Interaction-rotation driven localization-delocalization of eigenstate in Fock space: An exact diagonalization study on trapped Bose gas

M. A. H. Ahsan; Mohd Talib

Interaction-rotation driven localization-delocalization of eigenstate in Fock space: An exact diagonalization study on trapped Bose gas

T0 review · 2 major / 0 minor · reviewed 2026-07-03 · grok-4.3

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Pith's one-line read Increasing interactions and rotation drive delocalization of eigenstates in the Fock space of trapped bosons.

desk verdict Standard small-N ED numerics on rotating bosons; interaction and rotation both increase Fock-space spreading via IPR and entropies, but no scaling means the transition language is overstated. read the letter →

arxiv 2607.01888 v1 pith:ORJ7WTON submitted 2026-07-02 cond-mat.quant-gas

Mohd Talib , M. A. H. Ahsan This is my paper · ORCID

classification cond-mat.quant-gas

keywords localization-delocalizationtrappedbosonsexactdiagonalizationFockspacerotationeffectsinteractionstrengthentanglemententropyinverseparticipationratio

verification ladder T0 review T1 audit T2 compute T3 formal

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 investigates localization-delocalization transitions in the eigenstates of interacting bosons confined in a trap, both with and without rotation. Using exact diagonalization, it finds that stronger interactions cause eigenstates to spread more widely across the Fock space basis, lowering the inverse participation ratio while raising entropy measures. Rotation further promotes this delocalization, though the effect saturates at high angular momenta in smaller systems. These patterns intensify with more bosons, and delocalized states display stronger entanglement. The study unifies several measures to characterize how interaction, rotation, and particle number control the spread.

What carries the argument

Exact diagonalization within fixed total angular momentum subspaces, using the inverse participation ratio, Shannon entropy, and von Neumann entanglement entropy to quantify the spread of eigenstate weight in Fock space.

What would settle it

Performing the analysis for boson numbers large enough that the saturation effect either persists or disappears entirely would test whether the reported interplay holds.

Watch

Extended reading notes

Core claim

In the non-rotating case, a transition from localized to delocalized behavior is observed with increasing interaction strength. The transition is characterized by a decrease in IPR and a corresponding increase in entropy measures, indicating spread of eigenstate weight over all the basis states in the Hilbert space. In the presence of rotation, the system is driven further toward delocalization. For moderate angular momentum, the eigenstates exhibit partial spreading, while at higher angular momenta a saturation behavior emerges, where further increase in rotation has a limited effect on the localization properties. However, the saturation weakens with increasing system size, indicating a no

Load-bearing premise

The trends seen in small systems with feasible boson numbers reflect the core physics of the transition despite the rapid growth of the Hilbert space with particle number.

Editorial extensions

If this is right

The delocalization effect strengthens with increasing boson number due to growth in Hilbert space dimension.
Saturation of delocalization at high rotation becomes less pronounced in larger systems.
Localized states exhibit weaker entanglement while delocalized states show stronger entanglement.
The three measures provide a consistent characterization of the transition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Rotation could be tuned to control the degree of state delocalization in bosonic many-body systems.
The weakening of saturation with system size suggests that in the large-particle limit rotation may induce complete delocalization without bound.
These findings may connect to questions of ergodicity in rotating quantum gases.

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Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main point is that this paper runs exact diagonalization in fixed-Lz sectors for small numbers of trapped bosons and tracks ground-state spreading in Fock space with three measures: IPR, Shannon entropy, and von Neumann entropy. Stronger interactions reduce IPR and raise the entropies; adding rotation pushes the same direction, with a saturation at high angular momentum that becomes less pronounced as N grows.

What the work does reasonably is show that the three diagnostics move together and that the delocalization signal strengthens with N simply because the basis gets larger. That observation is straightforward and internally consistent for the sizes they can reach.

The soft spot is exactly the one in the stress-test note. All the reported trends are for very small N where Hilbert-space dimension is manageable. There is no finite-size scaling, no data collapse, and no attempt to extrapolate. The saturation behavior is noted but not quantified with any scaling form, so it is unclear whether the reported localization-delocalization crossover survives or changes character as N increases. Calling the finite-N spreading a “transition” therefore rests on an assumption that the small-system trends are representative, which the data do not test.

This is for people already working on few-body rotating gases or on Fock-space localization in bosonic systems who want concrete numerical benchmarks for the same measures. A reader outside that narrow slice will not find new methods or broadly applicable results. The calculations look solid within their scope and the authors are honest about the N-dependence they see, so the paper shows clear thinking even if the conclusions are limited by system size.

I would bring it to a reading group only if the group is focused on trapped rotating bosons. I would not cite it in my own work. It is worth sending to peer review in a specialized computational or few-body journal, where referees can check the implementation details and press on the finite-size interpretation.

Referee Report

2 major / 0 minor

Summary. The manuscript performs exact diagonalization of a trapped interacting Bose gas within fixed total angular momentum subspaces for small boson numbers N. It reports a localization-delocalization transition in Fock space as interaction strength increases (signaled by decreasing IPR and increasing Shannon and von Neumann entropies), with rotation further promoting delocalization; the effect strengthens with N, and a saturation at high angular momentum weakens with larger N. Three measures are shown to give consistent signals.

Significance. If the finite-size observations survive extrapolation, the work supplies a unified numerical characterization of how interaction, rotation, and particle number jointly control Fock-space spreading in small rotating Bose systems, with the internal consistency across IPR, information entropy, and entanglement entropy constituting a strength of the small-system study.

major comments (2)

[Abstract] Abstract: the central claim of an interaction-driven localization-delocalization transition (and its further enhancement by rotation) rests on trends observed for small finite N without finite-size scaling, data collapse, or extrapolation to N o∞; the reported strengthening with N and weakening saturation are noted but not quantified via scaling, leaving open whether the changes are crossovers rather than a transition whose location or character persists in the thermodynamic limit.
[Abstract] Abstract: the statement that 'the effect becomes more pronounced with increasing number of bosons due to the increase of the Hilbert space dimension' is presented as supporting evidence for a transition, yet no systematic study of how the apparent transition point or the saturation regime scales with N is provided, which is load-bearing for interpreting the results as a genuine delocalization transition.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of an interaction-driven localization-delocalization transition (and its further enhancement by rotation) rests on trends observed for small finite N without finite-size scaling, data collapse, or extrapolation to N→∞; the reported strengthening with N and weakening saturation are noted but not quantified via scaling, leaving open whether the changes are crossovers rather than a transition whose location or character persists in the thermodynamic limit.

Authors: The manuscript explicitly studies finite systems, as stated in the title, abstract, and introduction, because exact diagonalization is restricted to small N. We report the observed trends toward delocalization with interaction and rotation, and note that these trends strengthen with N, but we make no claim that the behavior constitutes a transition persisting in the thermodynamic limit. No finite-size scaling or extrapolation is performed. The internal consistency across IPR, Shannon entropy, and von Neumann entropy remains a strength for the accessible sizes. We will revise the abstract to emphasize the finite-N scope and remove any implication of thermodynamic-limit behavior. revision: partial
Referee: [Abstract] Abstract: the statement that 'the effect becomes more pronounced with increasing number of bosons due to the increase of the Hilbert space dimension' is presented as supporting evidence for a transition, yet no systematic study of how the apparent transition point or the saturation regime scales with N is provided, which is load-bearing for interpreting the results as a genuine delocalization transition.

Authors: The statement reflects our direct numerical observation that larger N (and thus larger Hilbert-space dimension) permits greater eigenstate spreading. Calculations were performed for multiple N, showing the effect becoming more pronounced. We acknowledge that no systematic scaling analysis of the apparent transition point or saturation regime with N is included. We will revise the abstract to present this as a finite-size observation rather than evidence for a thermodynamic transition. revision: partial

standing simulated objections not resolved

Whether the observed trends constitute a true transition whose character persists in the thermodynamic limit, as this requires finite-size scaling and extrapolation to N→∞ beyond the reach of exact diagonalization.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: direct numerical outputs from exact diagonalization

full rationale

The paper performs exact diagonalization in fixed-Lz subspaces to obtain eigenstates, then directly computes IPR, Shannon entropy, and von Neumann entropy from those states. All reported trends (IPR drop with interaction, further delocalization with rotation, saturation at high Lz) are numerical observations, not derived quantities that reduce to fitted inputs or self-citations by construction. No analytical steps, ansatze, or uniqueness theorems are invoked. The finite-size limitation noted by the skeptic is a validity concern, not circularity.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on standard non-relativistic quantum mechanics for identical bosons in a harmonic trap plus the numerical validity of exact diagonalization for small Hilbert spaces; no new free parameters, ad-hoc axioms, or invented entities are introduced.

assumptions (1)

standard math Standard construction of the bosonic Fock space and the many-body Hamiltonian for particles in a harmonic trap with contact interactions
Invoked throughout the abstract when describing the eigenspectrum and basis states.

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Cite this review

Pith. "Pith review of Interaction-rotation driven localization-delocalization of eigenstate in Fock space: An exact diagonalization study on trapped Bose gas." pith.science (2026). https://pith.science/paper/ORJ7WTON

@misc{pith2026260701888,
  author       = {Pith},
  title        = {Pith review of: Interaction-rotation driven localization-delocalization of eigenstate in Fock space: An exact diagonalization study on trapped Bose gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORJ7WTON}},
  note         = {Machine review of arXiv:2607.01888}
}

read the original abstract

We investigate the localization-delocalization transition and entanglement structure in a finite system of interacting bosons in non-rotating and rotating cases. The many-body eigenspectrum is obtained via exact diagonalization within subspaces of fixed total angular momentum, and the structure of the ground state is analyzed using the inverse participation ratio (IPR), the Shannon entropy (information entropy) and the von Neumann entanglement entropy. In the non-rotating case, a transition from localized to delocalized behavior is observed with increasing interaction strength. The transition is characterized by a decrease in IPR and a corresponding increase in entropy measures, indicating spread of eigenstate weight over all the basis states in the Hilbert space. The effect becomes more pronounced with increasing number of bosons due to the increase of the Hilbert space dimension. In the presence of rotation, the system is driven further toward delocalization. For moderate angular momentum, the eigenstates exhibit partial spreading, while at higher angular momenta a saturation behavior emerges, where further increase in rotation has a limited effect on the localization properties. However, the saturation weakens with increasing system size, indicating a nontrivial interplay between rotation and number of bosons. The consistent behavior of IPR, information entropy and von Neumann entanglement entropy demonstrates that these measures provide a unified characterization of the localization-delocalization transition. The results highlight the combined role of interaction strength, rotation and number of bosons in driving the system towards delocalized state. We observe a connection between localization-delocalization and entanglement, with localized states exhibiting weaker entanglement and delocalized states showing stronger entanglement.

Figures

Figures reproduced from arXiv: 2607.01888 by the authors.

**Figure 2.** FIG. 2. Plot of (a) inverse participation ratio, (b) informa [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Plot of (a) inverse participation ratio, (b) informa [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Plot of (a) inverse participation ratio, (b) informa [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed July 3, 2026 · model on record in the stance chip above.

[1] [1]

Beyond this inter- action strength, for stronger interaction, the entropy in- creases, reﬂecting enhanced spreading of eigenstate weight over all the basis states

669, indicating that the structure of the eigenstates does not change signiﬁcantly in this regime. Beyond this inter- action strength, for stronger interaction, the entropy in- creases, reﬂecting enhanced spreading of eigenstate weight over all the basis states. As the rotation is increased to higher angular momentum values, Lz = 8 and 12, the in- formati...

work page

[2] [2]

Richerme, Z.-X

P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Mon- roe, Non-local propagation of correlations in quantum sys- tems with long-range interactions, Nature 511, 198 (2014)

work page 2014

[3] [3]

Islam, R

R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entangle- ment entropy in a quantum many-body system, Nature 528, 77 (2015)

work page 2015

[4] [4]

A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schit- tko, P. M. Preiss, and M. Greiner, Quantum thermalization through entanglement in an isolated many-body system, Science 353, 794 (2016)

work page 2016

[5] [5]

K. X. Wei, C. Ramanathan, and P. Cappel- laro, Exploring localization in nuclear spin chains, Phys. Rev. Lett. 120, 070501 (2018)

work page 2018

[6] [6]

H. Dong, P. Zhang, C. B. Da˘ g, Y. Gao, N. Wang, J. Deng, X. Zhang, J. Chen, S. Xu, K. Wang, Y. Wu, C. Zhang, F. Jin, X. Zhu, A. Zhang, Y. Zou, Z. Tan, Z. Cui, Z. Zhu, F. Shen, T. Li, J. Zhong, Z. Bao, H. Li, Z. Wang, Q. Guo, C. Song, F. Liu, A. Chan, L. Ying, and H. Wang, Measuring the spectral form factor in many- body chaotic and localized phases of qu...

work page 2025

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E. P. Wigner, Random Matrices in Physics, SIAM Review 9, 1 (1967)

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T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Random-matrix physics: spectrum and strength ﬂuctuations, Rev. Mod. Phys 53, 385 (1981)

work page 1981