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arxiv: 2606.20387 · v1 · pith:BRP6KUOEnew · submitted 2026-06-18 · 🪐 quant-ph · cond-mat.stat-mech

Interaction geometry and ground-state properties of sparse quantum lattice models

Alex Gunning , Sebastian Schmid , Zhengxiao Liu , Sridevi Kuriyattil , Aydin Deger , Andrew J. Daley This is my paper

Pith reviewed 2026-06-26 17:03 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords long-range interactionsquantum lattice modelsphase transitionsinteraction geometrysparse graphsgeometric frustrationground-state properties
0
0 comments X

The pith

Phase structure and criticality in sparse long-range quantum models are governed by one effective-geometry principle.

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

The paper investigates graphs whose degree grows only logarithmically with system size and shows how their connectivity symmetry and frustration shape ground-state phases. Even-p power-of-p graphs retain the complex transitions seen in the power-of-two case, odd-p graphs are controlled by geometric frustration, and Fibonacci graphs supply a direct geometric link between short- and long-range limits. A sympathetic reader cares because the same principle appears to organise criticality across these families, offering a unified description for experimentally relevant long-range systems. The work therefore extends earlier models by demonstrating that interaction geometry, rather than microscopic details, organises the low-energy behaviour.

Core claim

Across power-of-p and Fibonacci graphs, phase structure and criticality are governed by the same effective-geometry principle. Even values of p produce rich phase diagrams inherited from the power-of-two model, while odd values are dominated by geometric frustration; Fibonacci graphs lack discrete self-similarity yet map short-range to long-range limits directly through their connectivity.

What carries the argument

The effective-geometry principle that organises phase structure and criticality according to symmetry and frustration in the interaction graph.

If this is right

  • Even-p graphs retain the rich phase structure of the power-of-two model.
  • Odd-p graphs are governed by geometric frustration that suppresses or reshapes transitions.
  • Fibonacci graphs exhibit a direct geometric mapping between short- and long-range limits without discrete self-similarity.
  • The same effective-geometry principle organises both phase structure and criticality across the models studied.

Where Pith is reading between the lines

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

  • The principle could be tested by constructing quantum simulators whose interaction graphs are tuned to match the even-p, odd-p, or Fibonacci families.
  • If the principle holds more generally, it may allow prediction of ground-state properties for other logarithmically sparse graphs not yet simulated.
  • Experimental platforms with tunable long-range couplings could directly map measured critical exponents onto the geometric features identified here.

Load-bearing premise

The specific graphs examined are representative of the broader class of sparse models whose degree grows logarithmically with system size.

What would settle it

Observation of qualitatively different phase behaviour or criticality in another sparse graph with logarithmic degree growth that cannot be accounted for by the same symmetry, frustration, or geometric-mapping rules would falsify the unifying claim.

Figures

Figures reproduced from arXiv: 2606.20387 by Alex Gunning, Andrew J. Daley, Aydin Deger, Sebastian Schmid, Sridevi Kuriyattil, Zhengxiao Liu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

We investigate how interaction geometry shapes the low-energy phases of sparse tunable long-range quantum models. We focus on a class of graphs whose degree grows logarithmically with system size, and show how symmetry and frustration in graph connectivity can drive, suppress, and reshape ground-state phase transitions. The central examples are power-of-$p$ graphs, where even and odd values of $p$ exhibit qualitatively distinct behaviour: even-$p$ graphs inherit the rich phase structure of the power-of-two model, while odd-$p$ graphs are governed by geometric frustration. Fibonacci graphs provide a contrasting case, lacking the discrete self-similarity of the power-of-$p$ family but exhibiting a direct geometric mapping between the short- and long-range limits. Across our models, we find that phase structure and criticality are governed by the same effective-geometry principle, unifying our framework for experimentally motivated long-range quantum systems.

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 how interaction geometry shapes low-energy phases in sparse tunable long-range quantum models on graphs whose degree grows logarithmically with system size. Central examples are power-of-p graphs (even p inherits rich phase structure from the power-of-two case; odd p is governed by geometric frustration) and Fibonacci graphs (lacking discrete self-similarity but possessing a direct short-to-long-range geometric mapping). The paper concludes that phase structure and criticality are controlled by a single effective-geometry principle that unifies the framework for experimentally motivated long-range systems.

Significance. If the effective-geometry principle is shown to control criticality and the examined graphs are representative, the work would supply a unifying lens for classifying ground states in sparse long-range quantum models, with direct relevance to experimental platforms that realize tunable long-range interactions.

major comments (1)
  1. [Abstract] Abstract (final sentence): the claim that 'phase structure and criticality are governed by the same effective-geometry principle, unifying our framework' is load-bearing on the assumption that the specific graphs (power-of-p even/odd p, Fibonacci) are representative of the broader class of logarithmically sparse graphs. No argument, additional examples, or mapping to other members of the class is supplied to justify the extrapolation from the reported cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify the scope of our claims. The single major comment concerns the generality of the effective-geometry principle. We address it directly below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the claim that 'phase structure and criticality are governed by the same effective-geometry principle, unifying our framework' is load-bearing on the assumption that the specific graphs (power-of-p even/odd p, Fibonacci) are representative of the broader class of logarithmically sparse graphs. No argument, additional examples, or mapping to other members of the class is supplied to justify the extrapolation from the reported cases.

    Authors: We agree that the abstract statement would benefit from explicit support for generality. The power-of-p (even/odd) and Fibonacci graphs were deliberately chosen as representative archetypes within the logarithmically sparse class: even-p cases preserve the hierarchical symmetry of the p=2 model, odd-p cases introduce geometric frustration, and Fibonacci graphs remove discrete self-similarity while retaining a direct short-to-long-range geometric correspondence. The effective-geometry principle itself is formulated in terms of the shared logarithmic degree growth and the resulting distribution of interaction ranges, which is the defining property of the class. Nevertheless, the manuscript does not contain an explicit mapping to additional members (e.g., random recursive graphs or other non-deterministic constructions). We will therefore add a short subsection in the Discussion that (i) states the minimal conditions on the degree sequence required for the principle to apply and (ii) sketches how the same geometric classification extends to two further logarithmically sparse families. The abstract sentence will be softened to reflect that the principle unifies the examined cases and is expected to apply more broadly under those conditions. These changes constitute a targeted major revision. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper reports numerical and analytical results on phase structure for explicitly enumerated graph families (power-of-p with even/odd p, Fibonacci). No equations are presented that define a quantity in terms of itself, no fitted parameters are relabeled as predictions, and the central effective-geometry principle is stated as an observed pattern across the studied cases rather than derived by reducing to prior self-citations or ansatzes. The representativeness concern raised in the skeptic note is an external-validity issue, not a circularity reduction. The derivation chain is therefore self-contained against the reported models.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.1-grok · 5698 in / 945 out tokens · 27608 ms · 2026-06-26T17:03:13.905008+00:00 · methodology

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Reference graph

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