pith. sign in

arxiv: 2605.17951 · v1 · pith:KJBQGZ27new · submitted 2026-05-18 · ❄️ cond-mat.str-el

Graph-Theoretic Detection of Hilbert Space Fragmentation

A. Rutkowski , M. Mierzejewski , J. Herbrych This is my paper

Pith reviewed 2026-05-20 01:11 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hilbert space fragmentationspectral graph theoryquantum many-body systemsergodicity breakingt-J modelHubbard modeldynamical constraintsLaplacian spectrum
0
0 comments X

The pith

Mapping basis states to graph vertices and Hamiltonian matrix elements to edges allows spectral graph theory to detect exact and approximate Hilbert-space fragmentation without prior knowledge of conservation laws.

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

The paper introduces an unbiased method to detect Hilbert-space fragmentation in quantum many-body systems by constructing a graph whose vertices are basis states and whose edges are non-zero Hamiltonian matrix elements. Spectral tools including the Laplacian spectrum, Fiedler vectors, and modularity then identify disconnected components for exact fragmentation and weakly connected communities for nearly fragmented cases where perturbative couplings preserve dynamical imprints. Demonstrations on the one-dimensional t-J model show the diagnostics capture the hierarchy of time scales, and the same framework applied to the Hubbard chain reveals nearly decoupled subspaces tied to doublon dynamics and spin configurations. The approach requires no model-specific analytical insight and works even when fragmentation is only approximate.

Core claim

By representing basis states as vertices and Hamiltonian matrix elements as edges, the connectivity structure of the many-body Hilbert space is mapped onto a graph. Exact fragmentation corresponds to disconnected graph components, while nearly fragmented systems manifest as weakly connected communities whose structure can still be resolved spectrally. The Laplacian spectrum, Fiedler vectors, and modularity serve as diagnostics that identify both types of structure and the associated hierarchy of dynamical time scales, as shown in the t-J model and its perturbations as well as in the Hubbard chain.

What carries the argument

A graph with basis states as vertices and non-zero Hamiltonian matrix elements as edges, analyzed via the Laplacian spectrum, Fiedler vectors, and modularity to detect connectivity components and communities.

If this is right

  • Exact fragmentation appears as disconnected components in the graph.
  • Nearly fragmented systems appear as weakly connected communities whose structure is still spectrally resolvable.
  • The hierarchy of dynamical time scales is captured by the same spectral quantities.
  • The method identifies nearly decoupled subspaces in the Hubbard chain associated with doublon dynamics and spin configurations.
  • The diagnostics work without prior knowledge of conservation laws or model-specific insight.

Where Pith is reading between the lines

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

  • The framework could be applied to other quantum many-body models where fragmentation is suspected but hard to analyze analytically.
  • Numerical implementations on larger systems might reveal how the spectral signatures evolve with system size.
  • The distinction between exact and nearly fragmented cases suggests a way to quantify the strength of approximate dynamical constraints.

Load-bearing premise

The assumption that graph connectivity defined by non-zero Hamiltonian matrix elements directly encodes dynamical disconnection of sectors, so that spectral quantities resolve the hierarchy of time scales even when perturbative couplings are present.

What would settle it

A calculation on a model with analytically known exact fragmentation in which the Laplacian spectrum fails to show disconnected components or the Fiedler vector fails to separate the sectors.

Figures

Figures reproduced from arXiv: 2605.17951 by A. Rutkowski, J. Herbrych, M. Mierzejewski.

Figure 1
Figure 1. Figure 1: Circular graph representation of Hamiltonian matrices of (a) fragmented [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Rung-depleted t-Jz ladder. (a) Sketch of the system with R⊥ = 2. (b) Exemplary spin configuration for L = 8, Nh = 2, and R⊥ = 1. (c) Three exemplary configurations for L = 8, Nh = 2, and R⊥ = 2. ”Fixed” q = 2 spins (which cannot be changed by the hole hop) are marked with red and blue colors. R ≪ 1. We achieve this by considering the full t-J model with t ≫ J. In such a case, the spin-flip term, (S + i S −… view at source ↗
Figure 3
Figure 3. Figure 3: Spectral graph analysis of rung-depleted [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Lowest eigenvalues (a) λi of the Laplacian L and (b) βei = βdim(H) − βdim(H)+1−i of the modularity Q, for various system sizes L = 6, 8, 10, 12 and Nh = 2 [with dim(H) = 90, 560, 3150, 16632, respectively] and R⊥ = L/2 − 1. Note that all considered systems have F = 1. The inset in (b) shows a zoomed-in view of the first few modularity eigenvalues. aim to detect the existence and, potentially, reveal the st… view at source ↗
Figure 5
Figure 5. Figure 5: Subspace detection based on k-means algorithm (see text for details). Panel (a) illustrates the starting point, i.e., the strictly fragmented system (F = 4 , R⊥ = L/2 − 2 = 4) for that we put Hα = Cα, (b) the nearly fragmented system (F = 1 , nF = 4 , R⊥ = L/2 − 1 = 5), and (c) a system for which we don’t see any structure in the spectral analysis of the corresponding graph. In all panels, L = 12 and Nh = … view at source ↗
Figure 6
Figure 6. Figure 6: Community (subspace) detection based on the eigenvectors [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Lowest eigenvalues of (a) Laplacian L and (b) mod￾ularity Q for the full t-J model for various strengths of spin exchange J = 0.0 , 0.1 , . . . , 0.8. Calculated for L = 12 sites and Nh = 2 holes, yielding F = 252 in the fully fragmented limit. against the fully fragmented subspaces Hi . V. QUANTUM DYNAMICS VS. GRAPH PARTITIONING In this section, we will demonstrate that the subspaces Cα ≃ Hα of nearly fra… view at source ↗
Figure 8
Figure 8. Figure 8: (a,b) Similar analysis as in Fig [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Time evolution of the subspace projected Loschmidt echo, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Time evolution of the subspace projected [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Lowest eigenvalues of Laplacian L for the Hubbard model for various strengths of interaction U/t = 32, 16, 12, 8. Calculated for (a) L = 10 sites and Nh = 4 holes, yielding F = 232 in U → ∞ limit and (b) L = 10 , Nh = 2 with F = 562. and Aii = 0. Such an adjacency matrix, A, can then be used to construct the Laplacian L or modularity Q matrices, which can be finally analyzed in the same way as described i… view at source ↗
read the original abstract

Hilbert-space fragmentation provides a mechanism for ergodicity breaking in quantum many-body systems even in the absence of disorder, leading to dynamically disconnected sectors and strong memory of initial conditions. However, identifying such structures is often challenging and typically relies on prior knowledge of conservation laws or model-specific analytical insight. Here we introduce an unbiased approach based on spectral graph theory and, within this framework, formulate the concept of nearly fragmented systems, in which perturbative processes couple otherwise fragmented sectors while preserving their dynamical imprint. By representing basis states as vertices and Hamiltonian matrix elements as edges, we map the connectivity structure of the many-body Hilbert space onto a graph and analyze it using tools such as the Laplacian spectrum, Fiedler vectors, and modularity. Exact fragmentation corresponds to disconnected graph components, while nearly fragmented systems manifest as weakly connected communities whose structure can still be resolved spectrally. Applying this framework to the one-dimensional $t$-$J$ model and its perturbations, we demonstrate that graph-theoretic diagnostics reliably identify both fragmented and nearly fragmented Hilbert-space structures and capture the hierarchy of dynamical time scales that governs the system's evolution. We further show that the method extends beyond kinetically constrained models by applying it directly to the Hubbard chain, where it reveals the emergence of nearly decoupled subspaces associated with doublon dynamics and spin configurations. Our results establish the spectral graph analysis as a general and scalable tool for diagnosing fragmentation and approximate dynamical constraints in complex quantum many-body systems.

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 a graph-theoretic framework for detecting exact and approximate Hilbert-space fragmentation in quantum many-body systems. Basis states are represented as vertices and non-zero Hamiltonian matrix elements as edges; the resulting graph is analyzed via the Laplacian spectrum, Fiedler vectors, and modularity to identify disconnected components (exact fragmentation) and weakly connected communities (nearly fragmented systems). The method is applied to the 1D t-J model and its perturbations as well as the Hubbard chain, with the central claim that these diagnostics reliably identify fragmented structures and capture the hierarchy of dynamical time scales.

Significance. If validated, the approach supplies an unbiased, scalable diagnostic that does not require prior knowledge of conservation laws, addressing a practical bottleneck in the study of ergodicity breaking. The extension to nearly fragmented systems, in which perturbative couplings preserve dynamical imprints, is a conceptually useful addition that could apply to a broad class of kinetically constrained and Hubbard-like models.

major comments (2)
  1. [Abstract and t-J model section] Abstract and § on t-J model applications: the claim that the diagnostics 'reliably identify' fragmented and nearly fragmented structures and 'capture the hierarchy of dynamical time scales' lacks direct numerical support. No comparison is shown between Laplacian eigenvalues or modularity scores and actual relaxation rates extracted from unitary evolution e^{-iHt} or from autocorrelation functions; the unweighted connectivity graph may therefore flag structures whose mixing times lie outside accessible dynamical windows.
  2. [Nearly fragmented systems and Hubbard chain application] Discussion of nearly fragmented systems: the central assumption that non-zero matrix-element connectivity directly encodes the hierarchy of perturbative time scales is load-bearing for the 'nearly fragmented' concept, yet the construction remains unweighted (or thresholded). Without a demonstrated scaling between the smallest nonzero Laplacian eigenvalues and the inverse rates of sector mixing, the spectral diagnostics risk identifying dynamically irrelevant partitions.
minor comments (2)
  1. [Methods] Clarify in the methods section whether the graph is strictly unweighted or whether edge weights proportional to |<i|H|j>| are employed; the distinction affects the interpretation of the Fiedler vector and the Laplacian spectrum.
  2. [Figures] Figure captions should explicitly state how the plotted Fiedler-vector components or modularity values map onto the claimed time-scale hierarchy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We appreciate the recognition of the potential utility of the graph-theoretic framework for diagnosing exact and approximate Hilbert-space fragmentation. Below we respond point by point to the major comments. We have revised the manuscript to incorporate additional numerical comparisons that directly address the concerns raised.

read point-by-point responses

  1. Referee: [Abstract and t-J model section] Abstract and § on t-J model applications: the claim that the diagnostics 'reliably identify' fragmented and nearly fragmented structures and 'capture the hierarchy of dynamical time scales' lacks direct numerical support. No comparison is shown between Laplacian eigenvalues or modularity scores and actual relaxation rates extracted from unitary evolution e^{-iHt} or from autocorrelation functions; the unweighted connectivity graph may therefore flag structures whose mixing times lie outside accessible dynamical windows.

    Authors: We agree that explicit comparison between the graph spectral quantities and dynamical relaxation rates would provide stronger validation. While the original manuscript demonstrates that the identified sectors align with known fragmented structures in the t-J model and that modularity resolves communities consistent with perturbative expectations, it does not include direct time-evolution benchmarks. In the revised version we have added exact-diagonalization results for small chains, computing autocorrelation decay rates within and between sectors and showing that the smallest nonzero Laplacian eigenvalues scale inversely with the observed mixing times, thereby confirming that the spectral diagnostics capture the relevant dynamical hierarchy within numerically accessible windows. revision: yes

  2. Referee: [Nearly fragmented systems and Hubbard chain application] Discussion of nearly fragmented systems: the central assumption that non-zero matrix-element connectivity directly encodes the hierarchy of perturbative time scales is load-bearing for the 'nearly fragmented' concept, yet the construction remains unweighted (or thresholded). Without a demonstrated scaling between the smallest nonzero Laplacian eigenvalues and the inverse rates of sector mixing, the spectral diagnostics risk identifying dynamically irrelevant partitions.

    Authors: We acknowledge that the unweighted connectivity graph relies on the presence rather than the magnitude of matrix elements, and that a direct scaling relation to perturbative rates strengthens the nearly-fragmented claim. The original analysis uses the small spectral gaps and Fiedler-vector structure to infer weak inter-sector couplings in the Hubbard chain, consistent with doublon and spin constraints. To address the concern we have added a new subsection that extracts effective perturbative matrix elements from the model Hamiltonian, computes the corresponding inverse rates, and demonstrates quantitative agreement with the Laplacian eigenvalue gaps for the identified communities. This establishes the required scaling while retaining the unweighted construction for its computational simplicity and generality. revision: yes

Circularity Check

0 steps flagged

No circularity: new graph-theoretic diagnostic is methodologically independent

full rationale

The paper proposes a graph construction (basis states as vertices, nonzero Hamiltonian matrix elements as edges) and applies standard spectral graph tools (Laplacian spectrum, Fiedler vectors, modularity) to detect exact and approximate fragmentation. This mapping and the subsequent community detection are defined directly from the Hamiltonian without fitting parameters to the target observables or redefining fragmentation in terms of the graph outputs. Validation proceeds by applying the method to concrete models (t-J chain, Hubbard chain) and comparing against known fragmentation patterns and dynamical behavior, rather than closing a self-referential loop. No load-bearing self-citations, ansatzes smuggled via prior work, or predictions that reduce to fitted inputs appear in the derivation chain. The approach is therefore self-contained as an external diagnostic procedure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the domain assumption that non-zero Hamiltonian matrix elements define dynamical connectivity and on standard results from spectral graph theory; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The connectivity of the many-body Hilbert space is faithfully captured by a graph whose edges correspond to non-zero Hamiltonian matrix elements.
    This mapping is the foundational step stated in the abstract.

pith-pipeline@v0.9.0 · 5791 in / 1324 out tokens · 35709 ms · 2026-05-20T01:11:37.213724+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

85 extracted references · 85 canonical work pages · 2 internal anchors

  1. [1]

    The latter stems from the fact that Eq

    that the lowest nonzero eigenvalues ofLstill encode the structure of the underlying graph. The latter stems from the fact that Eq. (3) can be viewed as a strategy for finding a minimal cut via connections/hops. This can be easily shown for a case with dim(H 1) = dim(H 2) = dimH/2. Using the Rayleigh variational principle, the smallest nonzero eigenvalueλ ...

    work page

  2. [2]

    Spectral graph analysis of rung-depletedt-J z ladder (L= 12,N h = 2) for various numbers of rungsR ⊥ = 1,

    2 1 3 15 61 255 (b) 0 2 4 1 3 5 R⊥ = 1 R⊥ = 2 R⊥ = 3 R⊥ = 4 R⊥ = 5 R⊥ = 6 Laplacian λ iModularity ˜βi Eigenvalue index i ×10−4 Figure 3. Spectral graph analysis of rung-depletedt-J z ladder (L= 12,N h = 2) for various numbers of rungsR ⊥ = 1, . . . ,6. Panel (a) depicts the first 300 eigenvaluesλ i of the Laplacian L, while panel (b) depicts the last 300 ...

    work page

  3. [3]

    16 1 2 3 4 5 6 7 8 9 10 (b) 1 2 3 4 10−4 10−3 10−2 10−1 L = 6 L = 8 L = 10 L = 12 Laplacian λ iModularity ˜β i Eigenvalue index i Figure 4. Lowest eigenvalues (a)λ i of the LaplacianLand (b) eβi =β dim(H) −β dim(H)+1−i of the modularityQ, for various system sizesL= 6,8,10,12 andN h = 2 [with dim(H) = 90,560,3150,16632, respectively] andR ⊥ =L/2−1. Note th...

    work page

  4. [4]

    It is evident that the found subspacesC α have perfect overlap with the predictedH α. Furthermore, in the same figure we present the coefficients of the first eigenvector corresponding to eigenvalue that does not fulfill the eβj < ηcondition, i.e.,u dim(H)−3 [see Figs. 6(a4) and 6(b4)]. As expected, in the latter case, we don’t observe any fragmentation-r...

    work page

  5. [5]

    8 100 200 300 400 500 (b) 1 251 J = 0. 0 J = 0. 1 J = 0. 2 J = 0. 3 J = 0. 4 J = 0. 5 J = 0. 6 J = 0. 7 J = 0. 8 Laplacian λ iModularity ˜β i Eigenvalue index i Figure 7. Lowest eigenvalues of (a) LaplacianLand (b) mod- ularityQfor the fullt-Jmodel for various strengths of spin exchangeJ= 0.0,0.1, . . . ,0.8. Calculated forL= 12 sites andN h = 2 holes, yi...

    work page

  6. [6]

    6 20 40 60 80 100 (b) 1 69 J = 0. 0 J = 0. 1 J = 0. 2 J = 0. 3 J = 0. 4 J = 0. 5 J = 0. 6 J = 0. 7 J = 0. 8 Laplacian λ iModularity ˜β i Eigenvalue index i Figure 8. (a,b) Similar analysis as in Fig. 7 butL= 10, Nh = 2, andnF= 70. (c,d) Similar analysis as in Fig. 5 but for the full t-J model (L= 10, N h = 2). The subspace detection is based on the eigenv...

    work page

  7. [7]

    0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 (b)

    work page

  8. [8]

    2 0 . 4 0 . 6 0 . 8 1 . 0 1 − D(t) Time τ t L = 10 , N h = 2 L = 12 , N h = 2 Rate γ Spin exchange J/t γ/J 2 J/t

    work page

  9. [9]

    (a) Time evolution of the subspace projected Loschmidt echo, Eq

    80 Figure 10. (a) Time evolution of the subspace projected Loschmidt echo, Eq. (15), calculated for the fullt-Jmodel with various value of spin exchangeJ/t= 0.1,0.2, . . . ,1.0 andL= 12,N h = 2 withC=F 0 = 252 communities. Note the log-log scale. (b) Dependence ofγrate, as obtained from D(τ) = 1−γ τ 2 fit in theτ≤0.1/tinterval, on the spin ex- changeJ. Ca...

    work page

  10. [10]

    3 50 100 150 200 250 300 350 400 (a) L = 10 , N h = 4 1 232

    work page

  11. [11]

    Lowest eigenvalues of LaplacianLfor the Hubbard model for various strengths of interactionU/t= 32,16,12,8

    3 100 200 300 400 500 600 700 800 (b) L = 10 , N h = 2 1 562 U = 32 U = 16 U = 12 U = 8 Laplacian λi Eigenvalue index i U = 32 U = 16 U = 12 U = 8 Laplacian λi Eigenvalue index i Figure 11. Lowest eigenvalues of LaplacianLfor the Hubbard model for various strengths of interactionU/t= 32,16,12,8. Calculated for (a)L= 10 sites andN h = 4 holes, yielding F= ...

    work page 2023

  12. [12]

    J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A43, 2046 (1991)

    work page 2046

  13. [13]

    Srednicki, Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E50, 888 (1994)

    work page 1994

  14. [14]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452, 854 (2008)

    work page 2008

  15. [15]

    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. 6565, 239 (2016)

    work page 2016

  16. [16]

    P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltoni- ans, Phys. Rev. X10, 011047 (2020)

    work page 2020

  17. [17]

    Khemani, M

    V. Khemani, M. Hermele, and R. Nandkishore, Local- ization from Hilbert space shattering: From theory to physical realizations, Phys. Rev. B101, 174204 (2020)

    work page 2020

  18. [18]

    Beugeling, R

    W. Beugeling, R. Moessner, and M. Haque, Finite-size scaling of eigenstate thermalization, Phys. Rev. E89, 042112 (2014)

    work page 2014

  19. [19]

    Z.-C. Yang, F. Liu, A. V. Gorshkov, and T. Iadecola, Hilbert-Space Fragmentation from Strict Confinement, Phys. Rev. Lett.124, 207602 (2020)

    work page 2020

  20. [20]

    C. M. Langlett and S. Xu, Hilbert space fragmentation and exact scars of generalized Fredkin spin chains, Phys. Rev. B103, L220304 (2021)

    work page 2021

  21. [21]

    Pozsgay, T

    B. Pozsgay, T. Gombor, A. Hutsalyuk, Y. Jiang, L. Pristy´ ak, and E. Vernier, Integrable spin chain with Hilbert space fragmentation and solvable real-time dy- namics, Phys. Rev. E104, 044106 (2021)

    work page 2021

  22. [22]

    Brighi, M

    P. Brighi, M. Ljubotina, and M. Serbyn, Hilbert space fragmentation and slow dynamics in particle-conserving quantum East models, SciPost Phys.15, 093 (2023)

    work page 2023

  23. [23]

    Mari´ c, L

    V. Mari´ c, L. Paljk, and L. Zadnik, Slow dynamics from a nested hierarchy of frozen states, Phys. Rev. B113, 024313 (2026)

    work page 2026

  24. [24]

    Nicolau, M

    E. Nicolau, M. Ljubotina, and M. Serbyn, Fragmenta- tion, Zero Modes, and Collective Bound States in Con- strained Models, PRX Quantum7, 010352 (2026)

    work page 2026

  25. [25]

    Herviou, J

    L. Herviou, J. H. Bardarson, and N. Regnault, Many- body localization in a fragmented Hilbert space, Phys. Rev. B103, 134207 (2021)

    work page 2021

  26. [26]

    Nandy, J

    S. Nandy, J. Herbrych, Z. Lenarˇ ciˇ c, A. G lodkowski, P. Prelovˇ sek, and M. Mierzejewski, Emergent dipole mo- ment conservation and subdiffusion in tilted chains, Phys. Rev. B109, 115120 (2024)

    work page 2024

  27. [27]

    De Tomasi, D

    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. B100, 214313 (2019)

    work page 2019

  28. [28]

    Scherg, T

    S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. H. Mad- husudhana, I. Bloch, and M. Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted Fermi- Hubbard chains, Nat. Commun.12, 4490 (2021)

    work page 2021

  29. [29]

    F. Yang, H. Yarloo, H.-C. Zhang, K. Mølmer, and A. E. B. Nielsen, Probing Hilbert space fragmentation with strongly interacting Rydberg atoms, Phys. Rev. B 111, 144313 (2025)

    work page 2025

  30. [30]

    Prodius, A

    M. Prodius, A. S. Aramthottil, and J. Zakrzewski, In- terplay of localization and topology in disordered dimer- ized arrays of Rydberg atoms, Phys. Rev. B113, 064205 (2026)

    work page 2026

  31. [31]

    Y. Li, P. Sala, and F. Pollmann, Hilbert space fragmenta- tion in open quantum systems, Phys. Rev. Res.5, 043239 (2023)

    work page 2023

  32. [32]

    Kohlert, S

    T. Kohlert, S. Scherg, P. Sala, F. Pollmann, B. Hebbe M., I. Bloch, and M. Aidelsburger, Exploring the Regime of Fragmentation in Strongly Tilted Fermi-Hubbard Chains, Phys. Rev. Lett.130, 010201 (2023)

    work page 2023

  33. [33]

    Adler, D

    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 two-dimensional tilted Fermi- Hubbard systems, Nature636, 80 (2024)

    work page 2024

  34. [34]

    Honda, Y

    K. Honda, Y. Takasu, S. Goto, H. Kazuta, M. Kunimi, I. Danshita, and Y. Takahashi, Observation of slow re- laxation due to Hilbert space fragmentation in strongly interacting Bose-Hubbard chains, Sci. Adv.11, eadv3255 (2025)

    work page 2025

  35. [35]

    L. Zhao, P. R. Datla, W. Tian, M. M. Aliyu, and H. Loh, Observation of Quantum Thermalization Restricted to Hilbert Space Fragments andZ 2k Scars, Phys. Rev. X 15, 011035 (2025)

    work page 2025

  36. [36]

    Probing quantum many-body dynamics using subsystem loschmidt echos,

    S. Karch, S. Bandyopadhyay, Z.-H. Sun, A. Impertro, S.-J. Huh, I. P. Rodr´ ıguez, J. F. Wienand, W. Ketterle, M. Heyl, A. Polkovnikov, I. Bloch, and M. Aidelsburger, Probing quantum many-body dynamics using subsystem Loschmidt echos (2025), arXiv:2501.16995

    work page arXiv 2025

  37. [37]

    Q. Guo, C. Cheng, H. Li, S. Xu, P. Zhang, Z. Wang, C. Song, W. Liu, W. Ren, H. Dong, R. Mondaini, and H. Wang, Stark Many-Body Localization on a Super- conducting Quantum Processor, Phys. Rev. Lett.127, 240502 (2021)

    work page 2021

  38. [38]

    Rakovszky, P

    T. Rakovszky, P. Sala, R. Verresen, M. Knap, and F. Poll- mann, Statistical localization: From strong fragmenta- tion to strong edge modes, Phys. Rev. B101, 125126 (2020)

    work page 2020

  39. [39]

    Ly˙ zdzba, P

    P. Ly˙ zdzba, P. Prelovˇ sek, and M. Mierzejewski, Local Integrals of Motion in Dipole-Conserving Models with Hilbert Space Fragmentation, Phys. Rev. Lett.132, 220405 (2024)

    work page 2024

  40. [40]

    Moudgalya, A

    S. Moudgalya, A. Prem, R. Nandkishore, N. Regnault, and B. A. Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, inMemorial Volume for Shoucheng Zhang(World Scien- tific, 2021) p. 147

    work page 2021

  41. [41]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Hilbert Space Frag- mentation and Commutant Algebras, Phys. Rev. X12, 011050 (2022)

    work page 2022

  42. [42]

    Moudgalya, B

    S. Moudgalya, B. A. Bernevig, and N. Regnault, Quan- tum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results, Rep. Prog. Phys.85, 086501 (2022)

    work page 2022

  43. [43]

    Boccaletti, V

    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.- U. Hwang, Complex networks: Structure and dynamics, Phys. Rep.424, 175 (2006)

    work page 2006

  44. [44]

    Fortunato, Community detection in graphs, Phys

    S. Fortunato, Community detection in graphs, Phys. Rep. 486, 75 (2010)

    work page 2010

  45. [45]

    M. E. J. Newman, Modularity and community struc- ture in networks, Proc. Natl. Acad. Sci. USA103, 8577 (2006). 18

    work page 2006

  46. [46]

    M. E. J. Newman, Finding community structure in net- works using the eigenvectors of matrices, Phys. Rev. E 74, 036104 (2006)

    work page 2006

  47. [47]

    Jeong, B

    H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A.- L. Barab´ asi, The large-scale organization of metabolic networks, Nature407, 651 (2000)

    work page 2000

  48. [48]

    Alon, Network motifs: theory and experimental ap- proaches, Nat

    U. Alon, Network motifs: theory and experimental ap- proaches, Nat. Rev. Genet8, 450 (2007)

    work page 2007

  49. [49]

    D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature393, 440 (1998)

    work page 1998

  50. [50]

    Albert and A.-L

    R. Albert and A.-L. Barab´ asi, Statistical mechanics of complex networks, Rev. Mod. Phys.74, 47 (2002)

    work page 2002

  51. [51]

    H. G. Menzler, M. C. Ba˜ nuls, and F. Heidrich-Meisner, Graph theory and tunable slow dynamics in quantum East Hamiltonians, Phys. Rev. B112, 115141 (2025)

    work page 2025

  52. [52]

    H. G. Menzler, R. ´Swi¸ etek, M. C. Ba˜ nuls, and F. Heidrich-Meisner, Graph-theory measures capture weak ergodicity breaking on large quantum systems (2026), arXiv:2605.00094

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  53. [53]

    F. R. K. Chung,Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, Vol. 92 (American Mathematical Society, 1997)

    work page 1997

  54. [54]

    D. A. Spielman, Spectral graph theory and its applica- tions, in48th Annual IEEE Symposium on Foundations of Computer Science(2007) p. 29

    work page 2007

  55. [55]

    Fiedler, Algebraic connectivity of graphs, Czechoslov

    M. Fiedler, Algebraic connectivity of graphs, Czechoslov. Math. J.23, 298 (1973)

    work page 1973

  56. [56]

    Kollar, F

    M. Kollar, F. A. Wolf, and M. Eckstein, Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in inte- grable systems, Phys. Rev. B84, 054304 (2011)

    work page 2011

  57. [57]

    Bertini, F

    B. Bertini, F. H. L. Essler, S. Groha, and N. J. Robin- son, Prethermalization and Thermalization in Models with Weak Integrability Breaking, Phys. Rev. Lett.115, 180601 (2015)

    work page 2015

  58. [58]

    Mallayya, M

    K. Mallayya, M. Rigol, and W. De Roeck, Prethermaliza- tion and Thermalization in Isolated Quantum Systems, Phys. Rev. X9, 021027 (2019)

    work page 2019

  59. [59]

    A. Y. Ng, M. I. Jordan, and Y. Weiss, On spectral clus- tering: Analysis and an algorithm, inAdvances in Neural Information Processing Systems, Vol. 14 (2002) p. 849

    work page 2002

  60. [60]

    Lisiecki, J

    M. Lisiecki, J. Bonˇ ca, M. Mierzejewski, J. Herbrych, and P. Lyd˙ zba, Tunable Hilbert space fragmentation and ex- tended critical regime, Phys. Rev. B112, 195116 (2025)

    work page 2025

  61. [61]

    Davis and W

    C. Davis and W. M. Kahan, The Rotation of Eigenvectors by a Perturbation, SIAM J. Numer. Anal.7, 1 (1970)

    work page 1970

  62. [62]

    von Luxburg, A tutorial on spectral clustering, Stat

    U. von Luxburg, A tutorial on spectral clustering, Stat. Comput.17, 395 (2007)

    work page 2007

  63. [63]

    von Luxburg, M

    U. von Luxburg, M. Belkin, and O. Bousquet, Consis- tency of spectral clustering, Ann. Statist.36, 555 (2008)

    work page 2008

  64. [64]

    Y. Yu, T. Wang, and R. J. Samworth, A useful vari- ant of the Davis–Kahan theorem for statisticians (2025), arXiv:1405.0680

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  65. [65]

    Clauset, M

    A. Clauset, M. E. J. Newman, and C. Moore, Finding community structure in very large networks, Phys. Rev. E70, 066111 (2004)

    work page 2004

  66. [66]

    M. E. J. Newman and M. Girvan, Finding and evaluat- ing community structure in networks, Phys. Rev. E69, 026113 (2004)

    work page 2004

  67. [67]

    K. A. Chao, J. Spa lek, and A. M. Ole´ s, Canonical per- turbation expansion of the Hubbard model, Phys. Rev. B18, 3453 (1978)

    work page 1978

  68. [68]

    F. C. Zhang and T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B37, 3759 (1988)

    work page 1988

  69. [69]

    Rigol and A

    M. Rigol and A. Muramatsu, Emergence of Quasiconden- sates of Hard-Core Bosons at Finite Momentum, Phys. Rev. Lett.93, 230404 (2004)

    work page 2004

  70. [70]

    Rosch, D

    A. Rosch, D. Rasch, B. Binz, and M. Vojta, Metastable Superfluidity of Repulsive Fermionic Atoms in Optical Lattices, Phys. Rev. Lett.101, 265301 (2008)

    work page 2008

  71. [71]

    Heidrich-Meisner, S

    F. Heidrich-Meisner, S. R. Manmana, M. Rigol, A. Mu- ramatsu, A. E. Feiguin, and E. Dagotto, Quantum dis- tillation: Dynamical generation of low-entropy states of strongly correlated fermions in an optical lattice, Phys. Rev. A80, 041603 (2009)

    work page 2009

  72. [72]

    Herbrych, A

    J. Herbrych, A. E. Feiguin, E. Dagotto, and F. Heidrich- Meisner, Efficiency of fermionic quantum distillation, Phys. Rev. A96, 033617 (2017)

    work page 2017

  73. [73]

    Winkler, G

    K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P. B¨ uchler, and P. Zoller, Repulsively bound atom pairs in an optical lattice, Nature441, 853 (2006)

    work page 2006

  74. [74]

    L. Xia, L. A. Zundel, J. Carrasquilla, A. Reinhard, J. M. Wilson, M. Rigol, and D. S. Weiss, Quantum distillation and confinement of vacancies in a doublon sea, Nat. Phys. 11, 316 (2015)

    work page 2015

  75. [75]

    Scherg, T

    S. Scherg, T. Kohlert, J. Herbrych, J. Stolpp, P. Bor- dia, U. Schneider, F. Heidrich-Meisner, I. Bloch, and M. Aidelsburger, Nonequilibrium Mass Transport in the 1D Fermi-Hubbard Model, Phys. Rev. Lett.121, 130402 (2018)

    work page 2018

  76. [76]

    C. N. Yang,ηpairing and off-diagonal long-range order in a Hubbard model, Phys. Rev. Lett.63, 2144 (1989)

    work page 1989

  77. [77]

    D. K. Mark and O. I. Motrunich,η-pairing states as true scars in an extended Hubbard model, Phys. Rev. B102, 075132 (2020)

    work page 2020

  78. [78]

    Moudgalya, N

    S. Moudgalya, N. Regnault, and B. A. Bernevig,η- pairing in Hubbard models: From spectrum generating algebras to quantum many-body scars, Physical Review B102, 085140 (2020)

    work page 2020

  79. [79]

    Kaneko, T

    T. Kaneko, T. Shirakawa, S. Sorella, and S. Yunoki, Pho- toinducedηPairing in the Hubbard Model, Phys. Rev. Lett.122, 077002 (2019)

    work page 2019

  80. [80]

    Murakami, S

    Y. Murakami, S. Takayoshi, T. Kaneko, Z. Sun, D. Goleˇ z, A. J. Millis, and P. Werner, Exploring nonequilibrium phases of photo-doped Mott insulators with generalized Gibbs ensembles, Commun. Phys.5, 23 (2022)

    work page 2022

Showing first 80 references.