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arxiv: 2604.12754 · v1 · submitted 2026-04-14 · ❄️ cond-mat.str-el · quant-ph

Unconventional entanglement scaling and quantum criticality in the long-range spin-one Heisenberg chain with single-ion anisotropy

Patrick Adelhardt , Sean R. Muleady , Kai P. Schmidt , Alexey V. Gorshkov This is my paper

Pith reviewed 2026-05-10 14:05 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords long-range interactionsquantum criticalityHeisenberg chainHaldane phasecontinuous symmetry breakingentanglement entropyspin-onesingle-ion anisotropy
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The pith

Long-range interactions cause critical exponents to vary continuously with decay rate at transitions out of the Haldane phase.

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

The paper determines the phase diagram of the spin-one Heisenberg chain with single-ion anisotropy under staggered antiferromagnetic power-law interactions. Long-range couplings stabilize continuous symmetry breaking phases that compete directly with the topological Haldane phase. Numerical matrix-product-state simulations and series expansions reveal that the critical boundaries show exponents varying continuously with the interaction decay exponent, accompanied by logarithmic corrections to entanglement entropy beyond short-range expectations. These results matter because they identify a concrete, experimentally accessible model where long-range forces reshape both symmetry breaking and topological regimes.

Core claim

The large-D-to-U(1)-CSB transition exhibits unconventional, continuously varying critical exponents as a function of the long-range decay exponent, with strong dependence on boundary conditions leading to distinct finite-size scalings for long-range potentials. The Haldane-to-U(1)-CSB transition likewise displays unconventional quantum criticality with distinct continuously varying critical exponents. Entanglement entropy scaling in the U(1) and SU(2) CSB phases includes logarithmic corrections beyond those from Goldstone modes.

What carries the argument

Staggered antiferromagnetic power-law interactions with tunable decay exponent, which stabilize CSB phases in competition with the Haldane phase and are analyzed through finite-size scaling of entanglement entropy and staggered magnetization.

Load-bearing premise

Matrix-product-state calculations and series expansions accurately capture long-range interactions and finite-size effects without significant truncation or convergence artifacts near the critical boundaries.

What would settle it

A measurement of how the scaling of entanglement entropy or staggered magnetization changes as the interaction decay exponent is tuned across the Haldane-to-U(1)-CSB boundary in an experimental atomic platform would confirm or refute the continuous variation of exponents.

Figures

Figures reproduced from arXiv: 2604.12754 by Alexey V. Gorshkov, Kai P. Schmidt, Patrick Adelhardt, Sean R. Muleady.

Figure 1
Figure 1. Figure 1: Quantum phase diagram as a function of the single [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spin-spin correlations ⟨S σ 1 S σ j ⟩ for the σ = x, z spin components, as well as the string-order correlator O z 1,j be￾tween site 1 and an arbitrary site j. The trivial large-D phase (a) is characterized by exponentially decaying correlations for all quantities. The Z2 AF phase (b) exhibits finite staggered spin correlations in the z component, the U(1) CSB phase (c) in the x (and y) component, and the … view at source ↗
Figure 3
Figure 3. Figure 3: Logarithmic coefficient b of the entangle￾ment entropy as a function of the decay exponent α for D = −0.5, 0, 0.5, and 1.25 in panels (a), (b), (c), and (d), respectively. Data points include errorbars from the extrap￾olation procedure. The ground state in the mz = 0 sector is used to compute the entanglement entropy, except in the Haldane phase, where the mz = 1 ground state is taken to avoid the residual… view at source ↗
Figure 4
Figure 4. Figure 4: Analysis of the Gaussian transition between the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Critical exponents ν (panel (a)) and β (panel (b)) and the corresponding errors from the extrapolation proce￾dure of the Ising transition between the Haldane and Z2 AF phase from data collapses of the staggered Mz magnetization curves for various decay exponents α. Gray lines represent the expected critical values for a 2D Ising transition. The critical exponents are close to the expected values but start … view at source ↗
Figure 6
Figure 6. Figure 6: Critical exponents zν (a), γ (b), ν (c), and Θ (d) and their errors obtained from the data-collapse extrapolation procedure for the large-D-to-U(1)-CSB transition determined from pCUT+MC (panels (a) and (b)), MPS and ED (panels (c) and (d)) as a function of the decay exponent σ = α − d, where d = 1. Gray lines indicate the expected long-range mean-field values from the long-range O(2) quantum rotor model a… view at source ↗
Figure 7
Figure 7. Figure 7: Critical exponents ν and β and their errors obtained from the scaling procedure in the different critical regimes as a function of the anisotropy parameter D. The D-axis is lin￾ear for D ≤ 1, while for D > 1 it is uniformly spaced in 1/D. The results are obtained from three different finite-size scaling procedures using MPS: standard data collapse in D and λ, biased collapse using the critical point λ pCUT… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the critical exponents ν (panel (a)) and β (panel (b)), including uncertainties from the scaling analysis, for the Haldane-to-U(1)-CSB and large-D-to-U(1)- CSB transitions as a function of the decay exponent σ = α−d, where d = 1. The exponents associated with the large-D-to￾U(1)-CSB transition display smoothly varying exponents with a sharp increase at σ ≈ 2, whereas those of the Haldane-to￾U… view at source ↗
Figure 9
Figure 9. Figure 9: Scaling behavior of the entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Constant term c of the entanglement en￾tropy as a function of the decay exponent α for D = −0.5, 0, 0.5, and 1.25 in panels (a), (b), (c), and (d), respectively. Data points include errorbars from the extrap￾olation procedure. The ground state in the mz = 0 sector is used to compute the entanglement entropy, except in the Haldane phase, where the mz = 1 ground state is taken to avoid the residual ln 2 con… view at source ↗
Figure 11
Figure 11. Figure 11: Representative data collapses probing the Ising, SU(2), and U(1) continuous symmetry breaking (CSB) transitions [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Scaling of the critical points Dc, αc, and λc = 1/Dc (first row), the critical exponents ν (second row), and β (third row) obtained from data collapses at α = 5, D ∈ {0, 0.4, 1.25}, and α = 1.75, plotted versus the inverse system size 1/Lmin. The minimal system size Lmin corresponds to the lower bound of a sliding window used for the collapses, with each window containing eight system sizes and shifted se… view at source ↗
read the original abstract

Long-range interactions can fundamentally reshape the low-energy properties of low-dimensional quantum matter, altering both continuous symmetry breaking and topological phenomena. However, their impact on the quantum criticality separating these regimes remains poorly understood. We determine the ground-state phase diagram and critical properties of the spin-one Heisenberg chain with single-ion anisotropy and staggered antiferromagnetic power-law interactions, using matrix-product state (MPS) calculations complemented by high-order series expansions (pCUT+MC). Such long-range, non-frustrated interactions circumvent the Hohenberg-Mermin-Wagner theorem, thereby stabilizing continuous symmetry breaking (CSB) phases in direct competition with the Haldane phase. We map out the resulting phase diagram and analyze the entanglement entropy scaling behavior in the U(1) and SU(2) CSB phases, finding logarithmic corrections beyond the short-range, universal contributions expected from linearly dispersed Goldstone modes. We further characterize all critical boundaries through finite-size scaling of either the entanglement entropy or the staggered magnetization. In particular, the large-D-to-U(1)-CSB transition exhibits unconventional, continuously varying critical exponents as a function of the long-range decay exponent with a strong dependence on the imposed boundary conditions leading to distinct finite-size scalings for sufficiently long-range potentials. Remarkably, the Haldane-to-U(1)-CSB transition likewise displays unconventional quantum criticality with distinct continuously varying critical exponents. Our work positions this model as a target for near-term atomic platforms with tunable long-range couplings and exhibiting natural single-ion anisotropy, offering a minimal playground for exploring the interplay between long-range interactions, continuous symmetry breaking, and topology.

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 studies the ground-state phase diagram of the spin-1 Heisenberg chain with single-ion anisotropy and staggered antiferromagnetic power-law interactions using matrix-product-state (MPS) simulations and high-order perturbative continuous unitary transformations combined with Monte Carlo (pCUT+MC). It reports that long-range interactions stabilize U(1) and SU(2) continuous symmetry breaking phases in competition with the Haldane phase, with entanglement entropy exhibiting logarithmic corrections beyond short-range expectations, and that both the large-D-to-U(1)-CSB and Haldane-to-U(1)-CSB transitions display unconventional quantum criticality characterized by continuously varying critical exponents that depend on the interaction decay exponent and on boundary conditions.

Significance. If the numerical results on convergence and finite-size scaling are robust, the identification of continuously varying exponents at these transitions would provide a concrete example of how long-range interactions can modify quantum criticality in systems hosting topological phases, with potential relevance to tunable atomic platforms. The complementary use of MPS and series expansions is a methodological strength.

major comments (2)
  1. The abstract and methods description provide no quantitative data on MPS bond-dimension extrapolation, truncation errors, or series-order stability for the long-range power-law terms when extracting the continuously varying exponents via finite-size scaling of entanglement entropy or staggered magnetization. This information is load-bearing for the central claim of unconventional criticality, as incomplete convergence could shift apparent critical points or scaling forms, especially for sufficiently long-range potentials where the paper itself notes distinct BC-dependent scalings.
  2. The finite-size scaling analysis for the large-D-to-U(1)-CSB transition (and similarly for the Haldane-to-U(1)-CSB boundary) reports continuously varying exponents as a function of the decay exponent, but does not include explicit checks (e.g., data collapse quality or extrapolation to infinite bond dimension) demonstrating that the observed BC dependence reflects asymptotic behavior rather than truncation artifacts in the long-range tail.
minor comments (2)
  1. Clarify the precise definition of the staggered magnetization observable used for scaling and how open versus periodic boundary conditions are implemented for the power-law interactions in the MPS code.
  2. The discussion of logarithmic corrections to entanglement entropy in the CSB phases would benefit from an explicit comparison to the expected form for Goldstone modes in the presence of long-range interactions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address each of the major comments below and will incorporate the suggested improvements in the revised version.

read point-by-point responses

  1. Referee: The abstract and methods description provide no quantitative data on MPS bond-dimension extrapolation, truncation errors, or series-order stability for the long-range power-law terms when extracting the continuously varying exponents via finite-size scaling of entanglement entropy or staggered magnetization. This information is load-bearing for the central claim of unconventional criticality, as incomplete convergence could shift apparent critical points or scaling forms, especially for sufficiently long-range potentials where the paper itself notes distinct BC-dependent scalings.

    Authors: We agree with the referee that providing quantitative convergence data is essential to support the claims of unconventional criticality. In the revised manuscript, we will include a new appendix with detailed information on MPS bond-dimension extrapolations, including plots of entanglement entropy and magnetization versus bond dimension for representative parameters, along with estimated truncation errors. For the pCUT+MC method, we will add data on the convergence with series order. These analyses show that the results are well-converged and the continuously varying exponents are reliable. revision: yes

  2. Referee: The finite-size scaling analysis for the large-D-to-U(1)-CSB transition (and similarly for the Haldane-to-U(1)-CSB boundary) reports continuously varying exponents as a function of the decay exponent, but does not include explicit checks (e.g., data collapse quality or extrapolation to infinite bond dimension) demonstrating that the observed BC dependence reflects asymptotic behavior rather than truncation artifacts in the long-range tail.

    Authors: We thank the referee for pointing this out. While our current analysis includes finite-size scaling for different system sizes, we will enhance the revised version by adding explicit data collapse plots for the scaling forms at the transitions, for various decay exponents and boundary conditions. Additionally, we will include extrapolations of the critical exponents to infinite bond dimension, confirming that the BC-dependent scalings are physical and not due to truncation in the long-range interactions. This will be supported by new figures and discussion. revision: yes

Circularity Check

0 steps flagged

No circularity in numerical MPS/pCUT+MC analysis of phase diagram and critical exponents

full rationale

The paper determines the phase diagram and critical properties exclusively through direct numerical computations: matrix-product state simulations for ground states and entanglement entropy, supplemented by high-order pCUT series expansions with Monte Carlo sampling for the long-range model. Critical boundaries are characterized via standard finite-size scaling of entanglement entropy or staggered magnetization, yielding extracted exponents as outputs of these calculations rather than any analytical derivation that reduces to fitted inputs, self-definitions, or self-citations by construction. No load-bearing steps invoke uniqueness theorems, ansatzes smuggled via prior work, or renaming of known results; the continuously varying exponents emerge from the computed observables under varying decay exponents and boundary conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of quantum many-body physics and the validity of the chosen numerical methods for long-range Hamiltonians; no new entities are postulated.

axioms (2)
  • domain assumption The model Hamiltonian with power-law interactions is well-defined and can be accurately approximated by matrix-product states and perturbative series expansions.
    Invoked throughout the abstract when stating that MPS and pCUT+MC determine the phase diagram and critical properties.
  • domain assumption Finite-size scaling of entanglement entropy and staggered magnetization reliably extracts critical exponents even for long-range potentials.
    Used to characterize all critical boundaries and to claim unconventional, continuously varying exponents.

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Forward citations

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

  1. Unconventional Quantum Criticality in Long-Range Spin-1 Chains: Insights from Entanglement Entropy and Bipartite Fluctuations

    cond-mat.str-el 2026-04 unverdicted novelty 7.0

    Quantum Monte Carlo study of long-range spin-1 chains finds unconventional quantum criticality at alpha_c = 2.48(2) with dynamic exponent z not equal to 1, characterized via entanglement entropy and bipartite fluctuations.

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