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arxiv: 2503.24362 · v4 · submitted 2025-03-31 · ❄️ cond-mat.str-el · quant-ph

Recursion method for out-of-equilibrium many-body dynamics: strengths and limitations

Ilya Shirokov , Viacheslav Khrushchev , Filipp Uskov , Ivan Dudinets , Igor Ermakov , Oleg Lychkovskiy This is my paper

Pith reviewed 2026-05-22 22:01 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords recursion methodquantum quenchLanczos coefficientsquench coefficientsmany-body dynamicsstrongly correlated systemsout-of-equilibrium dynamicsdynamical correlation functions
0
0 comments X

The pith

Extending the recursion method to quantum quenches requires non-universal quench coefficients that cannot be extrapolated, limiting the reliable timescale.

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

The paper examines whether the recursion method, already effective for dynamical correlation functions via universal Lanczos coefficients, can be applied to expectation values after a quantum quench. It establishes that quench dynamics introduces additional quench coefficients defined as overlaps between the Lanczos basis operators and the initial state. These coefficients lack any universal structure and vary strongly with the choice of initial state, ranging from decaying to irregular or growing sequences. As a result, extrapolation from early terms is unreliable, restricting the times at which the method produces accurate results compared to its use for correlations. The approach can nevertheless remain useful when the initial state produces a sufficiently regular sequence of coefficients.

Core claim

Dynamical correlation functions in the recursion method are fully determined by the Lanczos coefficients b_n, which in generic systems exhibit universal behavior that permits reliable extrapolation from the first few dozen explicitly computed terms. Quench dynamics requires in addition the quench coefficients c_n, which are the overlaps of the Lanczos basis operators with the initial state. Unlike the b_n, the c_n display no universal structure and cannot be reliably extrapolated; their sequences are highly state-dependent, and less regular sequences limit the accessible timescale, although the method stays competitive for favorable initial states.

What carries the argument

Quench coefficients c_n, the overlaps of Lanczos basis operators with the initial state, which carry the information needed for quench dynamics but lack the universal structure of the Lanczos coefficients b_n.

If this is right

  • The method produces accurate results only up to shorter times than it does for dynamical correlation functions.
  • The accessible timescale is determined by the regularity of the particular sequence of c_n for the chosen initial state.
  • Initial states that produce decaying c_n sequences allow the method to remain competitive with other approaches.
  • Extrapolation techniques that work for the Lanczos coefficients b_n cannot be applied to the quench coefficients c_n.

Where Pith is reading between the lines

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

  • The state dependence implies that the method's practical range can be extended by selecting initial states whose overlaps produce more regular coefficient sequences.
  • The contrast with correlation functions points to a general distinction between equilibrium and non-equilibrium dynamics within the same Lanczos operator space.
  • Hybrid schemes that combine the recursion method with other techniques may be needed to reach longer times in generic quench problems.

Load-bearing premise

The non-universal and state-dependent behavior of the quench coefficients c_n is an inherent obstacle to extrapolation in generic systems.

What would settle it

Explicit computation of a much larger number of quench coefficients in a generic interacting many-body model, followed by checking whether the sequence eventually follows a universal decaying pattern or continues to be irregular or growing.

Figures

Figures reproduced from arXiv: 2503.24362 by Filipp Uskov, Igor Ermakov, Ilya Shirokov, Ivan Dudinets, Oleg Lychkovskiy, Viacheslav Khrushchev.

Figure 1
Figure 1. Figure 1: FIG. 1. Time evolution of the total polarization along [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Quench dynamics of nonintegrable Ising models on (a) one-dimensional, (b) square, and (c) cubic lattices. Shown is [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

The recursion method, which solves coupled Heisenberg equations in a Lanczos operator basis, has recently emerged as a powerful nonperturbative tool for computing dynamical correlation functions in strongly correlated two- and three-dimensional quantum many-body systems. Motivated by this success, we investigate whether the method can be extended to expectation values of observables following a quantum quench. We find that such an extension encounters an obstacle absent in the computation of dynamical correlation functions. The latter are fully determined by the Lanczos coefficients $b_n$, which in generic systems exhibit universal behavior, enabling reliable extrapolation from the first few dozens of explicitly computed coefficients. In contrast, quench dynamics additionally requires "quench coefficients" $c_n$, defined as overlaps of Lanczos basis operators with the initial state. We show that, unlike the Lanczos coefficients, the quench coefficients display no universal structure and cannot be reliably extrapolated, thereby limiting the time up to which the method yields accurate results. The behavior of quench coefficients is highly state-dependent, ranging from decaying to irregular or even growing sequences; typically, the less regular the sequence $c_n$, the shorter the accessible timescale. Nevertheless, for favorable initial states, the method remains competitive with state-of-the-art approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates extending the recursion method to expectation values of observables after a quantum quench. It claims that dynamical correlation functions are fully determined by Lanczos coefficients b_n, which exhibit universal behavior in generic systems and permit reliable extrapolation from the first few dozen coefficients. In contrast, quench dynamics require additional 'quench coefficients' c_n (overlaps of Lanczos basis operators with the initial state), which lack universal structure, are highly state-dependent (decaying, irregular, or growing), and cannot be reliably extrapolated, thereby limiting the accessible timescale—though the method remains competitive for favorable initial states.

Significance. If the claimed non-universal and non-extrapolatable behavior of c_n holds across generic systems, the result would usefully delineate the strengths and limitations of the recursion method for out-of-equilibrium dynamics, informing when extrapolation is viable versus when the method's accuracy is inherently restricted by initial-state dependence.

major comments (1)
  1. Abstract: the central claim that c_n 'display no universal structure and cannot be reliably extrapolated' is asserted without any explicit examples, derivations, or numerical data. This is load-bearing for the conclusion that extrapolation is unreliable and that the accessible timescale is limited, as it leaves unresolved whether the behavior is inherent or an artifact of finite computation or specific models.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the central claim that c_n 'display no universal structure and cannot be reliably extrapolated' is asserted without any explicit examples, derivations, or numerical data. This is load-bearing for the conclusion that extrapolation is unreliable and that the accessible timescale is limited, as it leaves unresolved whether the behavior is inherent or an artifact of finite computation or specific models.

    Authors: The abstract summarizes the central results; the main text supplies the supporting material. It contains explicit numerical computations of c_n sequences for multiple models and initial states, together with derivations showing how these sequences enter the time evolution. The data illustrate decaying, irregular, and growing behaviors that persist upon increasing the number of Lanczos coefficients, indicating the limitation is intrinsic rather than a finite-computation artifact. We are prepared to add a brief reference to the relevant figures and sections in a revised abstract if the referee considers it useful. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract presents the central claim as an empirical contrast: Lanczos coefficients b_n exhibit universal behavior (referenced to prior literature on the recursion method) while quench coefficients c_n are state-dependent and lack universal structure, limiting extrapolation. No derivation chain is supplied in the available text, no parameters are fitted then renamed as predictions, and no self-citation is invoked to justify uniqueness or an ansatz. The limitation follows directly from the observed properties of c_n without reducing to a self-definitional or fitted-input construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work relies on the established recursion method and Lanczos basis from prior literature; abstract mentions no new free parameters, axioms, or invented entities.

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discussion (0)

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