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The paper claims that the many-body localization transition in the random-field XXZ chain is governed by rare, system-wide resonances in Hilbert space, and that even infinitesimal interactions destabilize the Anderson insulator at finite di

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A glass-theory freezing construction weights rare long-range resonances in the random-field XXZ chain, yielding a finite-size phase diagram with an ergodic phase, a rare-resonance-driven intermediate regime, and a robust MBL phase, plus apparent Anderson-insulator instability at infinitesimal intera

T0 review reviewed 2026-08-04 challenge →

load-bearing objection Extends a promising large-deviation method to the XXZ chain and maps three regimes, but the small-Δ delocalization claim is an upper-bound inference from an unregularized proxy, not a demonstration. the 3 major comments →

arxiv 2510.18545 v3 pith:456EO4QP submitted 2025-10-21 cond-mat.dis-nn cond-mat.stat-mechcond-mat.str-elquant-ph

Large deviations in the many-body localization transition: The case of the random-field XXZ chain

Greivin Alfaro Miranda , Fabien Alet , Giulio Biroli , Leticia F. Cugliandolo , Nicolas Laflorencie , Marco Tarzia This is my paper
classification cond-mat.dis-nn cond-mat.stat-mechcond-mat.str-elquant-ph
keywords many-body localizationrare resonancesHilbert space graphlarge deviationsAnderson insulatorrandom-field XXZ chainLandauer transmissionfreezing transition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper tries to show that the stability of the many-body localized (MBL) phase is controlled by rare, system-wide resonances in Hilbert space, and that these can be counted systematically by treating the sum of squared propagators between distant many-body states as a partition function of a glassy model. Using a Lagrange multiplier that acts like an effective temperature, the authors identify three regimes in the random-field XXZ chain: an ergodic phase, an intermediate phase where rare long-range resonances drive delocalization, and a stable MBL phase where such resonances cannot kill localization. Their central quantitative claim is that even an infinitesimal interaction strength destroys the Anderson insulator at finite disorder, with both MBL transition lines staying finite as Δ→0. If correct, the 1D Anderson insulator is non-perturbatively unstable to interactions, a conclusion perturbative local-integrals-of-motion constructions would miss. A sympathetic reader would care because this supplies a mechanism—rare resonant paths on the Hilbert-space graph—for the finite-size delocalization seen in many numerics, and a finite-size phase diagram that matches correlation-based estimates.

Core claim

The paper's central claim is that the statistics of off-diagonal resolvent elements between basis states at the 'equator' of Hilbert space, analyzed through a directed-polymer-in-random-media-type freezing construction, reveal three distinct regimes in the random-field XXZ chain. In the intermediate regime, delocalization from a random initial state is governed not by typical matrix elements but by rare outliers in the heavy-tailed distribution of |G0f|²—outliers that correspond to system-wide resonances. The authors further claim that in the Anderson basis the critical disorder for both the ergodic-to-intermediate and intermediate-to-MBL transitions remains finite down to the smallest inter

What carries the argument

The central object is the biased Hilbert-space Landauer transmission T0(β) = Σ_{f∈E} |G0f|^β, where G0f = ⟨f|(E−H)^{-1}|0⟩ is the unregularized resolvent between an initial basis state and basis states at the equator (zero overlap). Treating T0 as the partition function of a directed polymer in a random medium, the authors introduce a Lagrange multiplier β—an effective inverse temperature—and compare annealed and quenched free-energies φ_a and φ_q. The location of the minimum of φ_a gives the freezing temperature β*; whether β* lies above or below the physical value β=2, and whether the plateau value φ_a(β*) is positive or negative, decides between ergodic, rare-resonance-delocalized, and ge

Load-bearing premise

Everything rests on the assumption that the rare large outliers in the unregularized sum T0 = Σ|G0f|² are physical many-body resonances, not spurious poles of the resolvent that would disappear if a small imaginary regulator were kept.

What would settle it

Take the same disorder realizations and initial states, compute the properly regularized transmission T_FL with a small but finite imaginary part η (or the exact P_E from full diagonalization), and check whether the anomalies that drive the freezing analysis survive in the η→0 limit. If the rare outliers in T0 vanish under regularization, the intermediate rare-resonance regime and the finite-W delocalization at small Δ are overestimates.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim is right, many-body localization is stable only at disorder strong enough that even the rare resonant outliers of T0 decay exponentially; this defines an upper-bound MBL threshold.
  • The 1D Anderson insulator is non-perturbatively unstable to arbitrarily weak interactions at finite disorder, so perturbative expansions around non-interacting l-bits will fail to capture the leading delocalization mechanism.
  • The intermediate 'delocalization via rare resonances' region shrinks with system size, consistent with either a finite-size crossover or a genuine non-ergodic delocalized phase; the paper's finite-size data favor a crossover to a direct transition.
  • Eigenstates in the rare-resonance regime are highly heterogeneous: typical and rare disorder realizations differ strongly in wavefunction decay and transmission-path structure, so observables averaged over disorder will mix qualitatively different behaviors.

Where Pith is reading between the lines

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

  • If the unregularized outliers survive proper η regularization, the same freezing machinery could be applied to quasiperiodic (non-random) potentials to test whether the rare resonances come from disorder-strength fluctuations in real space or from the structure of Hilbert space itself.
  • The paper's MBL threshold is explicitly an upper bound; a regularized version of T0 with finite η might push W_MBL upward and shrink the intermediate region—a testable prediction for exact-diagonalization or Krylov studies at L ≈ 20–24.
  • The transmission-path visualization suggests a concrete observable for experiments: disorder realizations hosting cat-like nearly-degenerate eigenstate pairs should also show anomalous long-time imbalance revival, linking Hilbert-space rare events to real-space dynamics.
  • If the intermediate phase is only a prethermal crossover, the robust MBL phase boundary at large L would coincide with the avalanche lower bound; comparing the present W_MBL with thermal-bubble avalanche estimates for the same model is a direct test.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The manuscript studies the random-field XXZ chain at infinite time from a random basis state, using the unregularized Hilbert-space Landauer proxy T0 = Σ_{f∈E}|G0f|² (Eq. 21) and a β-dressed generalization T0(β) = Σ|G0f|^β (Eq. 32). By analogy with directed polymers in random media, the authors identify a freezing temperature β* from the minimum of the annealed free energy (Eq. 33), and classify parameters by β* versus 2 and by the sign of φ_a(β*). This yields a finite-size (W,Δ) phase diagram with ergodic, rare-resonance-delocalized, and MBL regimes, in both spin and Anderson bases. The central physical claim is that weak interactions destroy the Δ=0 Anderson insulator at finite W, with W_MBL and W_typ remaining finite down to Δ≈0.05. The paper also visualizes rare transmission paths on the Hilbert-space graph and contrasts typical versus rare disorder samples.

Significance. If established, the finite critical disorder at small Δ is an important non-perturbative statement, connecting rare Hilbert-space resonances to MBL destabilization and providing a framework for interpreting finite-size drifts. The method has a rigorous anchor on the Bethe lattice (Eqs. 22–25), passes external benchmarks on the random regular graph and the Rosenzweig–Porter model, and the two-basis comparison is an honest internal consistency check. The paper is also explicit about limitations, including the admission that the MBL threshold is an upper bound and that some rare events may be artifacts. These caveats, however, are in tension with the stronger 'demonstrate' language in the abstract.

major comments (3)
  1. [Sec. III.C, Eq. (21); Sec. V] The central claim that arbitrarily small Δ destroys the Anderson insulator at finite W rests on treating rare outliers of T0 as physical resonances. Section III.C states that the unregularized |G0f|² 'can no longer be strictly interpreted as probabilities' and that the poles of G are unregularized; Section V concedes that the MBL threshold is an upper bound and that 'some of the rare events... might be spurious artifacts.' The benchmarks quoted (RRG, Rosenzweig–Porter, transverse-field Ising; Refs. [68,109]) do not test the XXZ model at small Δ. In particular, Eq. (24) is exact only in the linearized/localized regime of the Bethe lattice; in the intermediate region where rare-resonance delocalization is claimed, the linearization fails and the outlier statistics of T0 are exactly what needs validation. Please provide a direct comparison with the regularized P_E (Eq. 16) or T_FL (Eq. 19)
  2. [Sec. III.D.2, Eq. (34)] The plateau-replacement construction φ̃_a(β)=φ_a(β*) for β≥β* assumes that Eq. (21) belongs to the same universality class as directed polymers in random media. Section III.D.2 explicitly calls this 'the key assumption underlying our approach' and notes that the connection is rigorously established only for single-particle Anderson localization on the Bethe lattice. Appendix C provides only qualitative correlation data over a path length of L/4; it does not quantitatively establish the ultrametric correlations required for the freezing construction. Since the W_MBL line is extracted from φ_a(β*), this assumption is load-bearing. A quantitative test—for example, verifying the predicted tail-exponent relation Eq. (36) across sizes and using it to independently predict β*—would make the construction falsifiable and materially strengthen the characterization of W_MBL.
  3. [Sec. IV.B, Sec. V, Abstract] The text states that W_ergo shifts rightward and W_MBL shifts leftward with increasing L, so the intermediate rare-resonance region shrinks; the 'first scenario' (no intermediate phase in the thermodynamic limit) is favored. Yet the abstract and Sec. IV.B assert a discontinuous departure from the Anderson insulator at Δ=0. With L≤22 and with the Anderson basis studied only down to Δ=0.05 (the spin basis is restricted to Δ≥0.25), the data do not exclude W_MBL→0 as Δ→0 or as L→∞. The claim should be reformulated as a finite-size/upper-bound statement, or accompanied by an explicit extrapolation in both L and Δ, before the word 'demonstrate' is used.
minor comments (3)
  1. [Footnote [77]] The footnote contains an unresolved '[?]' placeholder for the reference on the sign of the interaction at high energy. Please complete this citation.
  2. [Sec. V] There is a typographical artifact 'e E[lnP_E]' and 'e E[lnT_0]' where the 'e' appears to be a stray character. Please correct.
  3. [Fig. 17 caption] The caption contains the typo 'discante' for 'distance'. Also, the color/symbol scheme in Fig. 7 is difficult to parse in print; adding explicit symbols for the three phases would improve readability.

Circularity Check

1 steps flagged

Central proxy T0 is validated mainly by co-author self-citations (Ref. [68] and unpublished Ref. [109]); the formal phase-boundary definitions are not circular, so the small-∆ claim is at risk but not forced by construction.

specific steps
  1. self citation load bearing [Sec. III C after Eq. (21); Sec. III D 2; Refs. [68,109]]
    "The second argument supporting the choice of the typical value of T0 as an order parameter for MBL comes from the benchmark analysis presented in Ref. [68]. In that work, some of us employed this quantity ... A final argument in favor of our approximation is provided by direct numerical tests [109] ... we computed the probability to delocalize from a random initial state, using Eqs. (15) and (16) ... and compared it to the Landauer transmission T0 ... their covariance increases with system size and approaches unity in the strong-disorder regime [109]."

    The paper replaces the true delocalization probability PE (Eq. 16) with the unregularized sum T0 = Σ_f |G0f|^2 (Eq. 21), even though Sec. III C concedes that, without the imaginary regulator, |G0f|^2 is unbounded and 'can no longer be strictly interpreted as probabilities.' The equivalence PE ↔ T0 is therefore not a mathematical identity but is asserted to be validated by Ref. [68] (co-authored by Biroli and Tarzia) and by Ref. [109], listed as 'M. Tarzia, In preparation.' The Bethe-lattice argument preceding it is an analogy for tree-like graphs, not a derivation for the correlated Hilbert-space graph. Thus the central observable on which all three regimes and the finite small-∆ transitions rest is load-bearingly supported by self-citations, including an unpublished one, rather than by an

full rationale

I found no place where a predicted quantity is literally equal to a fitted input by construction: W_ergo, W_MBL and W_typ_MBL are defined as crossings of computed free-energy/annealed curves, not fitted to the target phase diagram, and the three regimes are read off those crossings. The paper also contains substantial independent numerical content (exact-diagonalization distributions, eigenstate amplitude analyses, Hilbert-space path visualizations) and cross-checks against Refs. [66,67], which use different real-space observables even though the author lists overlap. The main circularity concern is narrower: the proxy T0, whose outliers are interpreted as physical rare resonances, is justified by the authors' own prior work [68] and by an unpublished manuscript by one of the present authors [109]. Because the strongest abstract claim — that infinitesimal interactions destroy the Anderson insulator — depends on treating rare T0 outliers as genuine resonances, and the authors themselves state in Sec. V that 'some of the rare events we consider crucial for MBL destabilization might be spurious artifacts,' this self-referential validation is load-bearing. However, the concessions are explicit and the formal derivation does not reduce to its inputs by definition, so a score of 4 is appropriate rather than 6 or above.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 1 invented entities

The central boundary lines (W_ergo, W_MBL, W_typ_MBL) are not obtained by fitting any constant to the target phase diagram; they emerge from crossings of free-energy curves. The unseen cost is carried by procedural choices and by two asserted equivalences: (i) the DPRM universality of the unregularized transmission T_0 (Sec. III D 2), and (ii) the fidelity of T_0's rare outliers to genuine physical resonances (Sec. V: 'might be spurious artifacts'). The free parameters listed are hand-chosen numerical windows and the plateau-replacement rule; the invented entity is the auxiliary temperature β, a formal device with no physical referent. There are no fitted global/per-period constants of the kind that would make the phase-diagram agreement trivial.

free parameters (4)
  • Energy-window half-width η for initial-state selection = η = 64 (L ≥ 14), η = 32 (L ≤ 12)
    Chosen by hand in App. A (Eq. (A2)) to define the 'mid-spectrum' ensemble of initial basis states; all subsequent averages depend on this window.
  • Disorder grid spacing around phase-boundary crossings = ΔW = 1.5
    App. B: the W_MBL crossing is found on a grid with spacing ΔW = 1.5; combined with the flat direction in φ_a(β*,W), the crossing location is weakly constrained.
  • Equator-shell interpolation for L not divisible by 4 = average over target shells with q = ±2/L
    App. A: since exact q = 0 targets do not exist for L = 14, 18, 22, T_0 and ln T_0 are averaged between the two nearest shells; this prescription feeds every size trend.
  • Plateau-replacement rule φ̃_a(β) = φ_a(β*) for β ≥ β*
    Eq. (34): the central estimation rule for the large-L typical value of T_0, adopted ad hoc from the DPRM freezing solution (Eq. (30)); its validity for the XXZ model is the key assumption of Sec. III D 2, not a derived result.
axioms (6)
  • ad hoc to paper T_0 (Eq. (21)) belongs to the same universality class as DPRM/mean-field glassy models, so the freezing construction (Eqs. (33)-(35)) estimates the asymptotic typical value of T_0.
    Sec. III D 2: 'The key assumption underlying our approach is that Eq. (21) belongs to the same universality class...' — asserted; rigorous only for Anderson localization on the Bethe lattice; for XXZ it is supported only indirectly by the correlation tests of App. C.
  • domain assumption Ultrametric (DPRM-like) correlations of the weights ln|G_0f|.
    Required for the freezing/plateau logic (Sec. III D 2 cites Refs. [97,99]). App. C measures the connected correlations and finds them increasing with disorder but admits 'the exact behaviour in our case is difficult to assess definitively.'
  • domain assumption The typical value of T_0 and of the true delocalization probability P_E scale identically with L in the localized regime.
    Sec. III C: argued via Bethe-lattice linearization (Eqs. (22)-(25), exact for trees) and benchmarked in Ref. [68] and unpublished Ref. [109]; the proportionality test is not shown for the XXZ model in this paper.
  • domain assumption Eigenstates far in energy from mid-spectrum contribute negligibly to P_0→f, justifying the resolvent at energy E = TrH/N.
    Sec. III C, Eq. (18): the 'energy-resolved' reduction is standard for many-body systems; the dependence on the regulator η→0 is assumed benign.
  • domain assumption Mid-spectrum (infinite-temperature) properties of the zero-magnetization sector with periodic boundary conditions are representative of the thermodynamic limit.
    Sec. II: standard choice; the drift of W_ergo and W_MBL with L (Sec. IV B) shows the representativeness is not yet established at L ≤ 22.
  • standard math Imbrie's theorem and related stability results for the transverse-field Ising model (Refs. [48,55,56]) are correct background.
    Invoked in Sec. I as the rigorous anchor for the existence of an MBL phase in a related model; cited as published peer-reviewed work.
invented entities (1)
  • Auxiliary inverse temperature β (β-dressed transmission T_0(β) = Σ|G_0f|^β) no independent evidence
    purpose: Formal Lagrange multiplier controlling the weight of extreme outliers in the heavy-tailed propagator distribution; formally plays the role of an effective temperature (abstract, Sec. III D 2).
    No physical referent or independent prediction attaches to β; its usefulness is entirely contingent on the DPRM universality assumption. It is a computational device, listed here for completeness — no new physics entity is claimed.

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Cite this review

Pith. "Pith review of Large deviations in the many-body localization transition: The case of the random-field XXZ chain." pith.science (2026). https://pith.science/paper/456EO4QP

@misc{pith2026251018545,
  author       = {Pith},
  title        = {Pith review of: Large deviations in the many-body localization transition: The case of the random-field XXZ chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/456EO4QP}},
  note         = {Machine review of arXiv:2510.18545}
}
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read the original abstract

The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.

Figures

Figures reproduced from arXiv: 2510.18545 by Fabien Alet, Giulio Biroli, Greivin Alfaro Miranda, Leticia F. Cugliandolo, Marco Tarzia, Nicolas Laflorencie.

Figure 1
Figure 1. Figure 1: Probability distribution function for the infinite time probabilities, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Probability distributions of the delocalization proba [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Quantum transport on a network in a scattering [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sketch of the different scaling behavior with [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The annealed free-energy ϕa (dashed), the modified annealed free-energy ϕ˜a (solid) and the quenched free-energy ϕq (solid with triangular markers) in the spin (warm colors) and Anderson (cold colors) bases. Low (left panel), intermediate (middle panel) and large (right panel) disorder strengths. The sizes L are distinguished by the colors of the scale. The dashed gray lines show the relevant values at β =… view at source ↗
Figure 6
Figure 6. Figure 6: (From top to bottom) Disorder strength W depen￾dence of β⋆, ϕa(β⋆), and ϕq(β = 2) for different system sizes, in the spin basis (left panels) and the Anderson basis (right panels). The relevant values β⋆ = 2 and ϕ = 0 are cor￾respondingly indicated with an horizontal dashed-gray line. The crossing of the curves with these lines identify the posi￾tion of Wergo(L), WMBL(L) and Wtyp MBL(L), accordingly. The s… view at source ↗
Figure 7
Figure 7. Figure 7: Phase diagram of the system at the center of the energy spectrum, shown in the ∆– [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Hilbert space graphs for L = 8 shown in the spin basis (left) and the Anderson basis (right). The central vertex represents a random initial condition in the middle of the energy spectrum, that is a basis state of the Hamiltonian. All vertices of the graph are connected with black edges denoting the distance in the Hilbert space graph, given by number of applications of the Hamiltonian. In the spin basis w… view at source ↗
Figure 9
Figure 9. Figure 9: Probability distributions for the Hilbert space Landauer transmission [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Calculation of (a) β⋆ and (b) ϕa(β⋆) as functions of the correlation distance ζ S , for ∆ = 1 and L = 20. The values of the disorder strengths considered are shown in the legend. Horizontal gray dashed lines indicate the reference values β⋆ = 2 and ϕa(β⋆) = 0. These characteristic distances define crossover lines that separate different regimes in the W-ζ plane, as shown in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 11
Figure 11. Figure 11: Relevant regions in the W-ζ S,A plane for spin (left) and Anderson (right) bases, with ∆ = 1. The transition lines to the ergodic region (green) is determined by the condition β⋆(ζ S,A) = 2, while the inaccessible regions (shades of gray) are determined by ϕa(β⋆, ζS,A) = 0, for each system size used. The critical disorder strengths Wergo and WMBL are identified with dashed lines, colored according to thei… view at source ↗
Figure 12
Figure 12. Figure 12: Basis state amplitudes, within a given eigenstate, as a function of the correlation distance [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Scattering geometry to measure the reaction of the [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Rarefaction of paths in the Hilbert space graph (for the spin basis) for [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Inverse participation ratios ( [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The values of β⋆, ϕa(β⋆), and ϕq(β = 2) as a function of the cumulative number of disorder realizations NS over which the average E[· · · ] is taken over. For both spin (top panels) and Anderson (bottom) bases. for NS = 1, 2, . . . , NS = 64. We then consider the second half of this sequence, i.e., from NS = 32 to NS = 64, and compute the error bars as the difference between the maximum and minimum values… view at source ↗
Figure 17
Figure 17. Figure 17: Correlations of the equivalent polymer energy for the [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Calculation of (a) β⋆ and (b) ϕa(β⋆) as functions of the correlation distance ζ S , for ∆ = 1 and L = 20. The values of the disorder strengths considered are shown in the legend. Horizontal gray dashed lines indicate the reference values β⋆ = 2 and ϕa(β⋆) = 0. These lines are used to extract the corresponding characteristic correlation distances [PITH_FULL_IMAGE:figures/full_fig_p032_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Decay of the basis state probability as a function [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.