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arxiv: 2604.11420 · v2 · pith:KX6EMISLnew · submitted 2026-04-13 · ❄️ cond-mat.stat-mech · quant-ph

Exact Criterion for Ground-State Overlap Dominance after Quantum Quenches

Taisanul Haque This is my paper

Pith reviewed 2026-05-21 01:16 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords quantum quenchesground-state overlapBloch vectorsfree-fermion systemsdynamical quantum phase transitionsKitaev chainFisher zerosmomentum sectors
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The pith

For Hamiltonians factorizing into 2x2 momentum sectors, ground-state overlap is dominated by the final ground state if and only if initial and final Bloch vectors have positive dot product in every sector.

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

The paper derives the exact necessary and sufficient condition for when the initial ground state has the largest overlap with any final eigenstate after a sudden quench. It shows that a recent conjecture linking this dominance to staying inside the same equilibrium phase is false in general, even for translationally invariant free-fermion models. The condition is purely geometric and applies when the Hamiltonian breaks into independent two-level sectors labeled by momentum. This matters because it determines whether post-quench dynamics remain near the new ground state or populate excited states instead, and it also governs whether real-time dynamical phase transitions can appear.

Core claim

For Hamiltonians that factorize into independent 2×2 momentum sectors, the exact necessary-and-sufficient condition for ground-state overlap dominance is that the initial and final sector Bloch vectors must have positive dot product for every momentum. This result proves the conjecture in classes where same-phase quenches enforce this geometric condition, but gives explicit same-phase counterexamples in Kitaev chains, where excited final eigenstates can dominate the overlap distribution. The same obstruction controls real-time Fisher-zero crossings, allowing dynamical quantum phase transitions without crossing an equilibrium phase boundary.

What carries the argument

The dot product of the initial and final Bloch vectors within each independent 2×2 momentum sector, which fixes the overlap amplitude between the initial ground state and every final eigenstate.

If this is right

  • Same-phase quenches do not guarantee ground-state overlap dominance in all free-fermion models.
  • Kitaev-chain quenches furnish explicit counterexamples where excited states dominate the overlap even inside one equilibrium phase.
  • Dynamical quantum phase transitions via Fisher zeros can occur without any equilibrium phase boundary being crossed.
  • The geometric criterion replaces the phase-based conjecture for every system whose Hamiltonian decouples into independent 2x2 sectors.

Where Pith is reading between the lines

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

  • The same Bloch-vector test may supply a practical diagnostic for overlap control in engineered quenches on quantum simulators.
  • Extensions to weakly interacting systems could be tested by checking whether small interactions preserve the sign of the effective dot product.
  • The link to Fisher zeros suggests that dynamical criticality diagnostics might be rewritten directly in terms of initial-final vector geometry.

Load-bearing premise

The Hamiltonian must factorize into independent 2x2 momentum sectors so that each sector behaves as an isolated two-level system whose overlap is set only by the Bloch-vector dot product.

What would settle it

In the Kitaev chain, pick a same-phase quench where the Bloch vectors have negative dot product in at least one momentum sector and compute the full overlap distribution; if the largest overlap is with an excited state rather than the ground state, the criterion holds, while the opposite outcome would falsify it.

Figures

Figures reproduced from arXiv: 2604.11420 by Taisanul Haque.

Figure 1
Figure 1. Figure 1: FIG. 1. Physical same-phase regions and exact theorem-valid [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Detailed validity map for the finite-range Kitaev chain [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. DQPT diagnostics for two same-phase quenches of the finite-range [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

It was recently conjectured and verified for the transverse-field Ising model [Phys. Rev. B 113, 165102 (2026)] that, after a sudden quench within the same equilibrium phase, the initial ground state has its largest overlap with the final ground state. We show that this phase-based criterion is generally false, even in translationally invariant free-fermion systems. For Hamiltonians that factorize into independent $2\times 2$ momentum sectors, we derive the exact necessary-and-sufficient condition for ground-state overlap dominance: the initial and final sector Bloch vectors must have positive dot product for every momentum. This result proves the conjecture in classes where same-phase quenches enforce this geometric condition, but gives explicit same-phase counterexamples in Kitaev chains, where excited final eigenstates can dominate the overlap distribution. We further show that the same obstruction controls real-time Fisher-zero crossings, allowing dynamical quantum phase transitions without crossing an equilibrium phase boundary.

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 / 2 minor

Summary. The manuscript derives an exact necessary-and-sufficient condition for ground-state overlap dominance after a quantum quench in translationally invariant free-fermion systems whose Hamiltonians factorize into independent 2×2 momentum sectors: the initial and final Bloch vectors in each sector must have positive dot product. It shows that a recent phase-based conjecture fails in general, supplies explicit same-phase counterexamples in Kitaev chains where excited final states dominate the overlap, and links the geometric obstruction to real-time Fisher-zero crossings that permit dynamical quantum phase transitions without an equilibrium phase boundary.

Significance. If the central algebraic result holds, the paper supplies a parameter-free, geometrically transparent criterion that replaces the conjectured phase-based rule for a wide class of integrable models. The derivation rests on the product structure of the many-body overlap and the elementary two-level overlap formula, both standard in the BdG formalism. The counterexamples and Fisher-zero connection are direct consequences of the same geometry and constitute falsifiable predictions for future quench experiments.

major comments (1)
  1. [Abstract] Abstract, paragraph 3: the statement that the condition is necessary and sufficient is load-bearing for the entire claim; the manuscript must exhibit the explicit step that converts the per-sector overlap inequality |⟨g_i|g_f⟩| > |⟨g_i|e_f⟩| into the dot-product condition on the Bloch vectors, including the handling of the overall phase factor.
minor comments (2)
  1. The Kitaev-chain counterexamples should include a table or figure that lists the momenta where the dot product changes sign, together with the numerical values of the overlaps for the ground and first excited final states.
  2. Notation for the Bloch vectors (n_i(k) and n_f(k)) should be defined once in the main text before the central theorem, rather than only in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation for minor revision. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation of the central result.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 3: the statement that the condition is necessary and sufficient is load-bearing for the entire claim; the manuscript must exhibit the explicit step that converts the per-sector overlap inequality |⟨g_i|g_f⟩| > |⟨g_i|e_f⟩| into the dot-product condition on the Bloch vectors, including the handling of the overall phase factor.

    Authors: We agree that an explicit derivation of this conversion is necessary for full rigor and transparency. In the revised manuscript we will insert a dedicated paragraph (in the main text, immediately following the statement of the per-sector overlap) that proceeds as follows. For a single momentum sector the two-level Hamiltonian is H = (1/2) b · σ with unit Bloch vector b. The ground state |g⟩ satisfies ⟨g|σ|g⟩ = -b, so the overlap between initial and final ground states is |⟨g_i|g_f⟩| = √[(1 + b_i · b_f)/2]. The excited state |e_f⟩ is orthogonal to |g_f⟩, yielding |⟨g_i|e_f⟩| = √[(1 - b_i · b_f)/2]. The inequality |⟨g_i|g_f⟩| > |⟨g_i|e_f⟩| is therefore equivalent to b_i · b_f > 0. Any overall U(1) phase factor between the two bases can be absorbed into the definition of the states without affecting the moduli; alternatively, one may work in the real gauge where the eigenvectors are chosen real, rendering the phase factor unity. The many-body overlap is the product over sectors, so the global ground-state dominance holds if and only if the inequality is satisfied in every sector. We will also add a short footnote clarifying that the same algebra applies when the Bloch vectors are not normalized, replacing the dot product by the appropriate cosine of the angle between them. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation starts from the explicit assumption that the Hamiltonian factorizes into independent 2×2 momentum sectors, treats each as an isolated two-level system, and obtains the necessary-and-sufficient condition (positive Bloch-vector dot product in every sector) directly from the product form of the many-body overlap together with the elementary comparison of ground-to-ground versus ground-to-excited amplitudes inside a single sector. No parameter is fitted and then relabeled as a prediction, no uniqueness theorem is imported from prior self-work, and the cited conjecture on the transverse-field Ising model is used only as motivation rather than as a load-bearing premise for the new algebraic criterion. The result is therefore self-contained within the stated scope and the standard Bogoliubov-de Gennes structure of free-fermion Hamiltonians.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the standard decomposition of free-fermion Hamiltonians into independent 2x2 momentum sectors and the geometric representation of each sector by a Bloch vector; no new entities or fitted parameters are introduced.

axioms (1)
  • domain assumption Each momentum sector of the Hamiltonian can be written as an independent 2x2 matrix whose eigenstates are fully characterized by a Bloch vector on the unit sphere.
    Invoked in the abstract when the paper restricts attention to Hamiltonians that factorize into independent 2x2 momentum sectors.

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

  1. Comment on "Extension of the adiabatic theorem"

    cond-mat.stat-mech 2026-04 conditional novelty 7.0

    The conjecture that the postquench ground state always has the largest overlap with the initial ground state for same-phase quenches is disproved by an explicit gapped free-fermion counterexample.

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