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arxiv: 2605.06849 · v1 · submitted 2026-05-07 · 🪐 quant-ph

Tracing complex zeros of the quantum survival amplitude: How the energy distribution controls dynamical phase transitions

Jakub Novotn\'y , Jan St\v{r}ele\v{c}ek , Pavel Str\'ansk\'y , Pavel Cejnar This is my paper

Pith reviewed 2026-05-11 02:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords dynamical quantum phase transitionsLoschmidt amplitudecomplex zerosenergy distribution envelopequantum quenchesIsing modelBCS statessurvival probability
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The pith

The envelope of the energy distribution governs the zeros of the complex-time survival amplitude and their approach to the real axis as precursors of dynamical quantum phase transitions.

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

The paper develops a general framework to approximate the zeros of the complex-time survival amplitude in finite quantum systems. It demonstrates that the large-scale properties of these zeros are determined by the envelope of the energy distribution of the initial state, which can be built from chains of periodic zeros linked to dominant eigenstates. Zeros cross to the real-time axis when two or more eigenstates are equally populated at the envelope's maximum. This provides a finite-size precursor to dynamical quantum phase transitions and is validated in the Ising model and exactly for certain BCS quenches. The approach also explains short-time dynamics through a Gaussian model where dephasing deforms the zero pattern.

Core claim

The large-scale properties of the distribution of zeros are governed by the envelope of the energy distribution of the initial state and can be constructed from chains of periodic zeros associated with its dominant contributions. Zeros reach the real-time axis when two or more eigenstates become equally populated at the maximum of the envelope, providing a finite-size precursor of DQPTs. The framework becomes exact for BCS ground-state quenches in two-band models, and a minimal Gaussian model describes the short-time deformation of the zero pattern into the asymptotic two-level structure.

What carries the argument

The envelope of the energy distribution of the initial state, used to construct chains of periodic zeros via an approximation based on the stability of zeros of holomorphic functions.

If this is right

  • Accurate approximation of zero distributions in quenched Ising models with varying interaction ranges.
  • Exact construction for BCS ground-state quenches in two-band models.
  • Explanation of anomalous DQPTs via delayed zero approach in the Gaussian spectrum model due to slow dephasing.
  • Universal role of the energy envelope in shaping the entire zero distribution and dynamical critical behavior.

Where Pith is reading between the lines

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

  • The framework could be tested in other quench scenarios or with different initial states to see if the envelope always dominates.
  • In larger systems, the precursor might sharpen into true transitions controlled by the envelope peak.
  • Controlling the energy distribution shape could provide a way to tune the timing of dynamical transitions in experiments.

Load-bearing premise

The approximation framework based on the stability of zeros of holomorphic functions accurately captures the large-scale zero distribution and its connection to the energy envelope without significant errors from finite-size effects or higher-order contributions.

What would settle it

Numerical computation of the exact zeros for a finite Ising quench system compared to the predicted chains from the energy envelope, checking if deviations remain small as system size increases.

Figures

Figures reproduced from arXiv: 2605.06849 by Jakub Novotn\'y, Jan St\v{r}ele\v{c}ek, Pavel Cejnar, Pavel Str\'ansk\'y.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The left column shows a short quench of the ground [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Zero distribution (black dots) of the survival ampli [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Lines of zeros [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The first row shows the two types of zero distri [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
read the original abstract

Motivated by the advance of dynamical quantum phase transitions (DQPTs), we analyze the zeros of the complex-time survival (Loschmidt) amplitude in finite quantum systems and develop a general framework for their approximation based on the stability of zeros of holomorphic functions. We show that the large-scale properties of the distribution of zeros are governed by the envelope of the energy distribution of the initial state and can be constructed from chains of periodic zeros associated with its dominant contributions. In this picture, zeros reach the real-time axis when two or more eigenstates become equally populated at the maximum of the envelope, providing a finite-size precursor of DQPTs. We apply the method to quenched ground states in the Ising model with tunable interaction range and demonstrate close agreement between the approximate and exact distributions of zeros. We prove that the approximate construction becomes exact for BCS ground-state quenches in two-band models. To describe short-time dynamics, we introduce a minimal Gaussian model with a nearly equidistant spectrum. Slow dephasing continuously deforms the initial zero pattern into the asymptotic two-level structure, explaining anomalous DQPTs as a delayed approach of zeros to the real-time axis. Our results identify the energy envelope as the key ingredient shaping dynamical critical behavior and provide a universal interpretation of the whole zero distribution of the complex-time survival amplitude.

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

0 major / 2 minor

Summary. The paper develops a general approximation framework for the zeros of the complex-time survival (Loschmidt) amplitude in finite quantum systems, based on the stability properties of zeros of holomorphic functions. It claims that the large-scale distribution of these zeros is governed by the envelope of the initial state's energy distribution and can be constructed from chains of periodic zeros associated with its dominant eigenstate contributions. Zeros reach the real-time axis when two or more eigenstates become equally populated at the maximum of this envelope, acting as a finite-size precursor to dynamical quantum phase transitions (DQPTs). The framework is validated numerically on quenched Ising ground states with tunable interaction range (close agreement with exact distributions), proven exact for BCS ground-state quenches in two-band models, and supplemented by a minimal Gaussian model that captures short-time dynamics via slow dephasing deforming the zero pattern into the asymptotic two-level structure, thereby explaining anomalous DQPTs.

Significance. If the central claims hold, the work offers a universal interpretation of the entire zero distribution of the complex-time survival amplitude, with the energy envelope identified as the controlling factor for dynamical critical behavior. Strengths include the explicit construction from holomorphic zero stability, exact proofs for specific models (BCS two-band quenches), and numerical demonstrations that address finite-size effects rather than leaving them uncontrolled. The Gaussian minimal model provides insight into short-time anomalous behavior. This advances the understanding of DQPTs in finite systems beyond infinite-size or mean-field limits and could apply to other many-body quenches.

minor comments (2)
  1. Abstract: The phrase 'chains of periodic zeros associated with its dominant contributions' would benefit from a brief parenthetical example or reference to the relevant equation in the main text to improve immediate readability for non-specialists.
  2. The manuscript would be strengthened by adding a short discussion (perhaps in the conclusions) of the range of validity of the holomorphic stability approximation when the energy distribution has multiple comparable peaks rather than a single dominant envelope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed summary, positive significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no points requiring direct response or changes to the manuscript at this stage. We are pleased that the central claims regarding the energy envelope governing the zero distribution and its relation to finite-size precursors of DQPTs are viewed favorably, along with the exact results for BCS models and the Gaussian minimal model.

Circularity Check

0 steps flagged

Derivation self-contained from holomorphic zero stability applied to survival amplitude

full rationale

The paper derives the large-scale zero distribution from the stability properties of zeros of holomorphic functions applied to the complex-time survival amplitude, constructs explicit chains from dominant eigenstate contributions, and validates the envelope-governed picture by exact proof for BCS quenches in two-band models plus numerical agreement for Ising quenches. No step reduces by construction to a fitted parameter renamed as prediction, no load-bearing self-citation chain is invoked, and the Gaussian minimal model is introduced as an illustrative ansatz for short-time deformation rather than a hidden assumption. The central claim therefore rests on independent mathematical properties and explicit demonstrations rather than circular re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the stability of zeros of holomorphic functions as the basis for approximation and on the energy envelope as the controlling factor; no explicit free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Stability of zeros of holomorphic functions governs the approximation of Loschmidt amplitude zeros
    Invoked as the foundation for constructing chains of periodic zeros from the energy envelope.

pith-pipeline@v0.9.0 · 5555 in / 1251 out tokens · 31183 ms · 2026-05-11T02:26:08.720355+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Unified resonant-manifold framework for dynamical quantum phase transitions

    quant-ph 2026-05 unverdicted novelty 7.0

    A resonant-manifold framework unifies manifold and branch DQPTs by attributing them to resonances within the initial manifold and with a transitional manifold connected by low-order processes, shown in Z2 LGT quenches.

Reference graph

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