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arxiv: 2507.11276 · v3 · submitted 2025-07-15 · ❄️ cond-mat.str-el · cond-mat.stat-mech· physics.comp-ph

Diagnosing phase transitions through time-scale entanglement

Stefan Rohshap , Hirone Ishida , Frederic Bippus , Leonard M. Verhoff , Anna Kauch , Karsten Held , Hiroshi Shinaoka , Markus Wallerberger This is my paper

Pith reviewed 2026-05-19 04:46 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechphysics.comp-ph
keywords time-scale entanglementquantics tensor trainphase transitionsquantum criticalityHubbard modelIsing modelcorrelation functionsmany-body physics
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0 comments X

The pith

Time-scale entanglement in quantum correlators increases near phase transitions and becomes scale-invariant at quantum critical points.

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

The paper introduces time-scale entanglement as a diagnostic for quantum phase transitions by examining how imaginary time scales couple in many-body systems. It shows that this coupling, measured through the bond dimension in a tensor-train representation of correlation functions, strengthens near transitions and crossovers. At true quantum critical points the entanglement becomes independent of scale. The approach works across multiple models and appears largely insensitive to which observable is chosen, suggesting a general probe of criticality that does not rely on conventional order parameters.

Core claim

Time-scale entanglement, accessed via the bond dimension of an n-particle correlator in the quantics tensor train representation, is generically enhanced in the vicinity of phase transitions and crossovers; at quantum critical points it becomes scale-invariant. This enhancement holds across finite-size Hubbard rings, the transverse-field Ising model, the single-impurity Anderson model, and the Mott transition, and remains largely independent of the specific observable.

What carries the argument

Quantics tensor train representation of correlators, in which the bond dimension directly encodes the degree of coupling between different imaginary time scales.

If this is right

  • Time-scale entanglement provides a universal diagnostic that detects both continuous and crossover transitions without requiring knowledge of an order parameter.
  • At quantum critical points the scale-invariant behavior of the bond dimension offers a direct signature of criticality.
  • The method applies equally to finite-size systems, impurity models, and lattice models such as the Hubbard and Ising cases.
  • Because the enhancement is largely observable-independent, the same diagnostic can be reused across different correlation functions.

Where Pith is reading between the lines

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

  • The same bond-dimension growth might appear in classical statistical mechanics near thermal critical points if imaginary-time correlators are replaced by spatial ones.
  • Scale invariance at criticality suggests a possible link to renormalization-group flow, where relevant operators mix time scales.
  • The approach could be tested on non-equilibrium steady states or driven systems to see whether time-scale entanglement also diagnoses dynamical phase transitions.

Load-bearing premise

The bond dimension in the quantics tensor train representation of an n-particle correlator faithfully encodes coupling between temporal scales in a way that is largely independent of the chosen observable.

What would settle it

A direct computation on a model with a known phase transition showing that the bond dimension stays flat or decreases when approaching the transition, or varies strongly with the choice of correlator.

Figures

Figures reproduced from arXiv: 2507.11276 by Anna Kauch, Frederic Bippus, Hirone Ishida, Hiroshi Shinaoka, Karsten Held, Leonard M. Verhoff, Markus Wallerberger, Stefan Rohshap.

Figure 1
Figure 1. Figure 1: FIG. 1. Entanglement of exponentially di [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: (b), the spin susceptibility χS is depicted. The red dots mark the Kondo temperature TK = V √ Ue−πU/8V 2 estimated via poor man’s scaling [51, 52], which delineates the boundary between Kondo and local moment regimes, as also reflected in the spin susceptibility behavior. Dmax has a broad peak lo￾cated in the vicinity of the Kondo temperature, diagnosing the thermally driven Kondo to local moments regime c… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Four-site Hubbard ring with nearest-neighbor hopping only: [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Four-site Hubbard ring with next nearest-neighbor hopping: [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. QTT bond dimension [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Hubbard dimer [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Hubbard dimer [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Hubbard dimer [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Hubbard dimer [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Four-site Hubbard ring with nearest-neighbor hopping only: [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Various entanglement measures in comparison to the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Four-site Hubbard ring with nearest-neighbor [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Four-site Hubbard ring with next-nearest neighbor hopping: [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The local spin susceptibility [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Various entanglement measures in comparison to the [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. U slice of the [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
read the original abstract

Spatial entanglement of quantum states has become a central paradigm of many-body physics. Here, we unearth a fundamentally different form of entanglement, the entanglement between imaginary time scales. This time-scale entanglement is accessible through quantics tensor train diagnostics (QTTD), where the bond dimension of an $n$-particle correlator encodes the coupling between temporal scales. Our central result is that time-scale entanglement is generically enhanced in the vicinity of phase transitions and crossovers. At quantum critical points, it becomes scale-invariant. We demonstrate time-scale entanglement across a range of systems, including finite-size Hubbard rings, the transverse-field Ising model, the single-impurity Anderson model, and the Mott transition in the Hubbard model. Remarkably, the enhanced time-scale entanglement is largely independent of the specific observable, establishing QTTD as a universal and unbiased diagnostic of criticality.

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 paper introduces 'time-scale entanglement' as a diagnostic for phase transitions, defined via the bond dimension of n-particle correlators in the quantics tensor train (QTT) representation. The central claim is that this entanglement is generically enhanced near phase transitions and crossovers, becomes scale-invariant at quantum critical points, and is largely independent of the specific observable chosen. Demonstrations are presented for finite-size Hubbard rings, the transverse-field Ising model, the single-impurity Anderson model, and the Mott transition in the Hubbard model.

Significance. If substantiated, the result offers a potentially universal, observable-independent diagnostic for criticality based on temporal scale coupling in tensor networks. The multi-model numerical evidence and the claim of scale invariance at QCPs could complement existing entanglement measures in condensed-matter theory, provided the mapping from bond dimension to inter-scale entanglement is shown to be robust rather than correlator-specific.

major comments (2)
  1. [Demonstrations and QTTD definition] The central claim that time-scale entanglement is 'largely independent of the specific observable' (abstract) is load-bearing but rests on the untested assumption that QTT bond dimension encodes genuine cross-scale couplings rather than observable-specific decay rates or analytic structure. Explicit variation of the quantics discretization (binary/dyadic splitting) and truncation thresholds across correlators (e.g., single-particle Green's function vs. two-particle susceptibility) is required to address this.
  2. [Numerical results] § on numerical evidence: the manuscript provides demonstrations across models but lacks quantitative metrics, error bars, or systematic checks against post-hoc parameter choices in the QTT representation, weakening the support for generic enhancement and scale invariance at QCPs.
minor comments (2)
  1. [Methods] Clarify the precise definition of the quantics mapping and how bond dimension is extracted from the tensor train; notation for the time-scale entanglement measure could be made more explicit.
  2. [Abstract] The abstract would benefit from a brief mention of the range of bond-dimension values or scaling exponents observed, to give readers a quantitative sense of the effect size.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation and robustness of our results on time-scale entanglement.

read point-by-point responses

  1. Referee: [Demonstrations and QTTD definition] The central claim that time-scale entanglement is 'largely independent of the specific observable' (abstract) is load-bearing but rests on the untested assumption that QTT bond dimension encodes genuine cross-scale couplings rather than observable-specific decay rates or analytic structure. Explicit variation of the quantics discretization (binary/dyadic splitting) and truncation thresholds across correlators (e.g., single-particle Green's function vs. two-particle susceptibility) is required to address this.

    Authors: We agree that explicit tests of discretization and truncation choices would further substantiate the robustness of the QTTD bond dimension as a measure of cross-scale couplings. Our existing demonstrations already employ different correlators (single-particle Green's functions and two-particle susceptibilities) across multiple models and observe consistent qualitative behavior, which supports independence from specific analytic structure. In the revised manuscript we will add explicit comparisons varying binary versus dyadic quantics splittings and truncation thresholds for representative correlators in the transverse-field Ising model and Hubbard ring, confirming that the enhancement near transitions and scale invariance at criticality remain qualitatively unchanged. revision: yes

  2. Referee: [Numerical results] § on numerical evidence: the manuscript provides demonstrations across models but lacks quantitative metrics, error bars, or systematic checks against post-hoc parameter choices in the QTT representation, weakening the support for generic enhancement and scale invariance at QCPs.

    Authors: We acknowledge that adding quantitative metrics and systematic checks will improve the clarity and strength of the numerical evidence. In the revision we will incorporate error bars derived from QTT truncation and discretization convergence, along with systematic parameter scans (e.g., maximum bond dimension and splitting level) for key figures in the Hubbard and Ising models. These additions will quantify the stability of the reported bond-dimension trends and provide stronger support for the generic enhancement and scale-invariance claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central result is numerical observation across models

full rationale

The paper introduces QTTD by defining time-scale entanglement via the bond dimension of the quantics tensor train representation of correlators, then numerically computes this quantity for multiple models (Hubbard rings, Ising, Anderson, Mott transition) and observes enhancement near transitions and scale invariance at QCPs. This is an empirical demonstration rather than a derivation that reduces to its inputs by construction. No self-citations load-bear the central claim, no parameters are fitted and then relabeled as predictions, and the independence from specific observables is asserted from explicit cross-observable checks rather than assumed. The result does not rename a known pattern by fiat but reports a new diagnostic whose validity rests on the numerical evidence presented.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces time-scale entanglement as a new interpretive layer on top of standard tensor-train decompositions and relies on the domain assumption that bond dimension directly reflects temporal-scale coupling.

axioms (1)
  • domain assumption The quantics tensor train representation of correlators accurately captures multi-scale temporal correlations without dominant truncation artifacts.
    Invoked when equating bond dimension growth to enhanced time-scale entanglement near criticality.
invented entities (1)
  • time-scale entanglement no independent evidence
    purpose: Diagnostic quantity that signals proximity to phase transitions via coupling of imaginary-time scales.
    Newly defined concept whose independent existence outside the QTTD framework is not demonstrated.

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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. Adaptive Patching for Tensor Train Computations

    physics.comp-ph 2026-02 unverdicted novelty 7.0

    An adaptive patching method exploits block-sparse QTT structures to reduce computational costs for tensor contractions and enables efficient evaluation of bubble diagrams and Bethe-Salpeter equations.

Reference graph

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    0 𝜇 / U 0 .0 0.5 1.0 D m a x of G 1 1 11 ↑↑↑ ↑ , T = 1/30 ( b ) 60 80 100 120 FIG. 5. QTT bond dimension Dmax of different imaginary-time cor- relation functions for the Hubbard dimer as a function of electronic repulsion U and chemical potential µ/U: (a) Dmax for the one-particle Green’s function G↑↑ 11 at temperature T = β−1 = 1/ 50, and (b) for the two-...

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