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Page curve like dynamics in Interacting Quantum Systems

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Pith's one-line read The paper argues that a finite interacting XXZ spin chain draining into a large empty XXZ bath produces Page-curve-like entanglement—power-law growth, a peak at the Page time, and decay—and that a coarse-grained Boltzmann entropy tracks…

desk verdict A plausible interacting-system extension of the Page-curve program with clean early-time analytics and an interesting Boltzmann-entropy comparison; the abstract overclaims and the TEBD convergence evidence is missing. read the letter →

arxiv 2504.14675 v1 pith:O3PBK7M3 submitted 2025-04-20 quant-ph cond-mat.mes-hallcond-mat.quant-gascond-mat.stat-mechhep-th

Tamoghna Ray , Abhishek Dhar , Manas Kulkarni This is my paper · ORCID
classification quant-phcond-mat.mes-hallcond-mat.quant-gascond-mat.stat-mechhep-th
keywords PagecurveentanglemententropyXXZspinchainintegrableandnon-integrabledynamicsBoltzmanninhomogeneousquenchmatrixproductstatesblackholeinformationparadox
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 aims to show that Page-curve-like entanglement dynamics—growth, a maximum at a Page time, and subsequent decay—occurs in a genuinely interacting one-dimensional quantum system, not only in non-interacting or exactly solvable models. It studies a finite XXZ spin-1/2 chain (a chain of spin-1/2 particles with exchange and Ising-type interaction) coupled at one end to a much larger XXZ bath, starting from a product state with the bath empty and the system either fully filled or in a high-entropy, effectively infinite-temperature state. The authors argue that particle leakage from system to bath drives the rise and fall of entanglement, with power-law growth exponents that depend on the initial state, the anisotropy, and whether an integrability-breaking next-nearest-neighbor coupling is present. The one studied exception is a filled system with integrability-breaking coupling, where the initial state has large overlap with a localized eigenstate and the dynamics freeze. A sympathetic reader would care because the result suggests Page-like behavior is generic in interacting quantum matter and connects black-hole-inspired entropy dynamics to ordinary statistical mechanics.

What carries the argument

The key machinery is the inhomogeneous quench of an XXZ spin-1/2 chain in the equivalent fermionic representation, where the anisotropy $\Delta$ becomes the interaction strength. A finite system block, with or without an integrability-breaking next-nearest-neighbor coupling $J'$, is attached to a much larger empty XXZ bath, and the time evolution is performed with time-evolving block decimation on matrix product states. The central quantities are the von Neumann entanglement entropy across the system-bath bond, the bath particle number and its fluctuations, and a coarse-grained Boltzmann entropy built by fitting a grand-canonical density matrix to the instantaneous energy and magnetization of each spatial cell. The mechanism that produces the Page curve is particle transfer: entanglement grows while the system leaks particles into the empty bath, peaks when a size-dependent fraction of the particles has escaped, and decays as the system empties toward a product state.

What would settle it

Repeat the same quenches with larger bond dimensions (for example $\chi=200$, $300$, $400$) and compare the entanglement curves at and after the Page time; if the peak height, peak time, or post-Page power laws shift noticeably, the claimed Page-curve decay is a numerical truncation artifact. For small system-plus-bath sizes, the same comparison can be made against exact diagonalization, and disagreement after the Page time would falsify the central claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Page curve is not confined to non-interacting or exactly solvable fermionic models: an interacting XXZ spin chain coupled to a large integrable XXZ bath, initialized with the bath empty and the system either filled or in a high-entropy infinite-temperature state, produces system-bath entanglement that first grows as a power law, reaches a maximum at a Page time, and then decays as particles leave the system. For the filled initial state in an integrable system the growth goes roughly as $t^{0.30}$ for $\Delta=0.8J$ and $t^{0.31}$ for $\Delta=1.0J$; for the high-entropy initial state the exponents are near $0.8$, with post-Page decays that are much slower in the non-integrable case. The exception to the Page curve is the filled, non-integrable system: the initial state overlaps a nearly localized eigenstate of the full setup, so entanglement, particle number, and fluctuations saturate and the dynamics freeze. In addition, the paper claims a detailed correspondence between the fine-grained von Neumann entanglement entropy and a coarse-grained Boltzmann entropy: starting from the high-entropy state, the bath's Boltzmann entropy follows the entanglement before the Page time and the system's Boltzmann entropy follows it after; starting from the filled state, the system's Boltzmann entropy has Page-curve shape but lies above the entanglement, while the bath entropy keeps increasing.

Load-bearing premise

The load-bearing premise is that the time-evolving block decimation calculations with bond-dimension cutoff $\chi=150$ and Trotter step $\delta t=0.05$ remain accurate well beyond the Page time; the paper says convergence was checked but does not display the convergence data, so all exponents, Page times, and entropy comparisons rest on that unshown check.

Editorial extensions

If this is right

  • If the central claims hold, Page-curve-like entanglement dynamics is a property of interacting one-dimensional spin chains, not only of non-interacting fermion models or engineered open systems.
  • The reported power-law exponents (roughly $t^{0.30}$–$t^{0.31}$ for the filled integrable case and $t^{0.77}$–$t^{0.80}$ for the high-entropy case) provide quantitative targets for analytic theories of entanglement growth in interacting systems.
  • The time-resolved matching of bath Boltzmann entropy to entanglement before the Page time, and system Boltzmann entropy to entanglement after it, indicates that reduced density matrices in those regimes are close to the grand-canonical states set by their mean energy and particle number.
  • The freeze of the filled non-integrable system implies that in this setup Page-like behavior requires either integrability or a high-entropy initial state, because a large overlap with a localized eigenstate stops particle transfer and hence stops entanglement growth.

Reading between the lines

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

  • The finite-size scaling of the Page time ($t_{\rm Page}\sim L_S^{1.7}$ for $\Delta=0.8J$ and $\sim L_S^2$ for $\Delta=1.0J$ in the integrable filled case) is not explained by the paper; comparing these scalings to the known spin-transport exponents of the XXZ chain would test whether the peak location is set by hydrodynamic spreading or by single-particle reflection.
  • The paper's agreement between coarse-grained Boltzmann entropy and entanglement before or after the Page time suggests a stronger, untested probe: if the reduced state were truly the grand-canonical one, higher charge cumulants (full counting statistics of bath particles) should also match at those times.
  • The freezing of the filled non-integrable case is likely a finite-size bound-state effect; a direct extension would be to increase $L_S$ or add a weak perturbation and check whether the plateau height and saturation time scale with system size, which would tell whether the freeze survives in the thermodynamic limit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper studies the growth and decay of system-bath entanglement in a one-dimensional XXZ spin chain (integrable for J'=0, nonintegrable for J'=1) coupled to a much larger integrable XXZ bath. The system is initialized either in a filled (spin-polarized) state or in a high-entropy random state, while the bath is initially empty. The authors derive early-time perturbation results for the von Neumann entropy, the mean bath particle number, and its fluctuations, and then use TEBD with bond dimension 150 to extract intermediate-time power laws, Page times, and post-Page decay laws. They also introduce a grand-canonical Boltzmann entropy for the system and for spatial bins of the bath, and compare its time dependence with the entanglement entropy. The main claims are that Page-curve-like entanglement dynamics appears for the integrable filled and high-entropy initial states and for the nonintegrable high-entropy initial state, while the nonintegrable filled initial state freezes; and that for the high-entropy initial state the bath Boltzmann entropy tracks the entanglement before the Page time while the system Boltzmann entropy tracks it after.

Significance. If the numerical results are reliable, the paper would extend Page-curve-like entanglement dynamics from solvable noninteracting models to interacting integrable and nonintegrable spin chains, and would provide a rare comparison between fine-grained entanglement entropy and a coarse-grained Boltzmann entropy in a dynamical setting. The early-time perturbation theory in Sec. IV is clean, concrete, and checked against the numerics, which is a genuine strength. The model setup is transparent and the summary table of power-law exponents is useful. However, the central quantitative conclusions currently rest on TEBD data whose truncation errors are not documented, on fitting windows that are not defined, and on a qualitative notion of agreement between entropies; these issues must be resolved before the claims can be accepted.

major comments (4)
  1. [§V and Appendix C] The TEBD truncation is the single most load-bearing element of the paper, but no convergence evidence is provided. The manuscript fixes χ=150 and δt=0.05 in §V and states in Appendix C that convergence with increasing χ is checked, yet no convergence plots, discarded-weight data, or χ-dependence studies appear anywhere. Because the system-bath cut is represented by an MPS bond, the Schmidt rank at that cut is at most 150, so the computed SvN cannot exceed ln 150 ≈ 5.0; if the true entanglement approaches this cap in the intermediate regime, the observed maximum and subsequent decay—the defining Page-curve features—could be truncation artifacts. The power-law exponents in Table I, the Page-time scalings t_Page ∝ L_S^1.7 and L_S^2 in §V A, the post-Page decay laws, and the Boltzmann-entropy comparisons in §V A are all extracted from these truncated evolutions. I request convergence tests for SvN, ⟨Nbath⟩, and var(Nbath) for representative parameter sets, including discarded weights and several χ values above 150, and a demonstration that the reported SvN maxima lie below the truncation bound with a safe margin.
  2. [Abstract and §V B] The abstract states that 'in all the above-mentioned cases we obtain the Page curve like behavior in the entanglement,' but the body reports the opposite for one of the four main settings: for the non-integrable system (J'=1) starting from the filled state, §V B reports that SvN, ⟨Nbath⟩, and var(Nbath) saturate and the dynamics freeze, and Table I marks these cases as N.A. because they lack a Page curve. The conclusions in §VI correctly state this exception, so the abstract and the introductory summary overclaim. Please revise the abstract and §I to say that Page-curve-like behavior is obtained in the integrable filled, integrable high-entropy, and nonintegrable high-entropy cases, with freezing in the nonintegrable filled case.
  3. [§V A and Table I] The central quantitative results are the intermediate-regime power laws and Page-time scalings, but the fitting protocol is not specified. The text only says the fit is taken in the intermediate regime 'from t∼O(1/J) to t_Page' without giving the precise fitting window, the fitting function, the number of points, or how the standard errors in Table I were obtained. The Page-time scalings t_Page ≈ L_S^1.7 and ≈ L_S^2 in §V A are inferred from insets for L_S=5,10,20,40 with L_B=200, but the extracted t_Page values and the criterion for identifying t_Page are not given. Please define the fit windows, the extraction procedure for t_Page, and the statistical uncertainties, and show the fits on the data, for example as dashed lines in the figures.
  4. [§V A and Fig. 4] The claimed agreement between Boltzmann entropy and entanglement entropy in the infinite-temperature case is stated only qualitatively. The text says S_B^bath 'agrees' with SvN before the Page time and S_B^sys agrees after the Page time, but no quantitative measure (relative deviation, maximum absolute difference over the claimed time window, or a tolerance) is given, and no error bars from the thermodynamic inversion are provided. Since the grand-canonical construction in §III matches only the first moments of energy and magnetization and the manuscript itself notes that the state is not truly thermal, this agreement needs a quantitative criterion before it can be called agreement. Please quantify the match and identify the time window over which the two curves are within a stated tolerance.
minor comments (4)
  1. [Fig. 5 caption] The caption of Fig. 5 refers to λ=0.8J and 1.0J, but the model parameter is Δ, not λ.
  2. [§III, Eqs. (12)-(14)] The bath Boltzmann entropy is defined as a sum over bins with inter-bin bond Hamiltonians neglected; for the small bins used here (LS=10), this can introduce a systematic error that should at least be discussed or bounded.
  3. [Appendix B and §V B] The overlap calculations in Appendix B are performed with LS=10 and LB=6, but they are used in §V B to explain freezing observed for LB=200; the transfer of this spectral statement to large baths needs at least a few checks at larger LB.
  4. [Figs. 3, 6, and A2 captions] The expression 't^2/8 − (t^2/4) log t seems to be a good fit' appears as an empirical statement for SvN; since it is not derived analytically, please clarify its status and show the comparison in the figure rather than only in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Page-curve dynamics and Boltzmann-entropy comparison are empirical numerical findings, not constructed identities.

full rationale

I find no significant circularity. The central results are (i) Page-curve-like growth and decay of the system-bath von Neumann entropy in TEBD simulations, (ii) fitted power-law growth exponents in the intermediate regime, and (iii) a comparison between SvN and a coarse-grained Boltzmann entropy SB. None of these reduces to its own input by construction. SvN is computed directly from the reduced density matrix, while SB is obtained from a grand-canonical density matrix whose only constraints are the time-dependent first moments of energy and magnetization (Eqs. 8-11); the reported agreement between SvN and SB in certain time windows is an empirical observation, not an identity. The exponents in Table I are fits to the same time series they describe, but the paper presents them as measured scalings rather than as predictions derived from the fits. The early-time analytical results in Sec. IV are derived from a Taylor expansion and used to validate the numerics, but they do not by themselves produce the Page-time maximum or the post-Page decay. Self-citations (e.g., Refs. 37 and 50) provide motivation, comparison, and the coarse-graining framework, but they are not load-bearing in the sense that the paper's conclusions would collapse without them. Appendix C asserts convergence with increasing bond dimension chi without showing convergence plots, but that is a numerical-reliability concern, not a circularity in the derivation chain.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central calculations are numerical; the ledger contains no invented entities. The meaningful preconditions are numerical convergence of TEBD, equivalence of the random-circuit state to an infinite-temperature state, use of a first-moment grand-canonical entropy as the Boltzmann entropy, and extrapolation from a small spectral calculation to the actual bath size. The power-law exponents and Page-time scalings are fitted outputs, not independently derived predictions.

free parameters (3)
  • Intermediate-time power-law exponents for SvN, <Nbath>, and var(Nbath) = See Table I, e.g., SvN ~ t^0.30, t^0.80; decay ~ t^-1.93
    Fitted to TEBD curves over intermediate time windows; the windows are not precisely defined, so the exponents carry fitting uncertainty beyond the quoted standard errors.
  • Page-time scaling exponents = tPage ~ L^1.7 for Delta=0.8, tPage ~ L^2 for Delta=1.0
    Inferred from finite-size data in insets of Fig. 2; used to claim that power-law growth terminates at a Page time.
  • Coarse-graining cell size for Boltzmann entropy = LS (system size)
    The Boltzmann entropy is computed with cells of size LS and by summing per-bin entropies; the reported S_B/SvN agreement may depend on this choice, but no cell-size dependence study is provided.
assumptions (4)
  • domain assumption TEBD truncation at bond dimension chi=150 and Trotter step dt=0.05 accurately represents the time-evolved state for all reported times.
    All Page-curve and exponent results come from these simulations; Appendix C asserts convergence checks without showing them.
  • domain assumption The Haar-random-circuit state on LS sites is effectively an infinite-temperature state of the system.
    Used to represent the high-Boltzmann-entropy initial state in Sec. V and Appendix C; for LS=10, typicality is expected but not quantified.
  • domain assumption The grand-canonical density matrix fixed by the instantaneous first moments E and M gives the Boltzmann entropy of the macrostate, and ignoring inter-bin bond energies is harmless.
    Sec. III defines SB this way; it is not a first-principles derivation and higher moments are not matched.
  • domain assumption The overlap spectrum computed with LB=6 captures the localization or extended-state mechanism operative at LB=200.
    Appendix B uses LS=10, LB=6 to explain freezing and non-freezing in the actual simulations, which use LB=200.

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Pith. "Pith review of Page curve like dynamics in Interacting Quantum Systems." pith.science (2026). https://pith.science/paper/O3PBK7M3

@misc{pith2026250414675,
  author       = {Pith},
  title        = {Pith review of: Page curve like dynamics in Interacting Quantum Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O3PBK7M3}},
  note         = {Machine review of arXiv:2504.14675}
}
abstract

We study the dynamics of entanglement in a one-dimensional $XXZ$ spin-$1/2$ chain, with and without integrability-breaking interactions, that is connected to a bath. We start from a state where the system and bath are completely unentangled, and the bath is polarized spin-down. We consider two different initial states for the system - (i) a polarized spin-up state, and (ii) an infinite temperature state. In the particle representation of the spin chain, the polarized spin-up state corresponds to a filled state, while the polarized spin-down state corresponds to an empty state. Starting from these inhomogeneous quenches, in all the above-mentioned cases we obtain the Page curve like behavior in the entanglement. We report different power-law behavior in the growth of entanglement for different initial states and different kinds of baths (interacting and non-interacting). In an attempt to explore plausible deep connections between entanglement and Boltzmann entropy, we investigate the latter in both the filled and the infinite temperature case, for the system and the bath. For the filled case, the Boltzmann entropy of the system has the form of a Page curve but quantitatively deviates from the entanglement. On the other hand, the entropy of the bath keeps increasing. Remarkably, for the infinite temperature case, we find that the system and bath Boltzmann entropies agree with the entanglement entropy, after and before the Page time, respectively. Our findings are expected to hold for generic interacting quantum systems and could be of relevance to black hole physics.

Figures

Figures reproduced from arXiv: 2504.14675 by the authors.

Figure 1
Figure 1. Schematic diagram of the model. The system, denoted by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The entanglement entropy SvN(t), the mean ⟨Nbath⟩, and the fluctuation var(Nbath) of the particle number of the bath, for the integrable case (J ′ = 0), are plotted as a function of time for (a) ∆ = 0.8J and (b) ∆ = 1.0J with LS = 10 and LB = 200. We start from a filled state, and the bath is interacting. The early time data [t up to O(1/J)] shows excellent agreement with the analytical solutions given by Eq. (24) f… view at source ↗
Figure 3
Figure 3. SvN(t), ⟨Nbath⟩, and var(Nbath) of the particle number of the bath, for the integrable case (J ′ = 0), are plotted as a function of time for (a) ∆ = 0.8J and (b) ∆ = 1.0J with LS = 10 and LB = 200. We start from a high entropy state and the bath is interacting. The early time data [t up to O(1/J)] for ⟨Nbath⟩ and var(Nbath), shows excellent agreement with the analytical solutions given by Eqs. (37) and (38) respecti… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: SvN(t) and SB(t) of the system and bath for the integrable case (J ′ = 0), are plotted as a function of time for the polarized initial condition (a,b) and for the high entropy case (c,d) for the two values of ∆ = 0.8J and ∆ = 1.0J. We consider LS = 10 and LB = 200, and…
Figure 5
Figure 5. Figure 5: (a) and (b), we plot the density profiles at different time instances, and we note that only a few particles (≈ 1) decay into the bath. The dynamics of the system freeze beyond this point. This can be attributed to the fact that the filled initial state has a high over…
Figure 6
Figure 6. Figure 6: SvN(t), ⟨Nbath⟩ and var(Nbath) of the particle number of the bath, for the non-integrable case with J ′ = 1, are plotted as a function of time for (a) ∆ = 0.8J and (b) ∆ = 1.0J with LS = 10 and LB = 200. We start from a high-Boltzmann entropy state, and the bath is int…

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Forward citations

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Reference graph

Works this paper leans on

61 extracted references · 51 canonical work pages · cited by 2 Pith papers

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    (1)] and Sz sys = ∑0 i=−(LS−1) Sz i are the Hamiltonian and the total magnetization of the system

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    Compute the total energy Ebin(t) = ⟨ψ(t)|Hbin|ψ(t)⟩ and the total magnetization for each bin Mbin(t) = ⟨ψ(t)|Sz bin|ψ(t)⟩ as a function of time. The Hamilto- nian and the total magnetization of a bin are given by Hbin = J ∑ i∈bin Sx i Sx i+1 + Sy i Sy i+1 + ∆Sz i Sz i+1 , (12) Sz bin = ∑ i∈bin Sz i . (13) Note that we have ignored the bond Hamiltonian be-...

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