pith. machine review for the scientific record. sign in

arxiv: 2605.10191 · v2 · submitted 2026-05-11 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· cond-mat.str-el· quant-ph

Recognition: 2 theorem links

· Lean Theorem

Computing eigenpairs of quantum many-body systems with Polfed.jl

Rok Pintar , Konrad Pawlik , Rafa{\l} \'Swi\k{e}tek , Miroslav Hopjan , Jan \v{S}untajs , Jakub Zakrzewski , Piotr Sierant , Lev Vidmar
Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:51 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nncond-mat.str-elquant-ph
keywords polfed.jlpolynomial filtered exact diagonalizationmid-spectrum eigenpairsquantum many-body hamiltonianslanczos iterationsparse matricesjulia packagegpu acceleration
0
0 comments X

The pith

Polfed.jl computes mid-spectrum eigenpairs of quantum many-body Hamiltonians by applying polynomial spectral transformations on the fly inside Lanczos iterations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Polfed.jl, a Julia package implementing the Polynomially Filtered Exact Diagonalization algorithm for large sparse Hamiltonians. Mid-spectrum eigenpairs matter for non-equilibrium many-body physics but remain inaccessible due to exponential Hilbert-space growth. The method evaluates a polynomial spectral transformation during Lanczos steps to target chosen energies while keeping the original matrix sparse. This cuts memory use relative to standard diagonalization and enables GPU acceleration through repeated sparse matrix-vector products. Benchmarks on spin chains, fermions, and the quantum sun model show access to bigger system sizes than competing approaches.

Core claim

POLFED performs a polynomial spectral transformation evaluated on the fly within a Lanczos iteration to compute mid-spectrum eigenpairs while preserving the sparsity of the Hamiltonian and reducing memory costs compared to other diagonalization methods.

What carries the argument

The polynomial spectral transformation evaluated on the fly within a Lanczos iteration, which maps the target energy window so that sparse matrix-vector multiplications dominate the cost.

If this is right

  • Mid-spectrum eigenpairs become reachable for larger disordered spin-chain and fermionic systems than with conventional diagonalization.
  • GPU acceleration yields substantial speedups because the dominant operations remain sparse matrix-vector multiplications.
  • Flexible energy targeting and automatic spectral mapping optimization are available for structured Hamiltonians.
  • The quantum sun model Hamiltonian can be constructed directly to study many-body ergodicity-breaking transitions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same on-the-fly filtering approach could be applied to any large sparse matrix problem outside quantum many-body physics.
  • Users could combine POLFED output with time-evolution routines to study dynamics around targeted energies.
  • The method's scaling on GPU hardware suggests it may push accessible system sizes in studies of many-body localization or ergodicity breaking.

Load-bearing premise

The polynomial spectral transformation evaluated on the fly inside Lanczos maintains numerical stability and sufficient accuracy for mid-spectrum states without prohibitive overhead or convergence failures on the targeted sparse Hamiltonians.

What would settle it

Run POLFED on a Hamiltonian small enough for full exact diagonalization and check whether the returned mid-spectrum eigenpairs agree with the exact results to within machine precision.

Figures

Figures reproduced from arXiv: 2605.10191 by Jakub Zakrzewski, Jan \v{S}untajs, Konrad Pawlik, Lev Vidmar, Miroslav Hopjan, Piotr Sierant, Rafa{\l} \'Swi\k{e}tek, Rok Pintar.

Figure 1
Figure 1. Figure 1: Spectral transformation. (a) Normalized density of states ρ(E˜) of the un￾transformed XXZ model, see Eq. (54), with L = 18 sites. (b) Spectral transformation P K λ˜ (H˜), see Eq. (32), for different orders of polynomial expansion K = 10, 20, and 50 at target energy λ˜ = 0. The black horizontal line denotes the cutoff value Ω and 2δ denotes the width of the energy window whose transformed eigenvalues lie ab… view at source ↗
Figure 2
Figure 2. Figure 2: Sparsity of the untransformed and transformed Hamiltonian. Sparsity pattern of P K λ˜ (H˜) for different polynomial orders K, where sparsity is measured as the fraction of zero matrix elements. Panels (a), (b), (c), and (d) correspond to polynomial orders K = 1, 2, 5, and 10, respectively. For K = 1 the filter is linear in H˜, so the sparsity pattern coincides with that of the original Hamiltonian. Sparsit… view at source ↗
Figure 3
Figure 3. Figure 3: Density of states. We consider the J1–J2 model, see Eq. (55), for L = 20 at quarter filling (particle number Np = 5). The blue histogram shows results obtained from exact diagonalization (ED); black dots at the bin centers are shown to facilitate comparison with the KPM and Gaussian approximations. The KPM and Gaussian curves are computed from Eqs. (34) and (36), and are shown with orange and green lines, … view at source ↗
Figure 4
Figure 4. Figure 4: Schematic overview of the block Lanczos. The inputs are the initial Krylov block V1 and the polynomial spectral filter P K λ˜ (H˜). Once the eigenvectors of the Lanczos matrix are computed and the convergence criterion is fulfilled, we transform these eigenvectors back to the Hilbert-space basis via ui = V1:m ti , (43) where V1:m = [V1 , V2 , . . . , Vm] denotes the horizontal concatenation of the first m … view at source ↗
Figure 5
Figure 5. Figure 5: Schematic of the Clenshaw recurrence algorithm. This algorithm is used in POLFED to evaluate the polynomial filter on the fly. The inputs are the current Krylov block Vk and the rescaled Hamiltonian H˜; the output is the transformed block Wk = P K λ˜ (H˜) Vk . 4.5 Applications to GPUs A natural application of POLFED is its implementation on graphics processing units (GPUs). This is advantageous for several… view at source ↗
Figure 6
Figure 6. Figure 6: Supported target specifications in Polfed.jl. Illustrated on the den￾sity of states of the quantum sun model in rescaled spectral coordinates E˜ ∈ [−1, 1]. The options :middle and :maxdos target the center of the spectrum and the point of maximal density of states, respectively. The form (:rescaled, ε) specifies the target directly in rescaled coordinates, whereas (:unrescaled, E) uses the phys￾ical energy… view at source ↗
Figure 7
Figure 7. Figure 7: POLFED vs shift-and-invert, CPU time. CPU-time comparison between POLFED and shift-and-invert benchmarks. Panel (a) shows the total CPU time as a function of nnz, the number of nonzero off-diagonal matrix elements per row. Panel (b) shows the corresponding CPU time per targeted eigenvalue. Colors indicate the system size L, while marker shapes distinguish the different model classes (XXZ Eq. (54), J1–J2 Eq… view at source ↗
Figure 8
Figure 8. Figure 8: Decomposition of the POLFED CPU time into its main computational parts: [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: CPU time of POLFED for different number of eigenpairs. CPU time of POLFED as a function of the number of requested eigenpairs Nev for the XXZ model, Eq. (54), at half filling. Colors indicate the system size L. Panel (a) shows the total CPU time tCPU, while panel (b) shows the CPU time per requested eigenpair, tCPU/Nev. Matrix multiplication was performed with Julia’s standard LinearAlgebra func￾tion mul!(… view at source ↗
Figure 10
Figure 10. Figure 10: Benchmarks of the GPU implementation of POLFED. Main panel: breakdown of the total wall time into the main computational components for dif￾ferent system sizes L on the GPU node. Inset: comparison of the total wall time of the GPU and CPU implementations. The benchmarks were performed for the QREM model, Eq. (62). In all runs, 1500 eigenpairs were targeted at the maximum of the density of states. For calc… view at source ↗
Figure 11
Figure 11. Figure 11: Number of off-diagonal matrix elements per row as a function of system [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
read the original abstract

We present Polfed$.$jl, an open-source Julia package implementing the Polynomially Filtered Exact Diagonalization (POLFED) algorithm for computing mid-spectrum eigenvalues and eigenvectors (shortly, eigenpairs) of quantum many-body Hamiltonians. Access to such eigenpairs is essential for studying non-equilibrium many-body physics, but is hindered by the exponential growth of Hilbert-space dimension. POLFED addresses this challenge through a polynomial spectral transformation evaluated on the fly within a Lanczos iteration, preserving Hamiltonian sparsity and substantially reducing memory costs compared to other diagonalization methods. The package supports flexible energy targeting, automatic optimization of the spectral mapping for structured Hamiltonians, and GPU acceleration, which is particularly effective since the dominant computational cost reduces to repeated sparse matrix-vector multiplications. Benchmarks on disordered spin-chain and fermionic models demonstrate access to larger system sizes than alternative approaches, and CPU--GPU comparisons confirm significant speedups. In particular, we also provide code for constructing the quantum sun model Hamiltonian, a toy model of a many-body ergodicity-breaking transition. While our focus is on many-body Hamiltonians, Polfed$.$jl may be applied to any large sparse matrix.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Polfed.jl, an open-source Julia package implementing the Polynomially Filtered Exact Diagonalization (POLFED) algorithm. It computes mid-spectrum eigenpairs of large sparse quantum many-body Hamiltonians by applying a polynomial spectral transformation on the fly inside a Lanczos iteration. This preserves Hamiltonian sparsity, reduces memory footprint relative to full diagonalization or shift-invert methods, supports flexible energy targeting and automatic spectral mapping optimization, and includes GPU acceleration. Benchmarks on disordered spin chains and fermionic models claim access to larger Hilbert-space dimensions than alternative approaches, and the package ships code for the quantum sun model Hamiltonian.

Significance. If the numerical stability and accuracy claims hold, the work supplies a practical, memory-efficient tool for accessing mid-spectrum states in many-body systems that are otherwise inaccessible, directly supporting studies of non-equilibrium dynamics and ergodicity breaking. The open-source implementation, GPU support (reducing cost to repeated sparse matvecs), and provision of the quantum sun model code are concrete strengths that enhance reproducibility and usability.

major comments (2)
  1. [Benchmarks] Benchmarks section: the reported access to larger system sizes for disordered spin-chain and fermionic models is presented without quantitative controls on eigenvector accuracy, residual norms, or convergence behavior of the Lanczos iteration when the polynomial filter degree is increased. This is load-bearing because the central claim of reliable mid-spectrum eigenpairs rests on the on-the-fly polynomial transformation not amplifying round-off errors in dense spectra.
  2. [Method] Method description: the automatic optimization of the spectral mapping for structured Hamiltonians is described at a high level but lacks explicit pseudocode or parameter choices showing that the procedure remains free of ad-hoc fitting parameters that could affect the claimed parameter-free character of the filtered Lanczos step.
minor comments (2)
  1. [Abstract] Abstract: the notation 'Polfed$.$jl' appears to be a LaTeX artifact and should be rendered consistently as Polfed.jl throughout.
  2. [Benchmarks] The manuscript would benefit from a short table comparing memory scaling and wall-time per eigenpair against standard sparse eigensolvers (e.g., ARPACK, FEAST) for the same models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have revised the manuscript to incorporate additional details and controls as requested.

read point-by-point responses

  1. Referee: [Benchmarks] Benchmarks section: the reported access to larger system sizes for disordered spin-chain and fermionic models is presented without quantitative controls on eigenvector accuracy, residual norms, or convergence behavior of the Lanczos iteration when the polynomial filter degree is increased. This is load-bearing because the central claim of reliable mid-spectrum eigenpairs rests on the on-the-fly polynomial transformation not amplifying round-off errors in dense spectra.

    Authors: We agree that quantitative controls on accuracy and convergence are essential to support the reliability claims. In the revised manuscript we have added residual norm tables for representative eigenvectors across the benchmarked system sizes, plots of Lanczos residual convergence versus iteration count for increasing polynomial degrees, and direct comparisons of eigenvector accuracy (via overlap with reference shift-invert results where feasible). These additions demonstrate that round-off amplification remains controlled for the filter degrees used, consistent with the on-the-fly evaluation preserving sparsity and numerical stability. revision: yes

  2. Referee: [Method] Method description: the automatic optimization of the spectral mapping for structured Hamiltonians is described at a high level but lacks explicit pseudocode or parameter choices showing that the procedure remains free of ad-hoc fitting parameters that could affect the claimed parameter-free character of the filtered Lanczos step.

    Authors: We thank the referee for highlighting this point. The revised Methods section now includes explicit pseudocode for the automatic spectral mapping optimization. The procedure relies solely on the analytically known spectral bounds of the Hamiltonian (e.g., from the Pauli-string structure) together with a fixed, small number of Chebyshev iterations for the mapping; no user-tuned or fitting parameters are introduced. This preserves the parameter-free nature of the subsequent filtered Lanczos iteration. revision: yes

Circularity Check

0 steps flagged

No circularity in POLFED derivation or implementation

full rationale

The paper presents POLFED.jl as a direct implementation of polynomial spectral transformation evaluated on-the-fly inside a Lanczos iteration for sparse many-body Hamiltonians. No equations, fitted parameters, or central claims reduce by construction to the paper's own inputs. The method is described as preserving sparsity via repeated sparse matvecs with no self-definitional loops, no load-bearing self-citations, and no imported uniqueness theorems. Benchmarks are presented as empirical validation rather than predictions forced by the algorithm definition itself. This is a standard algorithmic description with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on standard numerical linear algebra (Lanczos iteration and polynomial spectral transformations) with no new free parameters, axioms, or invented entities introduced beyond the software implementation itself.

pith-pipeline@v0.9.0 · 5550 in / 1172 out tokens · 65419 ms · 2026-05-14T21:51:51.690829+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

103 extracted references · 103 canonical work pages · 4 internal anchors

  1. [1]

    H. Q. Lin, Exact diagonalization of quantum-spin models, Phys. Rev. B 42, 6561 (1990), doi:10.1103/PhysRevB.42.6561

    work page doi:10.1103/physrevb.42.6561 1990

  2. [2]

    Simulations of pure and doped low-dimensional spin-1/2 gapped systems

    N. Laflorencie and D. Poilblanc, Simulations of pure and doped low-dimensional spin-1/2 gapped systems, In U. Schollw \"o ck, J. Richter, D. J. J. Farnell and R. F. Bishop, eds., Quantum Magnetism, vol. 645 of Lecture Notes in Physics, pp. 227--252. Springer, Berlin, Heidelberg, doi:10.1007/BFb0119595 (2004), cond-mat/0408363

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bfb0119595 2004

  3. [3]

    A. W. Sandvik, Computational studies of quantum spin systems, AIP Conf. Proc. 1297, 135 (2010), doi:10.1063/1.3518900, 1101.3281

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.3518900 2010

  4. [4]

    QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains

    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), doi:10.21468/SciPostPhys.2.1.003, 1610.03042

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.21468/scipostphys.2.1.003 2017

  5. [5]

    G. Lami, M. Collura and N. Ranabhat, Beginner's lecture notes on quantum spin chains, exact diagonalization and tensor networks (2025), 2503.03564

    work page arXiv 2025

  6. [6]

    XDiag: Exact Diagonalization for Quantum Many-Body Systems

    A. Wietek, L. Staszewski, M. Ulaga, P. L. Ebert, H. Karlsson, S. Sarkar, L. Shackleton, A. Sinha and R. D. Soares, XDiag : Exact diagonalization for quantum many-body systems (2026), 2505.02901

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  7. [7]

    Bethe, Zur theorie der metalle, Zeitschrift f \"u r Physik 71(3), 205 (1931), doi:10.1007/BF01341708

    H. Bethe, Zur theorie der metalle, Zeitschrift f \"u r Physik 71(3), 205 (1931), doi:10.1007/BF01341708

    work page doi:10.1007/bf01341708 1931

  8. [8]

    E. H. Lieb and W. Liniger, Exact analysis of an interacting bose gas. i. the general solution and the ground state, Phys. Rev. 130, 1605 (1963), doi:10.1103/PhysRev.130.1605

    work page doi:10.1103/physrev.130.1605 1963

  9. [9]

    E. H. Lieb, Exact analysis of an interacting bose gas. ii. the excitation spectrum, Phys. Rev. 130, 1616 (1963), doi:10.1103/PhysRev.130.1616

    work page doi:10.1103/physrev.130.1616 1963

  10. [10]

    C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions. i. proof of bethe's hypothesis for ground state in a finite system, Phys. Rev. 150, 321 (1966), doi:10.1103/PhysRev.150.321

    work page doi:10.1103/physrev.150.321 1966

  11. [11]

    C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions. ii. properties of the ground-state energy per lattice site for an infinite system, Phys. Rev. 150, 327 (1966), doi:10.1103/PhysRev.150.327

    work page doi:10.1103/physrev.150.327 1966

  12. [12]

    E. H. Lieb and F. Y. Wu, Absence of mott transition in an exact solution of the short-range, one-band model in one dimension, Phys. Rev. Lett. 20, 1445 (1968), doi:10.1103/PhysRevLett.20.1445

    work page doi:10.1103/physrevlett.20.1445 1968

  13. [13]

    Gaudin, The Bethe Wavefunction, Cambridge University Press, doi:10.1017/CBO9781107053885 (2014)

    M. Gaudin, The Bethe Wavefunction, Cambridge University Press, doi:10.1017/CBO9781107053885 (2014)

    work page doi:10.1017/cbo9781107053885 2014

  14. [14]

    Anderson, Z

    E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide , Society for Industrial and Applied Mathematics, Philadelphia, PA, third edn., doi:10.1137/1.9780898719604 (1999)

    work page doi:10.1137/1.9780898719604 1999

  15. [15]

    Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , J

    C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , J. Res. Natl. Bur. Stand. B 45, 255 (1950), doi:10.6028/jres.045.026

    work page doi:10.6028/jres.045.026 1950

  16. [16]

    Lambin and J

    P. Lambin and J. P. Gaspard, Continued-fraction technique for tight-binding systems. a generalized-moments method, Phys. Rev. B 26, 4356 (1982), doi:10.1103/PhysRevB.26.4356

    work page doi:10.1103/physrevb.26.4356 1982

  17. [17]

    Benoit, G

    C. Benoit, G. Poussigue and A. Assaf, Dynamic and scattering properties of the sierpinski gasket, Journal of Physics: Condensed Matter 4(12), 3153 (1992), doi:10.1088/0953-8984/4/12/011

    work page doi:10.1088/0953-8984/4/12/011 1992

  18. [18]

    Haydock, V

    R. Haydock, V. Heine and M. J. Kelly, Electronic structure based on the local atomic environment for tight-binding bands, Journal of Physics C: Solid State Physics 5(20), 2845 (1972), doi:10.1088/0022-3719/5/20/004

    work page doi:10.1088/0022-3719/5/20/004 1972

  19. [19]

    Aichhorn, M

    M. Aichhorn, M. Daghofer, H. G. Evertz and W. von der Linden, Low-temperature lanczos method for strongly correlated systems, Phys. Rev. B 67, 161103 (2003), doi:10.1103/PhysRevB.67.161103

    work page doi:10.1103/physrevb.67.161103 2003

  20. [20]

    Jakli c c and P

    J. Jakli c c and P. Prelov s s ek, Lanczos method for the calculation of finite-temperature quantities in correlated systems, Phys. Rev. B 49, 5065 (1994), doi:10.1103/PhysRevB.49.5065

    work page doi:10.1103/physrevb.49.5065 1994

  21. [21]

    J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón and D. Sánchez-Portal, The siesta method for ab initio order-n materials simulation, Journal of Physics: Condensed Matter 14(11), 2745 (2002), doi:10.1088/0953-8984/14/11/302

    work page doi:10.1088/0953-8984/14/11/302 2002

  22. [22]

    S. T. Pantelides, The electronic structure of impurities and other point defects in semiconductors, Rev. Mod. Phys. 50, 797 (1978), doi:10.1103/RevModPhys.50.797

    work page doi:10.1103/revmodphys.50.797 1978

  23. [23]

    Jakli c and P

    J. Jakli c and P. Prelov s ek, Finite-temperature properties of doped antiferromagnets, Advances in Physics 49(1), 1 (2000), doi:10.1080/000187300243381, https://doi.org/10.1080/000187300243381

    work page doi:10.1080/000187300243381 2000

  24. [24]

    Dagotto, Correlated electrons in high-temperature superconductors, Rev

    E. Dagotto, Correlated electrons in high-temperature superconductors, Rev. Mod. Phys. 66, 763 (1994), doi:10.1103/RevModPhys.66.763

    work page doi:10.1103/revmodphys.66.763 1994

  25. [25]

    Weiße, G

    A. Weiße, G. Wellein, A. Alvermann and H. Fehske, The kernel polynomial method, Reviews of Modern Physics 78(1), 275–306 (2006), doi:10.1103/revmodphys.78.275

    work page doi:10.1103/revmodphys.78.275 2006

  26. [26]

    J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991), doi:10.1103/PhysRevA.43.2046

    work page doi:10.1103/physreva.43.2046 2046

  27. [27]

    Srednicki,Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994), doi:10.1103/PhysRevE.50.888

    work page doi:10.1103/physreve.50.888 1994

  28. [28]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems , Rev. Mod. Phys. 83, 863 (2011), doi:10.1103/RevModPhys.83.863

    work page doi:10.1103/revmodphys.83.863 2011

  29. [29]

    D’Alessio, Y

    L. D'Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016), doi:10.1080/00018732.2016.1198134

    work page doi:10.1080/00018732.2016.1198134 2016

  30. [30]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015), doi:10.1146/annurev-conmatphys-031214-014726

    work page doi:10.1146/annurev-conmatphys-031214-014726 2015

  31. [31]

    D. A. Abanin, E. Altman, I. Bloch and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement , Rev. Mod. Phys. 91, 021001 (2019), doi:10.1103/RevModPhys.91.021001

    work page doi:10.1103/revmodphys.91.021001 2019

  32. [32]

    Laflorencie, Quantum entanglement in condensed matter systems, Phys

    N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rep. 646, 1 (2016), doi:10.1016/j.physrep.2016.06.008

    work page doi:10.1016/j.physrep.2016.06.008 2016

  33. [33]

    Alet and N

    F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics , Comptes Rendus Physique 19(6), 498 (2018), doi:10.1016/j.crhy.2018.03.003

    work page doi:10.1016/j.crhy.2018.03.003 2018

  34. [34]

    Sierant, M

    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), doi:10.1088/1361-6633/ad9756

    work page doi:10.1088/1361-6633/ad9756 2025

  35. [35]

    Serbyn, D

    M. Serbyn, D. A. Abanin and Z. Papi\' c , Quantum many-body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021), doi:10.1038/s41567-021-01230-2

    work page doi:10.1038/s41567-021-01230-2 2021

  36. [36]

    Chandran, T

    A. Chandran, T. Iadecola, V. Khemani and R. Moessner, Quantum many-body scars: A quasiparticle perspective, Annu. Rev. Condens. Matter Phys. 14, 443 (2023), doi:10.1146/annurev-conmatphys-031620-101617

    work page doi:10.1146/annurev-conmatphys-031620-101617 2023

  37. [37]

    Moudgalya, B

    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), doi:10.1088/1361-6633/ac73a0

    work page doi:10.1088/1361-6633/ac73a0 2022

  38. [38]

    Pietracaprina, N

    F. Pietracaprina, N. Macé, D. J. Luitz and F. Alet, Shift-invert diagonalization of large many-body localizing spin chains, SciPost Physics 5(5) (2018), doi:10.21468/scipostphys.5.5.045

    work page doi:10.21468/scipostphys.5.5.045 2018

  39. [39]

    Sierant, M

    P. Sierant, M. Lewenstein and J. Zakrzewski, Polynomially filtered exact diagonalization approach to many-body localization, Phys. Rev. Lett. 125, 156601 (2020), doi:10.1103/PhysRevLett.125.156601

    work page doi:10.1103/physrevlett.125.156601 2020

  40. [40]

    Saad, Numerical Methods for Large Eigenvalue Problems, SIAM, Philadelphia, revised edn., doi:10.1137/1.9781611970739 (2011)

    Y. Saad, Numerical Methods for Large Eigenvalue Problems, SIAM, Philadelphia, revised edn., doi:10.1137/1.9781611970739 (2011)

    work page doi:10.1137/1.9781611970739 2011

  41. [41]

    Fang and Y

    H.-r. Fang and Y. Saad, A filtered Lanczos procedure for extreme and interior eigenvalue problems , SIAM J. Sci. Comput. 34(4), A2220 (2012), doi:10.1137/110836535

    work page doi:10.1137/110836535 2012

  42. [42]

    Pieper, M

    A. Pieper, M. Kreutzer, M. Galgon, A. Alvermann, H. Fehske, G. Hager, G. Wellein and B. Lang, High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations , J. Comput. Phys. 325, 226 (2016), doi:10.1016/j.jcp.2016.08.027

    work page doi:10.1016/j.jcp.2016.08.027 2016

  43. [43]

    G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, fourth edn., ISBN 9781421407944 (2013)

    work page 2013

  44. [44]

    F. M. A. Avella, ed., Strongly Correlated Systems: Numerical Methods, Springer Berlin Heidelberg, ISBN 9783642351068, doi:10.1007/978-3-642-35106-8 (2013)

    work page doi:10.1007/978-3-642-35106-8 2013

  45. [45]

    D. C. Sorensen, Implicit application of polynomial filters in a k -step Arnoldi method , SIAM J. Matrix Anal. Appl. 13(1), 357 (1992), doi:10.1137/0613025

    work page doi:10.1137/0613025 1992

  46. [46]

    R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods , SIAM, Philadelphia, doi:10.1137/1.9780898719628 (1998)

    work page doi:10.1137/1.9780898719628 1998

  47. [47]

    G. H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues , In J. R. Rice, ed., Mathematical Software III , pp. 361--377. Academic Press, New York, doi:10.1016/B978-0-12-587260-7.50018-2 (1977)

    work page doi:10.1016/b978-0-12-587260-7.50018-2 1977

  48. [48]

    B. N. Parlett, The symmetric eigenvalue problem, SIAM (1998)

    work page 1998

  49. [49]

    B. N. Parlett and D. S. Scott, The Lanczos algorithm with selective orthogonalization , Math. Comp. 33(145), 217 (1979), doi:10.1090/S0025-5718-1979-0514820-3

    work page doi:10.1090/s0025-5718-1979-0514820-3 1979

  50. [50]

    H. D. Simon, Analysis of the symmetric Lanczos algorithm with reorthogonalization methods , Linear Algebra Appl. 61, 101 (1984), doi:10.1016/0024-3795(84)90025-9

    work page doi:10.1016/0024-3795(84)90025-9 1984

  51. [51]

    Saad, On the rates of convergence of the lanczos and the block-lanczos methods, SIAM Journal on Numerical Analysis 17(5), 687 (1980)

    Y. Saad, On the rates of convergence of the lanczos and the block-lanczos methods, SIAM Journal on Numerical Analysis 17(5), 687 (1980)

    work page 1980

  52. [52]

    R. R. Underwood, An iterative block lanczos method for the solution of large sparse symmetric eigenproblems, Tech. Rep. STAN-CS-75-496, Stanford University, Department of Computer Science (1975)

    work page 1975

  53. [53]

    Ericsson and A

    T. Ericsson and A. Ruhe, The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems , Math. Comp. 35(152), 1251 (1980), doi:10.1090/S0025-5718-1980-0583502-2

    work page doi:10.1090/s0025-5718-1980-0583502-2 1980

  54. [54]

    M. W. Long, P. Prelov s s ek, S. El Shawish, J. Karadamoglou and X. Zotos, Finite-temperature dynamical correlations using the microcanonical ensemble and the lanczos algorithm, Phys. Rev. B 68, 235106 (2003), doi:10.1103/PhysRevB.68.235106

    work page doi:10.1103/physrevb.68.235106 2003

  55. [55]

    Ericsson and A

    T. Ericsson and A. Ruhe, The spectral transformation lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems, Mathematics of Computation 35(152), 1251 (1980)

    work page 1980

  56. [56]

    W. E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9(1), 17 (1951), doi:10.1090/qam/42792

    work page doi:10.1090/qam/42792 1951

  57. [57]

    D. J. Luitz, N. Laflorencie and F. Alet, Many-body localization edge in the random-field heisenberg chain, Phys. Rev. B 91, 081103 (2015), doi:10.1103/PhysRevB.91.081103

    work page doi:10.1103/physrevb.91.081103 2015

  58. [58]

    Serbyn, Z

    M. Serbyn, Z. Papi c \' c and D. A. Abanin, Thouless energy and multifractality across the many-body localization transition, Phys. Rev. B 96, 104201 (2017), doi:10.1103/PhysRevB.96.104201

    work page doi:10.1103/physrevb.96.104201 2017

  59. [59]

    Mac\'e, F

    N. Mac\'e, F. Alet and N. Laflorencie, Multifractal scalings across the many-body localization transition, Phys. Rev. Lett. 123, 180601 (2019), doi:10.1103/PhysRevLett.123.180601

    work page doi:10.1103/physrevlett.123.180601 2019

  60. [60]

    S untajs, J

    J. S untajs, J. Bon c a, T. Prosen and L. Vidmar, Ergodicity breaking transition in finite disordered spin chains, Phys. Rev. B 102, 064207 (2020), doi:10.1103/PhysRevB.102.064207

    work page doi:10.1103/physrevb.102.064207 2020

  61. [61]

    Colbois and N

    J. Colbois and N. Laflorencie, Breaking the chains: Extreme value statistics and localization in random spin chains, Phys. Rev. B 108, 144206 (2023), doi:10.1103/PhysRevB.108.144206

    work page doi:10.1103/physrevb.108.144206 2023

  62. [62]

    Colbois, F

    J. Colbois, F. Alet and N. Laflorencie, Interaction-driven instabilities in the random-field xxz chain, Phys. Rev. Lett. 133, 116502 (2024), doi:10.1103/PhysRevLett.133.116502

    work page doi:10.1103/physrevlett.133.116502 2024

  63. [63]

    Laflorencie, J

    N. Laflorencie, J. Colbois and F. Alet, Cat states carrying long-range correlations in the many-body localized phase, Phys. Rev. B 112, 224207 (2025), doi:10.1103/gtfr-nblw

    work page doi:10.1103/gtfr-nblw 2025

  64. [64]

    P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl. 23(1), 15 (2001), doi:10.1137/S0895479899358194

    work page doi:10.1137/s0895479899358194 2001

  65. [65]

    Bollh\" o fer and Y

    M. Bollh\" o fer and Y. Notay, JADAMILU : a software code for computing selected eigenvalues of large sparse symmetric matrices , Comput. Phys. Commun. 177(12), 951 (2007), doi:10.1016/j.cpc.2007.08.013

    work page doi:10.1016/j.cpc.2007.08.013 2007

  66. [66]

    Rodriguez, L

    A. Rodriguez, L. J. Vasquez, K. Slevin and R. A. R\" o mer, Critical parameters from a generalized multifractal analysis at the Anderson transition , Phys. Rev. Lett. 105, 046403 (2010), doi:10.1103/PhysRevLett.105.046403

    work page doi:10.1103/physrevlett.105.046403 2010

  67. [67]

    Rodriguez, L

    A. Rodriguez, L. J. Vasquez, K. Slevin and R. A. R\" o mer, Multifractal finite-size scaling and universality at the Anderson transition , Phys. Rev. B 84, 134209 (2011), doi:10.1103/PhysRevB.84.134209

    work page doi:10.1103/physrevb.84.134209 2011

  68. [68]

    SILVER and H

    R. SILVER and H. R\" O DER, Densities of states of mega-dimensional hamiltonian matrices, International Journal of Modern Physics C 05(04), 735 (1994), doi:10.1142/S0129183194000842, https://doi.org/10.1142/S0129183194000842

    work page doi:10.1142/s0129183194000842 1994

  69. [69]

    Silver, H

    R. Silver, H. Roeder, A. Voter and J. Kress, Kernel polynomial approximations for densities of states and spectral functions, J. Comput. Phys. 124(1), 115–130 (1996), doi:10.1006/jcph.1996.0048

    work page doi:10.1006/jcph.1996.0048 1996

  70. [70]

    C. W. Clenshaw, A note on the summation of Chebyshev series , Math. Comp. 9(51), 118 (1955), doi:10.1090/S0025-5718-1955-0071856-0

    work page doi:10.1090/s0025-5718-1955-0071856-0 1955

  71. [71]

    B. N. Parlett, The Symmetric Eigenvalue Problem, Society for Industrial and Applied Mathematics, Philadelphia, PA, doi:10.1137/1.9781611971163 (1998)

    work page doi:10.1137/1.9781611971163 1998

  72. [72]

    I. S. Dhillon and B. N. Parlett, Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices, Linear Algebra and its Applications 387, 1 (2004), doi:10.1016/j.laa.2003.12.025

    work page doi:10.1016/j.laa.2003.12.025 2004

  73. [73]

    J. W. Demmel, O. A. Marques, B. N. Parlett and C. V \"o mel, Performance and accuracy of LAPACK 's symmetric tridiagonal eigensolvers , SIAM Journal on Scientific Computing 30(3), 1508 (2008), doi:10.1137/070688778

    work page doi:10.1137/070688778 2008

  74. [74]

    Sai Aramthottil, A

    A. Sai Aramthottil, A. Emami Kopaei, P. Sierant, L. Vidmar and J. Zakrzewski, False signatures of non-ergodic behavior in disordered quantum many-body systems, New Journal of Physics 27(12), 125001 (2025), doi:10.1088/1367-2630/ae20b3

    work page doi:10.1088/1367-2630/ae20b3 2025

  75. [75]

    P. R. N. Falcão, J. Zakrzewski and P. Sierant, Fermionic magic resources in disordered quantum spin chains, doi:10.48550/arXiv.2602.00245 (2026), 2602.00245

    work page doi:10.48550/arxiv.2602.00245 2026

  76. [76]

    A. S. Aramthottil, T. Chanda, P. Sierant and J. Zakrzewski, Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains, Phys. Rev. B 104, 214201 (2021), doi:10.1103/PhysRevB.104.214201

    work page doi:10.1103/physrevb.104.214201 2021

  77. [77]

    P. R. N. Falc\ ao, A. S. Aramthottil, P. Sierant and J. Zakrzewski, Many-body localization crossover is sharper in a quasiperiodic potential, Phys. Rev. B 110, 184209 (2024), doi:10.1103/PhysRevB.110.184209

    work page doi:10.1103/physrevb.110.184209 2024

  78. [78]

    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), doi:10.1103/PhysRevLett.133.196302

    work page doi:10.1103/physrevlett.133.196302 2024

  79. [79]

    Prodius, A

    M. Prodius, A. S. Aramthottil and J. Zakrzewski, Interplay of localization and topology in disordered dimerized arrays of rydberg atoms, Phys. Rev. B 113, 064205 (2026), doi:10.1103/lx2h-w9fl

    work page doi:10.1103/lx2h-w9fl 2026

  80. [80]

    Turkeshi, A

    X. Turkeshi, A. Dymarsky and P. Sierant, Pauli spectrum and nonstabilizerness of typical quantum many-body states, Phys. Rev. B 111, 054301 (2025), doi:10.1103/PhysRevB.111.054301

    work page doi:10.1103/physrevb.111.054301 2025

Showing first 80 references.