Interacting Anderson Quantum Sun Model

• The interacting Anderson Quantum Sun model is a disordered many-body spin system defined by a localized spin‐1/2 chain coupled to a small ergodic ‘sun’ via exponentially decaying, U(1)-symmetric XY interactions.
• The model exhibits various regimes, from conventional localized phases with Poisson spectral statistics to unconventional phases featuring volume-law entanglement alongside intermediate spectral behavior.
• Numerical diagnostics of spectral statistics, entanglement scaling, and rare-event correlations provide insights into how the bath induces rare resonances and decouples eigenstate thermalization from spectral chaos.

The interacting Anderson Quantum Sun (AQS) model is a disordered many-body spin system in which a localized spin-$1/2$ chain is coupled to a small, fully ergodic “sun” through exponentially decaying, $U(1)$-symmetric XY interactions. It was introduced to probe how an ergodic inclusion destabilizes Anderson localization and to test avalanche-based scenarios for delocalization. In addition to standard localized and ergodic phases, the model displays two nonstandard regimes: one with volume-law entanglement coexisting with intermediate spectral statistics, and another with Poisson level statistics together with sub-volume entanglement and rare-event-dominated systemwide correlations, interpreted as ergodic instabilities (Pawlik et al., 24 Jul 2025).

1. Model architecture and defining scales

The AQS Hamiltonian decomposes into chain, bath, and coupling terms,

$$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$

The localized chain is a disordered spin-$1/2$ chain with open boundary conditions,

$$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$

where the fields are i.i.d. uniform random variables $h_j \in [1-W,1+W]$ with $W=0.5$, and $J \ge 0$ controls intra-chain hopping. The model preserves the $U(1)$ symmetry $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$ (Pawlik et al., 24 Jul 2025).

The ergodic bath, or “sun,” contains $U(1)$0 spins and is represented by a Gaussian Orthogonal Ensemble random matrix restricted to conserve $U(1)$1,

$U(1)$2

with $U(1)$3, $U(1)$4 i.i.d. standard Gaussian, and all entries that violate $U(1)$5 symmetry set to zero. The chain and bath are connected through exponentially decaying XY couplings,

$U(1)$6

where $U(1)$7 is a randomly chosen sun-spin index, and $U(1)$8 with $U(1)$9 sets the decay profile (Pawlik et al., 24 Jul 2025).

Two intrinsic length scales organize the physics. The first is the sun–chain coupling decay length,

$$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$0

The second is the average Anderson localization length of the noninteracting chain,

$$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$1

Their competition controls whether localization survives, becomes unstable, or is replaced by thermal behavior. This suggests that the adjective “Anderson” refers to the localized-chain sector that seeds the model’s nonergodic behavior (Pawlik et al., 24 Jul 2025).

2. Numerical framework and observables

The main numerical analysis is performed in the largest $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$2 symmetry sector, $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$3 for even $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$4 and $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$5 for odd $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$6. Energies are taken from a mid-spectrum window $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$7 with $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$8, where $$H = H_{\text{chain}} + H_{\text{bath}} + H_{\text{coupling}}.$$9 is rescaled to $1/2$0, and at least $1/2$1 eigenvalues are retained per realization. Full exact diagonalization is used up to $1/2$2, and POLFED extends the calculations to $1/2$3. Disorder averaging ranges from $1/2$4 realizations for $1/2$5 to $1/2$6 for the largest sizes; finite-size critical points $1/2$7 are extrapolated using

$1/2$8

(Pawlik et al., 24 Jul 2025).

Three classes of diagnostics are central. The first is spectral statistics, based on level spacings $1/2$9 and adjacent-gap ratios

$$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$0

The corresponding benchmarks are $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$1 for Poisson and $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$2 for GOE statistics. Intermediate values of $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$3, stable under finite-size extrapolation, are treated as evidence for a genuinely non-GOE spectral regime rather than crossover noise (Pawlik et al., 24 Jul 2025).

The second class is entanglement diagnostics. For a bipartition $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$4 containing the sun plus the $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$5 chain spins closest to the sun, the von Neumann entropy is

$$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$6

It is normalized by the Page entropy $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$7, and its system-size dependence is parameterized as $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$8, with $$H_{\text{chain}} = \sum_{j=1}^L h_j S_j^z + J \sum_{j=1}^{L-1} (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y),$$9 for area law, $h_j \in [1-W,1+W]$0 for sub-volume scaling, and $h_j \in [1-W,1+W]$1 for volume law. Complementary information comes from participation entropies

$h_j \in [1-W,1+W]$2

with scaling $h_j \in [1-W,1+W]$3. In the AQS, $h_j \in [1-W,1+W]$4 in the volume-law phases, while $h_j \in [1-W,1+W]$5 in the localized and sub-volume regimes (Pawlik et al., 24 Jul 2025).

The third class concerns spatial correlations and rare events. Connected correlators are defined by

$h_j \in [1-W,1+W]$6

with typical decay extracted from

$h_j \in [1-W,1+W]$7

A rescaled observable,

$h_j \in [1-W,1+W]$8

is used to expose fat-tailed distributions at small $h_j \in [1-W,1+W]$9, which signal rare-event-dominated correlations. The ratio $W=0.5$0 provides an additional instability indicator: in the AQS sub-volume regime it decreases with $W=0.5$1, unlike the $W=0.5$2 behavior reported for Anderson or strongly disordered XXZ chains (Pawlik et al., 24 Jul 2025).

The study is deliberately static. Thermalization is inferred from ETH-consistent eigenstate diagnostics rather than direct quench dynamics, although supplementary analyses involving Thouless time and spectral form factor are mentioned (Pawlik et al., 24 Jul 2025).

3. Phase structure and anomalous regimes

At fixed disorder $W=0.5$3, the phase diagram in the $W=0.5$4 plane contains four regimes, labeled $W=0.5$5 through $W=0.5$6 (Pawlik et al., 24 Jul 2025).

Regime	Spectral statistics	Entanglement scaling
[A]	Poisson	Area law, $W=0.5$7
[B]	Poisson	Sub-volume, $W=0.5$8
[C]	Intermediate	Volume law, $W=0.5$9
[D]	GOE	Volume law, $J \ge 0$0

Regime $J \ge 0$1 is the conventional localized phase: Poisson spectrum, area-law entanglement, and no indication of extensive thermalization. Regime $J \ge 0$2 is the conventional ergodic phase: GOE level statistics and volume-law entanglement consistent with ETH. The AQS departs from the standard MBL/ETH dichotomy through regimes $J \ge 0$3 and $J \ge 0$4. In $J \ge 0$5, Poisson statistics persist while eigenstates already display sub-volume entanglement and rare-event-dominated systemwide correlations. In $J \ge 0$6, the eigenstates have volume-law entanglement and $J \ge 0$7, yet $J \ge 0$8 remains strictly between Poisson and GOE (Pawlik et al., 24 Jul 2025).

For small $J \ge 0$9, corresponding to strong disorder $U(1)$0, the spectral and eigenstate transitions nearly coincide at $U(1)$1, as in the $U(1)$2 Quantum Sun limit $U(1)$3. For larger $U(1)$4, a separation opens between the eigenstate transition $U(1)$5 and the spectral transition $U(1)$6: volume-law entanglement appears already at $U(1)$7, whereas GOE-like level repulsion only emerges at the larger scale $U(1)$8. This creates regime $U(1)$9, where eigenstate thermalization and spectral chaos are decoupled (Pawlik et al., 24 Jul 2025).

Inside the Poisson sector, the crossover between $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$0 and $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$1 is summarized by an empirical instability line,

$S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$2

This marks the onset of sub-volume entanglement and ergodic instabilities while the spectrum remains Poisson. A plausible implication is that the AQS supports an extended precursor regime in which the bath seeds long-range resonances before producing global spectral repulsion (Pawlik et al., 24 Jul 2025).

4. Rare events, avalanches, and the decoupling of ETH diagnostics

The physical interpretation centers on the competition between $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$3, which measures how far the bath effectively reaches into the chain, and $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$4, which measures how readily the Anderson-localized chain can hybridize. Increasing either parameter promotes delocalization. The empirical line $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$5 is consistent with the onset of rare resonant spots seeded by the bath, which correlate distant degrees of freedom without immediately generating full spectral level repulsion (Pawlik et al., 24 Jul 2025).

This mechanism is most visible in regime $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$6. There, $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$7 typically still decays exponentially, yet the histograms of $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$8 develop pronounced power-law tails at small $S^z_{\mathrm{tot}} = \sum_{\text{all spins}} S^z$9, involving a finite fraction of eigenstates. These tails are described as clear rare-event dominance, or Griffiths-like behavior, together with systemwide correlations. Simultaneously, the decrease of $U(1)$00 with $U(1)$01 is interpreted as an ergodic-like tendency inside a phase whose spectrum remains Poisson (Pawlik et al., 24 Jul 2025).

Regime $U(1)$02 sharpens the distinction between eigenstate and spectral criteria. Its eigenstates are ETH-like in the sense that $U(1)$03 and $U(1)$04, but the level statistics remain intermediate rather than GOE. The paper explicitly interprets this as a separation of eigenstate thermalization from spectral chaos, comparing the regime to mixed phase-space models and integrable-but-delocalized quadratic systems. In that sense, the AQS violates the usual “lockstep” expectation according to which Poisson spectra pair with area-law entanglement and GOE spectra with volume-law entanglement (Pawlik et al., 24 Jul 2025).

The resulting picture refines avalanche-based intuition. Rather than a single sharp conversion from localized to thermal behavior in all diagnostics at once, the AQS exhibits a staged destabilization: first rare-event-dominated correlations and sub-volume entanglement, then ETH-like volume-law eigenstates without full random-matrix spectral statistics, and only later complete GOE behavior. This suggests that in bath-coupled localized systems, different ETH indicators can separate over substantial parameter windows (Pawlik et al., 24 Jul 2025).

5. Relation to Quantum Sun models and localization theory

The AQS extends the earlier Quantum Sun construction by adding intra-chain XY hopping $U(1)$05. In the $U(1)$06 Quantum Sun limit $U(1)$07, a single critical coupling $U(1)$08 separates localization and ergodicity, with relatively small finite-size effects. The AQS retains this scale at small $U(1)$09 but develops a richer two-step thermalization structure once $U(1)$10 is increased (Pawlik et al., 24 Jul 2025).

Earlier work on Quantum Sun models established that these star-graph systems can support a sharp ergodicity-breaking phase transition and an energy-dependent many-body mobility edge. In the original and $U(1)$11-preserving Quantum Sun models, the critical coupling at $U(1)$12 is near $U(1)$13, and the mobility-edge line $U(1)$14 rises away from the spectral center because the many-body density of states is exponentially suppressed in the tails (Pawlik et al., 2023). The AQS does not replace that picture; rather, it introduces a localized-chain sector whose Anderson length $U(1)$15 competes explicitly with the bath reach $U(1)$16, thereby exposing unconventional regimes between fully localized and fully ergodic behavior (Pawlik et al., 24 Jul 2025).

A complementary rigorous result concerns the small-$U(1)$17 limit of the Quantum Sun model. For sufficiently small $U(1)$18, one can prove Fock-space many-body localization and Poisson spectral statistics, with eigenstates overwhelmingly aligned with product $U(1)$19-configurations beyond a random finite core. In that analysis the physically expected transition scale is $U(1)$20, although the proof applies only for $U(1)$21 (Roeck et al., 16 Jun 2025). This rigorous localized regime is consistent with the AQS phase $U(1)$22, but the AQS findings indicate that adding an Anderson-localized chain and finite $U(1)$23 creates intermediate structures not captured by the simpler lockstep localization picture (Pawlik et al., 24 Jul 2025).

The comparison to XXZ chains is also diagnostic. At comparable parameters, the AQS sub-volume regime $U(1)$24—Poisson statistics together with sub-volume entanglement and rare-event-dominated correlations—is reported to be absent in XXZ. The paper attributes this difference to the special role of the hub-and-spoke bath coupling and the noninteracting localization seed provided by the chain (Pawlik et al., 24 Jul 2025).

6. Finite-size constraints, experimental relevance, and open problems

The numerical evidence is based on relatively small total sizes, $U(1)$25, with a bath as small as $U(1)$26. The authors nevertheless emphasize that finite-size effects are modest enough to support extrapolations of $U(1)$27 and $U(1)$28, especially because the underlying Quantum Sun geometry is known to exhibit limited finite-size drift. Even so, the ultimate fate of regime $U(1)$29 in the thermodynamic limit remains open: the data show robust ergodic instabilities, but whether an eventual avalanche fully thermalizes that region is not settled (Pawlik et al., 24 Jul 2025).

Another limitation is diagnostic rather than numerical. The analysis is static, so the dynamical properties and timescales associated with the unconventional phases are not directly computed. Open questions include avalanche growth, Thouless time across the distinct regimes, and a microscopic theory for why intermediate spectral statistics in $U(1)$30 coexist with ETH-like eigenstates (Pawlik et al., 24 Jul 2025).

The proposed experimental setting is a platform with tunable hub-and-spoke couplings and controllable disorder, such as trapped-ion chains with ancilla spins acting as the bath. The relevant observables are entanglement scaling $U(1)$31, spectral $U(1)$32-statistics obtained from spectroscopy or Floquet quasienergies, and long-range connected spin correlations $U(1)$33 together with their fat-tailed distributions. These observables map directly onto the distinctions among regimes $U(1)$34–$U(1)$35 (Pawlik et al., 24 Jul 2025).

In summary, the interacting Anderson Quantum Sun model is significant because it exposes two distinct failures of the conventional MBL/ETH lockstep. It realizes a Poisson-spectral phase with sub-volume entanglement and rare-event-dominated correlations, and a volume-law phase with intermediate rather than GOE spectral statistics. This combination makes it a useful benchmark for studying how ergodic inclusions destabilize Anderson-localized structures, how avalanche scenarios unfold in finite systems, and how different diagnostics of thermalization can decouple in interacting disordered matter (Pawlik et al., 24 Jul 2025).