025).
  88. A. Morningstar, V. Khemani, and D. A. Huse, Kinetically constrained freezing transition in a dipole-conserving system, Phys. Rev. B 101, 214205 (2020).
  89. Z.-C. Yang, F. Liu, A. V. Gorshkov, and T. Iadecola, Hilbert-space fragmentation from strict confinement, Phys. Rev. Lett. 124, 207602 (2020).
  90. A. Bastianello, U. Borla, and S. Moroz, Fragmentation and emergent integrable transport in the weakly tilted Ising chain, Phys. Rev. Lett. 128, 196601 (2022).
  91. I.-C. Chen and T. Iadecola, Emergent symmetries and slow quantum dynamics in a Rydberg-atom chain with confinement, Phys. Rev. B 103, 214304 (2021).
  92. B. Mukherjee, D. Banerjee, K. Sengupta, and A. Sen, Minimal model for Hilbert space fragmentation with local constraints, Phys. Rev. B 104, 155117 (2021).
  93. B. Mukherjee, Z. Cai, and W. V. Liu, Constraint-induced breaking and restoration of ergodicity in spin-1 PXP models, Phys. Rev. Res. 3, 033201 (2021).
  94. D. Hahn, P. A. McClarty, and D. J. Luitz, Information dynamics in a model with Hilbert space fragmentation, SciPost Phys. 11, 074 (2021).
  95. K. Lee, A. Pal, and H. J. Changlani, Frustration-induced emergent Hilbert space fragmentation, Phys. Rev. B 103, 235133 (2021).
  96. J. Richter and A. Pal, Anomalous hydrodynamics in a class of scarred frustration-free Hamiltonians, Phys. Rev. Res. 4, L012003 (2022).
  97. B. Pozsgay, T. Gombor, A. Hutsalyuk, Y. Jiang, L. Pristyák, and E. Vernier, Integrable spin chain with Hilbert space fragmentation and solvable real-time dynamics, Phys. Rev. E 104, 044106 (2021).
  98. L. Zadnik and M. Fagotti, The folded spin-1/2 XXZ model: I. Diagonalisation, jamming, and ground state properties, SciPost Phys. Core 4, 010 (2021).
  99. L. Zadnik, K. Bidzhiev, and M. Fagotti, The folded spin-1/2 XXZ model: II. Thermodynamics and hydrodynamics with a minimal set of charges, SciPost Phys. 10, 099 (2021).
  100. C. Stahl, O. Hart, A. Khudorozhkov, and R. Nandkishore, Strong Hilbert space fragmentation and fractons from subsystem and higher-form symmetries, Phys. Rev. B 112, 104316 (2025).
  101. D. A. Huse, R. Nandkishore, and V. Oganesyan, Phenomenology of fully many-body-localized systems, Phys. Rev. B 90, 174202 (2014).
  102. J.-Y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
  103. R. Vosk, D. A. Huse, and E. Altman, Theory of the many-body localization transition in one-dimensional systems, Phys. Rev. X 5, 031032 (2015).
  104. K. Agarwal, E. Altman, E. Demler, S. Gopalakrishnan, D. A. Huse, and M. Knap, Rare-region effects and dynamics near the many-body localization transition, Ann. Phys. 529, 1600326 (2017).
  105. 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).
  106. 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).
  107. L. F. Santos, M. I. Dykman, M. Shapiro, and F. M. Izrailev, Strong many-particle localization and quantum computing with perpetually coupled qubits, Phys. Rev. A 71, 012317 (2005).
  108. P. Sierant and J. Zakrzewski, Challenges to observation of many-body localization, Phys. Rev. B 105, 224203 (2022).
  109. D. J. Luitz, N. Laflorencie, and F. Alet, Extended slow dynamical regime close to the many-body localization transition, Phys. Rev. B 93, 060201(R) (2016).
  110. B. Dabholkar and F. Alet, Ergodic and non-ergodic properties of disordered SU(3) chains, arXiv:2403.00442.
  111. G. A. Miranda, F. Alet, G. Biroli, L. F. Cugliandolo, N. Laflorencie, and M. Tarzia, Large deviations in the many-body localization transition: The case of the random-field XXZ chain, arXiv:2510.18545.
  112. 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).
  113. Y. Bahri, R. Vosk, E. Altman, and A. Vishwanath, Localization and topology protected quantum coherence at the edge of hot matter, Nat. Commun. 6, 7341 (2015).
  114. T. Thiery, F. Huveneers, M. Müller, and W. De Roeck, Many-body delocalization as a quantum avalanche, Phys. Rev. Lett. 121, 140601 (2018).
  115. W. De Roeck and F. Huveneers, Stability and instability towards delocalization in many-body localization systems, Phys. Rev. B 95, 155129 (2017).
  116. P. Sierant, D. Delande, and J. Zakrzewski, Many-body localization due to random interactions, Phys. Rev. A 95, 021601(R) (2017).
  117. P. Sierant, M. Lewenstein, A. Scardicchio, and J. Zakrzewski, Stability of many-body localization in Floquet systems, Phys. Rev. B 107, 115132 (2023).
  118. P. R. N. Falcão, A. S. Aramthottil, P. Sierant, and J. Zakrzewski, Many-body localization crossover is sharper in a quasiperiodic potential, Phys. Rev. B 110, 184209 (2024).
  119. F. Buccheri, A. De Luca, and A. Scardicchio, Structure of typical states of a disordered Richardson model and many-body localization, Phys. Rev. B 84, 094203 (2011).
  120. C. L. Baldwin, C. R. Laumann, A. Pal, and A. Scardicchio, The many-body localized phase of the quantum random energy model, Phys. Rev. B 93, 024202 (2016).
  121. G. Mossi and A. Scardicchio, Ergodic and localized regions in quantum spin glasses on the Bethe lattice, Philos. Trans. R. Soc. A 375, 20160424 (2017).
  122. A. De Luca and A. Scardicchio, Ergodicity breaking in a model showing many-body localization, Europhys. Lett. 101, 37003 (2013).
  123. A. Altland and T. Micklitz, Field theory approach to many-body localization, Phys. Rev. Lett. 118, 127202 (2017).
  124. A. Prakash, J. H. Pixley, and M. Kulkarni, Universal spectral form factor for many-body localization, Phys. Rev. Res. 3, L012019 (2021).
  125. I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, and A. L. Burin, Spectral diffusion and scaling of many-body delocalization transitions, Ann. Phys. 529, 1600360 (2017).
  126. J. Z. Imbrie, Diagonalization and many-body localization for a disordered quantum spin chain, Phys. Rev. Lett. 117, 027201 (2016).
  127. J. Z. Imbrie, On many-body localization for quantum spin chains, J. Stat. Phys. 163, 998 (2016).
  128. D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Localization-protected quantum order, Phys. Rev. B 88, 014206 (2013).
  129. V. Ros, M. Müller, and A. Scardicchio, Integrals of motion in the many-body localized phase, Nucl. Phys. B 891, 420 (2015).
  130. J. Z. Imbrie, V. Ros, and A. Scardicchio, Local integrals of motion in many-body localized systems, Ann. Phys. 529, 1600278 (2017).
  131. A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Constructing local integrals of motion in the many-body localized phase, Phys. Rev. B 91, 085425 (2015).
  132. M. Serbyn, Z. Papić, and D. A. Abanin, Local conservation laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013).
  133. T. O'Brien, D. Abanin, G. Vidal, and Z. Papic, Explicit construction of local conserved operators in disordered many-body systems, Phys. Rev. B 94, 144208 (2016).
  134. W. De Roeck, L. Giacomin, F. Huveneers, and O. Prosniak, Absence of normal heat conduction in strongly disordered interacting quantum chains, arXiv:2408.04338.
  135. D. Sels and A. Polkovnikov, Thermalization of dilute impurities in one-dimensional spin chains, Phys. Rev. X 13, 011041 (2023).
  136. M. Žnidarič, T. Prosen, and P. Prelovšek, Many-body localization in the Heisenberg XXZ magnet in a random field, Phys. Rev. B 77, 064426 (2008).
  137. Á. L. Corps, R. A. Molina, and A. Relaño, Signatures of a critical point in the many-body localization transition, SciPost Phys. 10, 107 (2021).
  138. A. Morningstar, D. A. Huse, and J. Z. Imbrie, Many-body localization near the critical point, Phys. Rev. B 102, 125134 (2020).
  139. B. De, P. Sierant, and J. Zakrzewski, On intermediate statistics across many-body localization transition, J. Phys. A: Math. Theor. 55, 014001 (2022).
  140. P. Sierant and J. Zakrzewski, Level statistics across the many-body localization transition, Phys. Rev. B 99, 104205 (2019).
  141. A. Goremykina, R. Vasseur, and M. Serbyn, Analytically solvable renormalization group for the many-body localization transition, Phys. Rev. Lett. 122, 040601 (2019).
  142. L. Zhang, B. Zhao, T. Devakul, and D. A. Huse, Many-body localization phase transition: A simplified strong-randomness approximate renormalization group, Phys. Rev. B 93, 224201 (2016).
  143. R. Modak and T. Nag, Many-body localization in a long-range model: Real-space renormalization-group study, Phys. Rev. E 101, 052108 (2020).
  144. E. Altman and R. Vosk, Universal dynamics and renormalization in many-body-localized systems, Annu. Rev. Condens. Matter Phys. 6, 383 (2015).
  145. A. C. Potter, R. Vasseur, and S. A. Parameswaran, Universal properties of many-body delocalization transitions, Phys. Rev. X 5, 031033 (2015).
  146. T. Thiery, M. Müller, and W. De Roeck, A microscopically motivated renormalization scheme for the MBL/ETH transition, arXiv:1711.09880.
  147. A. Stampiggi, Spectral decimation, Zenodo, 2026, https://doi.org/10.5281/zenodo.20349048.

Outline

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Phys. Rev. B 113, 245121– Published 9 June, 2026
  • Received 5 March 2026
  • Revised 25 May 2026
  • Accepted 27 May 2026
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