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Time dynamics with matrix product states: Many-body localization transition of large systems revisited

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper argues that TDVP time evolution with insufficient bond dimension makes systems in the many-body localization crossover look spuriously delocalized, and that correcting this shifts the critical disorder of large Heisenberg…

desk verdict The TDVP entanglement overshoot at small bond dimension in the MBL crossover is a real, benchmarked finding worth knowing; the paper's specific Wc≈4.2 estimate is too threshold-dependent to settle the disagreement with the earlier Wc≈5.5. read the letter →

arxiv 1908.06524 v3 pith:BOMKKWFC submitted 2019-08-18 cond-mat.stat-mech cond-mat.dis-nncond-mat.str-elquant-ph

Titas Chanda , Piotr Sierant , Jakub Zakrzewski This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nncond-mat.str-elquant-ph
keywords many-bodylocalizationmatrixproductstatesTDVPtDMRGentanglemententropyimbalancedecaycriticaldisorderHeisenbergspinchain
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 compares two matrix-product-state time-evolution methods for the disordered Heisenberg spin chain and finds that the "better" method depends on the regime. Benchmarking against exact Chebyshev propagation for L=26, it shows that TDVP with too-small bond dimension makes systems look spuriously delocalized: the imbalance decays too fast and the entanglement entropy is overestimated at long times. tDMRG makes the opposite error, underestimating the entropy and giving a predictable lower bound. Using converged data (χ=384) for L=50 and L=200, a power-law fit of the imbalance in t∈[100,200] yields a critical disorder $W_c = 4.2 \pm 0.3$, substantially lower than the previous TDVP-based estimate $W_c \approx 5.5$. The authors stress that the fit form and the threshold used to define the transition are the main sources of uncertainty.

What carries the argument

The central objects are matrix product states of bond dimension $\chi$ evolved by two algorithms: tDMRG, a TEBD variant using the Sornborger-Stewart decomposition that grows the bond dimension and truncates by singular values, and TDVP (one- and two-site), which projects the Hamiltonian onto the tangent space of the MPS manifold. The load-bearing mechanism is the different error accumulation: tDMRG's truncation discards the most entangled components, systematically underestimating entanglement entropy, while TDVP's tangent-projection error in disordered systems biases the state toward delocalized-looking dynamics and overestimates entanglement entropy at long times when $\chi$ is insufficient. Exact Chebyshev propagation for L=26 serves as the benchmark that reveals which method is trusted.

What would settle it

Compare the long-time imbalance decay for L=26 at W=3.5 using exact Chebyshev propagation out to t≈$10^{4}$ with a large number of disorder realizations and perform a formal model comparison between power-law and logarithmic fits; if the logarithmic fit is preferred, the β-threshold definition of Wc loses its foundation. Alternatively, run L=50 TDVP at χ=512 and check whether the entanglement entropy still exceeds the tDMRG lower bound for t>350, which would confirm the overestimation mechanism.

Watch

Extended reading notes

Core claim

The central claim is that unconverged TDVP results in the MBL crossover are qualitatively wrong in a specific direction: with insufficient bond dimension $\chi$, the imbalance decays faster than the exact result while the half-chain entanglement entropy overshoots the true value at long times, so the data look "more delocalized" rather than simply under-resolved. tDMRG, by contrast, converges monotonically and its entanglement entropy serves as a lower bound. On this basis the paper revisits the earlier L=100 TDVP study, showing that its $W_c \approx 5.5$ estimate is inflated by the combination of insufficient $\chi$ and a permissive $\beta$ threshold, and estimates $W_c = 4.2 \pm 0.3$ from imbalance decay for L=50 and L=200, with only weak system-size dependence. The paper also reports that open boundary conditions shift the level-statistics critical disorder to $W_c = 3.29(9)$, and argues that level-statistics and finite-time dynamics probes need not yield the same transition point.

Load-bearing premise

The estimate $W_c = 4.2 \pm 0.3$ rests on the assumption that the imbalance decays as a power law $t^{-\beta}$ on the delocalized side; the paper concedes this form has no real theoretical foundation, and a logarithmic decay would shift the inferred transition toward larger disorder.

Editorial extensions

If this is right

  • In the MBL crossover, TDVP results must be checked for bond-dimension convergence before drawing physical conclusions; a single $\chi$ value can place the system on the wrong side of the transition.
  • The critical disorder of the disordered Heisenberg chain, as inferred from finite-time imbalance dynamics, is near $W_c \approx 4.2 \pm 0.3$ and depends only weakly on system size between L=50 and L=200, contrary to the strong size dependence reported previously.
  • Because power-law and logarithmic fits of the imbalance are nearly indistinguishable on accessible time scales, critical-disorder estimates from curve fitting carry an irreducible ambiguity that should be reported.
  • tDMRG's entanglement entropy is a reliable lower bound, so bracketing the true entropy between tDMRG and converged TDVP data gives a practical convergence test.
  • The flowing-$\beta$ analysis indicates that W=4.5 is already on the localized side, supporting imbalance saturation in the thermodynamic limit.
  • Existing TDVP-based studies of entanglement growth in the MBL crossover may overestimate growth exponents if their bond dimension was insufficient; re-analyzing published data with tDMRG lower bounds could revise reported exponents.
  • The strong boundary-condition dependence of the level-statistics critical disorder (3.29 with open boundaries versus 3.72 with periodic boundaries) suggests that open-boundary finite-size scaling is not a clean thermodynamic probe, which may explain some discrepancies between level-statistics and dynamics-based transition estimates.
  • A direct test of the paper's convergence picture would be to compute operator spreading or density propagators with both methods at matched $\chi$; the method-dependent direction of error should carry over if the mechanism is general.
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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

2 major / 3 minor

Summary. The paper compares two matrix-product-state time-evolution algorithms, tDMRG and TDVP, on the disordered Heisenberg chain, using exact Chebyshev propagation for L=26 as a benchmark. It reports that TDVP is more accurate than tDMRG in the delocalized regime, but that in the MBL crossover tDMRG gives more predictable, controllable errors, while TDVP with insufficient bond dimension can spuriously accelerate imbalance decay and overestimate the long-time entanglement entropy. The paper then studies larger systems (L=50, 200) and, from power-law fits of the imbalance decay in t∈[100,200] with a hand-chosen threshold β=0.02, estimates Wc≈4.2±0.3, arguing that the previous Wc≈5.5 estimate of Ref. [41] is affected both by insufficient bond dimension and by the β cutoff. It also reports an open-boundary-condition level-statistics estimate Wc=3.29(9) and argues that time-dynamics and level-statistics estimates need not coincide.

Significance. If the algorithm-comparison result holds, it is a valuable caution for the many TDVP-based studies of MBL dynamics: unconverged TDVP data in the crossover region are biased in a counter-intuitive direction, making the system appear more delocalized, and the entanglement entropy can be overestimated at long times. The benchmark against exact Chebyshev evolution for L=26 with multiple bond dimensions is a clear strength, as is the authors' explicit discussion of the fitting ambiguities in the transition-point estimate. The quantitative claim Wc≈4.2±0.3, however, is not on the same footing as the algorithm comparison, because it depends on an assumed power-law decay and a hand-chosen threshold whose sensitivity the authors themselves demonstrate.

major comments (2)
  1. [V, Fig. 15] The central quantitative claim that the transition lies at Wc≈4.2±0.3, contradicting the Wc≈5.5 estimate of Ref. [41], is not supported by the analysis as presented. The threshold β=0.02 is chosen by hand, and the text explicitly states that using the β≈0.01 cutoff of Ref. [41] shifts the apparent Wc close to 5 even for the L=26 Chebyshev data. Since Fig. 13 shows that a logarithmic fit matches the power-law fit to within RMS error up to t=1000, and the authors state that the power-law decay has "no real theoretical foundation," the difference between Wc≈4.2 and Wc≈5.5 is dominated by the fitting convention rather than by the corrected convergence behavior of TDVP. The quoted error bar ±0.3 does not include this systematic uncertainty. The manuscript should either remove or substantially weaken the quantitative Wc claim, report a range that explicitly includes the threshold sensitivity (e.g., roughly 4 to 5), or provide an objective, data-driven criterion for β_c.
  2. [IV, Figs. 7–10 and 16] The χ→∞ extrapolation used in the flowing-β analysis is uncontrolled. A third-order polynomial in 1/χ is fitted to five data points with no stated uncertainty for the extrapolated value, and the authors note that at W=3 the TDVP and tDMRG extrapolations agree only up to moderate times and deteriorate at longer times. The flowing-β analysis in Fig. 16, which is used to argue that W=4.5 is above Wc, relies on these extrapolated data up to t=450. Please provide a systematic error estimate for the extrapolation (for example, testing higher-order fits or varying the set of χ values included) or restrict the conclusions to statements that do not depend on the extrapolated regime.
minor comments (3)
  1. [Abstract and I] "state-of-art estimate" should read "state-of-the-art estimate" in both the abstract and the introduction.
  2. [IV, Fig. 7 and Fig. 10 captions] The statement "Fitting errors are less than 10^-10" is unclear: if this refers to the residual of the polynomial fit in I(t), the value is implausibly small for a five-point fit, and if it refers to the extrapolated value, the basis should be stated. Please clarify or correct.
  3. [V, Fig. 14] The main text says "Shaded regions are discrepancies appearing in the delocalized regime," while the Fig. 14 caption says the exact Chebyshev data lie within a shaded area for L=26; please reconcile these descriptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the algorithm-comparison claims are benchmarked against exact Chebyshev dynamics, and the Wc estimate rests on an openly disclosed fitting threshold rather than a circular reduction.

full rationale

The paper's central technical claims about TDVP versus tDMRG are validated against an external, independent method: exact time evolution via Chebyshev polynomial expansion for L=26. The statements that TDVP at insufficient bond dimension overestimates the entanglement entropy at long times and makes the data look spuriously delocalized are supported by direct comparison with those exact results (Figs. 2-5), so they are not circular. The Wc≈4.2±0.3 estimate is derived by fitting the imbalance decay to a power law and reading off where the exponent β crosses a chosen threshold β=0.02. This is an openly described fitting procedure, not a hidden parameter renamed as a prediction. The paper explicitly flags the limitations: it states there is 'no real theoretical foundation' for the power-law conjecture, it admits the ambiguity of power-law versus logarithmic fits on the available time windows, and it acknowledges that 'a small shift of the fitted curves' or a different β cutoff would shift Wc, explicitly noting that β≈0.01 would move the apparent critical disorder close to 5. Those are limitations and sensitivity statements, not circular reductions. The self-citations present in the reference list are not load-bearing for the main derivation; the previous TDVP study [41] with which the paper disagrees is by a different group, and no uniqueness theorem or ansatz is smuggled in via self-citation. The paper is therefore self-contained against external benchmarks, and no circularity step can be identified from the quoted equations or text.

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

The paper contains no invented physical entities. Its central quantitative conclusions depend on several hand-chosen fitting parameters and two ad hoc assumptions (power-law decay form and 1/χ extrapolation), all acknowledged in the text. These choices, not hidden physics, carry most of the uncertainty in the Wc estimate.

free parameters (4)
  • Critical decay threshold βc = 0.02
    Hand-chosen in Sec. V to separate power-law decay from stationary imbalance; the paper notes that using β≈0.01 shifts Wc toward 5.
  • Gaussian width σ̃ in flowing-β analysis = 60
    Chosen in Sec. V to smooth oscillatory and fluctuating behavior; affects the time dependence of β(t) and the inferred localization crossover.
  • Time fit window = Jt ∈ [100,200] (and [100,500] for Chebyshev)
    Selected as a compromise in Sec. V between long-time convergence and oscillation effects; different windows change fitted exponents.
  • 1/χ extrapolation polynomial degree = 3
    Third-order polynomial in 1/χ is used to extrapolate imbalance to χ→∞ in Figs. 7, 10 and 16 without a physical model for the χ dependence.
assumptions (4)
  • domain assumption The disordered Heisenberg spin chain Hamiltonian (Eq. 3) is a valid model of the MBL transition.
    Standard model, cited [46,47]; all physical conclusions about MBL depend on it.
  • domain assumption Chebyshev polynomial expansion of the time-evolution operator is a numerically exact reference for L=26.
    Used as the benchmark in Sec. III; the paper does not quantify the truncation error but treats it as exact.
  • ad hoc to paper Imbalance decay on the delocalized side follows a power law t^{-β} in the fitted time window.
    The paper states 'there exists, as far as we know, no real theoretical foundation for this conjecture' (Sec. V) yet uses the power-law fit to define Wc.
  • ad hoc to paper Finite-χ results extrapolate to χ→∞ as a third-order polynomial in 1/χ.
    Used in Sec. IV and Fig. 16; extrapolated curves inherit this assumption and it is not derived.

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Cite this review

Pith. "Pith review of Time dynamics with matrix product states: Many-body localization transition of large systems revisited." pith.science (2026). https://pith.science/paper/BOMKKWFC

@misc{pith2026190806524,
  author       = {Pith},
  title        = {Pith review of: Time dynamics with matrix product states: Many-body localization transition of large systems revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOMKKWFC}},
  note         = {Machine review of arXiv:1908.06524}
}
read the original abstract

We compare accuracy of two prime time evolution algorithms involving Matrix Product States - tDMRG (time-dependent density matrix renormalization group) and TDVP (time-dependent variational principle). The latter is supposed to be superior within a limited and fixed auxiliary space dimension. Surprisingly, we find that the performance of algorithms depends on the model considered. In particular, many-body localized systems as well as the crossover regions between localized and delocalized phases are better described by tDMRG, contrary to the delocalized regime where TDVP indeed outperforms tDMRG in terms of accuracy and reliability. As an example, we study many-body localization transition in a large size Heisenberg chain. We discuss drawbacks of previous estimates [Phys. Rev. B 98, 174202 (2018)] of the critical disorder strength for large systems.

Figures

Figures reproduced from arXiv: 1908.06524 by the authors.

Figure 1
Figure 1. Imbalance I(t) as a function of time in the delo￾calized regime W = 3 for the system-size L = 26 obtained with TDVP for χ = 256 and 200 disorder realizations for two different time steps as indicated in the figure. Only t = 25n, n being a positive integer, points are plotted for clarity. The differences in the disordered averaged imbalance are of the order of 10−4 indicating a full convergence with the time step tak… view at source ↗
Figure 2
Figure 2. Imbalance I(t) as a function of time in the delo￾calized regime, W = 3, for L = 26 spin chain for different dimensions (χ) of the auxiliary space as indicated in the leg￾end. Here we take same 200 disorder realizations for all the techniques. Note that TDVP consistently leads to faster than exact decay (left), while tDMRG (right) reveals a false sat￾uration for too small χ. The magnitude of the error, ∆I(t), for the… view at source ↗
Figure 3
Figure 3. Entanglement entropy growth in time in the middle [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: Time evolution of the imbalance for disorder [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: Extrapolation of the imbalance for W = 3 to χ → ∞ by fitting a third-order polynomial of 1/χ to the data obtained for selected times and same values of χ as presented in [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 8
Figure 8. Figure 8: The entanglement entropy growth in the mid [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 11
Figure 11. Figure 11: The entanglement entropy error for L = 50 and W = 4 corresponding to imbalance depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]
Figure 12
Figure 12. Figure 12: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]
Figure 14
Figure 14. Figure 14: The power law (t −β ) β coefficient obtained from fits of imbalance obtained from both TDVP and tDMRG in t ∈ [100, 200] interval for L = 26 left and L = 50 right for χ = 384 for 200 disorder realizations. The error-bars correspond to 2σ-error obtained from statistical…
Figure 16
Figure 16. Figure 16: Flowing power-law exponent β(t) as derived for different disorder values (indicated in the figure) from the TDVP data for L = 50 and 800 disorder realizations. Here we also extrapolate the data to χ → ∞ to estimate the behavior of β(t) at infinite bond dimension. Obse…
Figure 17
Figure 17. Figure 17: The average level spacing ratio r as a function of disorder strength W for various system sizes L with open boundary condistions. The inset shows collapse of the data upon rescaling of disorder strength according to W → (W − WC )L 1/ν. The obtained value of critical d…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.