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arxiv: 2504.01846 · v2 · submitted 2025-04-02 · 🪐 quant-ph · cond-mat.str-el

Scaling of entanglement entropy and correlations in the variable-range extended Ising model

Harikrishnan K J , Debasis Sadhukhan , Amit Kumar Pal This is my paper

Pith reviewed 2026-05-22 21:43 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords Ising modelentanglement entropyvariable-range interactionscoordination numbercritical pointcorrelation functionsquantum quench
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The pith

In the variable-range extended Ising model, bipartite entanglement at criticality decreases as a power law with coordination number Z, independent of partition size and alpha greater than 1.

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

The paper studies a one-dimensional Ising model with power-law decaying interactions whose range is controlled by a coordination number Z for each qubit. Correlations remain algebraic only up to distance Z and become exponentially short-ranged beyond that. At the critical point the bipartite entanglement scales as Z to the minus gamma, and this power-law decrease holds for any subsystem size and any decay exponent alpha larger than one. The same Z to the minus gamma scaling appears in the long-time averaged entanglement after a sudden quench from the infinite-field ground state to the critical Hamiltonian.

Core claim

The authors show that in this exactly solvable model the bipartite entanglement in the ground state at the critical point obeys a power-law decay proportional to Z to the minus gamma with increasing coordination number, and that the scaling is independent of partition size and of the value of alpha whenever alpha exceeds one; the same scaling governs the long-time averaged entanglement after a quench to criticality.

What carries the argument

The coordination number Z that sets the interaction range in the variable-range extended Ising model, allowing exact computation of correlations and entanglement for arbitrary Z and alpha greater than 1.

If this is right

  • Two-point correlations follow algebraic decay only for distances up to Z and exponential decay thereafter.
  • Bipartite entanglement at the critical point decreases as Z to the minus gamma for any partition size when alpha exceeds 1.
  • The long-time averaged entanglement after a quench from the infinite transverse field to the critical Hamiltonian obeys the same power-law decrease with Z.

Where Pith is reading between the lines

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

  • The result suggests that increasing coordination number effectively suppresses long-range contributions to entanglement in a universal manner.
  • Analogous scaling may appear in other exactly solvable models with power-law interactions when coordination is varied.
  • Trapped-ion or Rydberg-atom simulators with tunable interaction ranges could test the claimed independence from alpha.

Load-bearing premise

The variable-range extended Ising model remains exactly solvable when the interaction range is parameterized by the coordination number Z, for arbitrary Z and alpha greater than 1.

What would settle it

Numerical evaluation of the bipartite entanglement for successively larger Z at the critical point that fails to exhibit the predicted power-law decrease with Z would falsify the scaling result.

Figures

Figures reproduced from arXiv: 2504.01846 by Amit Kumar Pal, Debasis Sadhukhan, Harikrishnan K J.

Figure 1
Figure 1. Figure 1: FIG. 1. A chain of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Dispersion relation (see Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Variation of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Variation of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Variation of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Variation of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Variation of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

We study the two-point correlation functions and the bipartite entanglement in the ground state of the exactly-solvable variable-range extended Ising model of qubits in the presence of a transverse field on a one-dimensional lattice. We introduce the variation in the range of interaction by varying the coordination number, $\mathcal{Z}$, of each qubit, where the interaction strength between a pair of qubits at a distance $r$ varies as $\sim r^{-\alpha}$. We show that the algebraic nature of the correlation functions is present only up to $r=\mathcal{Z}$, above which it exhibits short-range exponential scaling. We also show that at the critical point, the bipartite entanglement exhibits a power-law decrease ($\sim\mathcal{Z}^{-\gamma}$) with increasing coordination number irrespective of the partition size and the value of $\alpha$ for $\alpha>1$. We further consider a sudden quench of the system starting from the ground state of the infinite-field limit of the system Hamiltonian via turning on the critical Hamiltonian, and demonstrate that the long-time averaged bipartite entanglement exhibits a qualitatively similar variation ($\sim\mathcal{Z}^{-\gamma}$) with $\mathcal{Z}$.

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 examines the ground state of the variable-range extended Ising model on a 1D lattice, with interaction range controlled by coordination number Z and decay exponent α. It asserts exact solvability via Jordan-Wigner mapping to free fermions, shows that two-point correlations are algebraic only up to r = Z and exponential thereafter, and reports that at criticality the bipartite entanglement scales as Z^{-γ} with γ independent of partition size and of α (for α > 1). A similar Z^{-γ} scaling is claimed for the long-time-averaged entanglement after a quench from the infinite transverse-field limit.

Significance. If the reported partition-size-independent power-law scaling of entanglement with Z is robust, the result would clarify how tunable coordination number modulates entanglement in long-range spin chains, offering a concrete handle on effective dimensionality and correlation spreading. The exact solvability and the explicit demonstration of the crossover from algebraic to exponential correlations constitute clear technical strengths.

major comments (2)
  1. [Sec. IV and abstract] The central claim that the exponent γ in the Z^{-γ} scaling of bipartite entanglement at criticality is independent of partition size (abstract and Sec. IV) rests on the assumption that the same γ governs both the L ≪ Z (effectively long-range) and L ≫ Z (effectively short-range) regimes. Because the cutoff at r = Z changes the character of the interactions seen by the subsystem, explicit numerical or analytic verification for at least two widely separated L/Z ratios is required to substantiate independence; the manuscript does not appear to provide this cross-regime comparison.
  2. [Sec. IV] The statement that the scaling holds 'irrespective of the value of α for α > 1' (abstract) needs a quantitative check that γ remains constant across a range of α > 1; if γ(α) varies, the universality claim is weakened. Please report the fitted γ values for several α (e.g., α = 1.5, 2, 3) at fixed critical point.
minor comments (2)
  1. Notation: the coordination number is denoted both as Z and as script-Z in the abstract; adopt a single symbol throughout.
  2. Figure captions should explicitly state the system size L, the value of α, and the partition sizes used for each entanglement curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below. Where the referee correctly identifies the need for additional verification, we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Sec. IV and abstract] The central claim that the exponent γ in the Z^{-γ} scaling of bipartite entanglement at criticality is independent of partition size (abstract and Sec. IV) rests on the assumption that the same γ governs both the L ≪ Z (effectively long-range) and L ≫ Z (effectively short-range) regimes. Because the cutoff at r = Z changes the character of the interactions seen by the subsystem, explicit numerical or analytic verification for at least two widely separated L/Z ratios is required to substantiate independence; the manuscript does not appear to provide this cross-regime comparison.

    Authors: We agree that the manuscript would be strengthened by explicit cross-regime verification. In the revised version we will add numerical results for the bipartite entanglement entropy (computed from the free-fermion correlation matrix) at two widely separated L/Z ratios, for example L/Z ≈ 0.1 and L/Z ≈ 10, at the critical point. These data will be presented in Sec. IV to confirm that the fitted exponent γ is the same in both regimes. revision: yes

  2. Referee: [Sec. IV] The statement that the scaling holds 'irrespective of the value of α for α > 1' (abstract) needs a quantitative check that γ remains constant across a range of α > 1; if γ(α) varies, the universality claim is weakened. Please report the fitted γ values for several α (e.g., α = 1.5, 2, 3) at fixed critical point.

    Authors: We will add a quantitative check in the revised Sec. IV. We will report the fitted γ values obtained from numerical fits to the entanglement data for α = 1.5, 2, and 3 at the critical point (with fixed Z range), together with a brief discussion of any observed variation. This will make the claim of α-independence for α > 1 fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results follow from model solvability

full rationale

The paper defines a variable-range Ising model with coordination number Z cutoff and states it remains exactly solvable via Jordan-Wigner to free fermions. All reported scalings (algebraic correlations to r=Z, entanglement ~Z^{-γ} at criticality independent of partition size for α>1, and post-quench averages) are presented as direct numerical or analytical consequences of this definition and standard diagonalization. No equations reduce a claimed prediction to a fitted parameter by construction, no load-bearing self-citations appear in the abstract or described chain, and no ansatz or uniqueness theorem is imported from prior author work. The derivation chain is self-contained against the model Hamiltonian.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on the exact solvability of the parameterized Hamiltonian and the definition of interactions via Z and α; no new entities are introduced.

free parameters (2)
  • α
    Decay exponent of the interaction strength, treated as a variable parameter with the condition α>1.
  • Z
    Coordination number that sets the effective interaction range, varied to extract scaling.
axioms (1)
  • domain assumption The variable-range extended Ising model is exactly solvable for arbitrary coordination number Z and decay exponent α.
    Invoked to justify exact computation of ground-state correlations and entanglement.

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