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arxiv: 2605.24645 · v1 · pith:2TMGBXSZnew · submitted 2026-05-23 · 🪐 quant-ph

Geometric phases of reduced states in the transverse-field Ising chain

Chiragkumar R. Vasani , Erik Sj\"oqvist This is my paper

Pith reviewed 2026-06-30 12:56 UTC · model grok-4.3

classification 🪐 quant-ph
keywords geometric phasesreduced density matricesquantum phase transitionstransverse-field Ising modelinterferometric phaseUhlmann phasemixed states
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The pith

The interferometric geometric phase from two-site reduced states of the transverse-field Ising chain marks the quantum phase transition more reliably than the Uhlmann phase.

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

The paper computes two mixed-state geometric phases from two-site reduced density matrices of the transverse-field Ising chain. Coordinated local unitary rotations are applied to these reduced states, and the resulting phases are tracked as the transverse field strength crosses the known critical value. The interferometric phase displays a clear signature of the transition while the Uhlmann phase does not. A sympathetic reader cares because the result suggests that local geometric quantities can capture global many-body critical behavior without requiring the full chain state.

Core claim

In the transverse-field Ising model, after coordinated local unitary rotations are applied to two-site reduced density matrices, the interferometric geometric phase reliably indicates the quantum phase transition across the critical point, whereas the Uhlmann phase does not.

What carries the argument

two-site reduced density matrices of the Ising chain, subjected to coordinated local unitary rotations, from which the interferometric geometric phase and Uhlmann phase are extracted

Load-bearing premise

That geometric phases computed from locally rotated two-site reduced states faithfully reflect the global quantum phase transition of the full chain.

What would settle it

A calculation on larger finite chains in which the interferometric phase fails to show any distinguishing feature exactly at the known critical field value.

Figures

Figures reproduced from arXiv: 2605.24645 by Chiragkumar R. Vasani, Erik Sj\"oqvist.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) shows the interferometric GP deviation [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) shows the Uhlmann phase deviation [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) shows the interferometric GP deviation [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

Geometric phases have been extensively investigated in a wide range of quantum systems, often revealing deep connections to the underlying topology of many-body states. In this work, we examine two geometric phases defined for mixed quantum states$-$the interferometric geometric phase and the Uhlmann phase$-$extracted from two-site reduced density matrices of the transverse-field Ising model with nearest-neighbor interacting spins. By applying coordinated local unitary rotations to the spins, we compute the geometric phases associated with the two-site states across the critical point. We find that the interferometric phase is a more reliable indicator of the quantum phase transition in this model than the Uhlmann phase.

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

2 major / 2 minor

Summary. The manuscript computes the interferometric geometric phase and the Uhlmann phase from two-site reduced density matrices of the transverse-field Ising chain after coordinated local unitary rotations on the spins. It reports that the interferometric phase exhibits clearer signatures of the quantum phase transition at the critical point than the Uhlmann phase.

Significance. If the local phases after the chosen rotations are shown to faithfully track global observables, the result would provide a concrete example of how geometric phases of reduced states can serve as local diagnostics for quantum phase transitions in integrable spin chains, extending prior work on geometric phases in many-body systems.

major comments (2)
  1. [abstract, §3] The central claim that the interferometric phase is 'a more reliable indicator' of the QPT rests on the unverified assumption that coordinated local unitaries applied to two-site RDMs preserve the non-analyticities associated with the global ground-state transition (abstract and §3). No derivation or numerical test is provided showing that these phases reproduce the known divergence in the second derivative of the ground-state energy or the order-parameter jump at λ=1.
  2. [§4] At criticality the TFIM has power-law correlations; the manuscript does not demonstrate that the chosen local rotations commute with the global ℤ₂ symmetry or that the resulting two-site phases remain sensitive to the long-range entanglement structure rather than being artifacts of the marginalization (no section derives the fidelity between the local phase signatures and full-chain observables such as the magnetization or correlation length).
minor comments (2)
  1. [§2] Notation for the interferometric phase (e.g., definition of the parallel transport condition) should be stated explicitly with reference to the original literature rather than assumed.
  2. [Fig. 2] Figure captions should include the precise value of the transverse field at which each curve is evaluated and the system size used for the RDM extraction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [abstract, §3] The central claim that the interferometric phase is 'a more reliable indicator' of the QPT rests on the unverified assumption that coordinated local unitaries applied to two-site RDMs preserve the non-analyticities associated with the global ground-state transition (abstract and §3). No derivation or numerical test is provided showing that these phases reproduce the known divergence in the second derivative of the ground-state energy or the order-parameter jump at λ=1.

    Authors: We agree that the manuscript would benefit from an explicit link between the local geometric phases and the known non-analyticities of the global ground state. Our numerical results already indicate a sharper signature of the transition in the interferometric phase, but we did not provide a direct comparison to the second derivative of the ground-state energy or the magnetization jump. In the revised manuscript we will add a short derivation justifying why the chosen coordinated rotations preserve the critical non-analyticities together with numerical plots overlaying the phase derivatives against the energy second derivative and order parameter across λ=1. revision: yes

  2. Referee: [§4] At criticality the TFIM has power-law correlations; the manuscript does not demonstrate that the chosen local rotations commute with the global ℤ₂ symmetry or that the resulting two-site phases remain sensitive to the long-range entanglement structure rather than being artifacts of the marginalization (no section derives the fidelity between the local phase signatures and full-chain observables such as the magnetization or correlation length).

    Authors: The rotations are constructed to be invariant under the global ℤ₂ symmetry of the TFIM, but the manuscript indeed lacks an explicit verification of this property and of the fidelity with global observables. We will add a brief symmetry analysis of the rotations and a quantitative comparison of the interferometric-phase signatures with the magnetization and correlation length in the revised version to confirm that the local phases capture the essential critical behavior rather than marginalization artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: computations are direct and self-contained.

full rationale

The paper extracts interferometric and Uhlmann phases from two-site reduced density matrices of the transverse-field Ising chain after applying coordinated local unitaries, then compares their behavior across the critical point. No equations or claims reduce a derived quantity to a fitted parameter by construction, no self-citations are invoked as load-bearing uniqueness theorems, and the reported finding that one phase is a more reliable indicator rests on explicit numerical evaluation rather than definitional equivalence or imported ansatze. The derivation chain is therefore independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5633 in / 984 out tokens · 22911 ms · 2026-06-30T12:56:35.226535+00:00 · methodology

discussion (0)

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