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arxiv: 2605.30540 · v1 · pith:G5DYMTQLnew · submitted 2026-05-28 · ❄️ cond-mat.quant-gas · gr-qc

Entanglement entropy of an acoustic black hole

P.C. van de Graaf , H.T.C. Stoof This is my paper

Pith reviewed 2026-06-28 23:45 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas gr-qc
keywords entanglement entropyacoustic black holeHawking radiationphonon pairsvolume lawanalog gravityquantum correlations
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The pith

Entanglement entropy of an acoustic black hole scales linearly with subregion size for large regions due to horizon phonon pairs.

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

The paper introduces a numerical method to compute entanglement entropy in an acoustic black hole. For sufficiently large subregions this entropy grows linearly with the subregion size, producing a volume law rather than the area law seen in many other quantum systems. The linear growth traces to non-separable long-distance correlations created when phonon pairs are produced at the horizon. The system is locally thermal, so the volume-scaling portion of the entropy closely matches the thermal entropy carried by the outgoing Hawking radiation.

Core claim

The entanglement entropy of sufficiently large subregions scales linearly with size and thus shows a volume law instead of an area law. The origin of this scaling can be traced back to the non-separable long-distance correlations due to the production of phonon pairs at the horizon. The system is shown to be locally thermal, such that the part of the entanglement entropy scaling with volume is well approximated by the thermal entropy of the outgoing Hawking radiation.

What carries the argument

Numerical computation of entanglement entropy that captures non-separable long-distance correlations generated by phonon pair production at the horizon.

If this is right

  • The volume-law scaling replaces the usual area law once subregions exceed a threshold size set by the horizon correlations.
  • The volume contribution to entropy equals the thermal entropy of Hawking radiation because the system is locally thermal.
  • Long-range correlations from horizon phonon pairs are the mechanism that produces the volume law.
  • The numerical method provides a concrete way to extract these correlations in analog black-hole systems.

Where Pith is reading between the lines

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

  • If the same numerical approach works in other analog systems, it could be used to test whether volume-law entanglement appears whenever horizons produce paired excitations.
  • The local thermality result suggests that entanglement entropy measurements in cold-atom setups could serve as a proxy for Hawking-radiation properties without needing direct temperature probes.
  • The finding that horizon correlations dominate over short-range ones implies that entanglement-based probes of analog gravity may need to focus on scales much larger than the healing length.

Load-bearing premise

The numerical method accurately captures the long-distance non-separable correlations from phonon pair production at the horizon without significant artifacts from finite system size or discretization choices.

What would settle it

A calculation for large subregions that finds entanglement entropy scaling with boundary area rather than volume, or that deviates strongly from the thermal entropy of the outgoing radiation, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.30540 by H.T.C. Stoof, P.C. van de Graaf.

Figure 1
Figure 1. Figure 1: A sketch of the setup of an acoustic black hole [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The dispersion relation of the continuous Bogoliubov theory in the supersonic (left), and the subsonic (right) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The dispersion relation of the modified Bogoliubov theory in the supersonic (left), and the subsonic (right) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The occupation numbers of outgoing Hawking pairs in the supersonic (left) and subsonic (right) regions as a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The density-density correlation function of the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The entanglement entropy of a finite-size [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The logarithmic scaling coefficient α of the homogeneous Bogoliubov theory as a function of the relative length scale ξ/a, and the dimensionless gap parameter ˜ω0 ≡ ω0a/cr, in the limit when l ≪ cr/ω0 ≪ L. This behavior can be explained qualitatively using the long-distance correlations in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: The entanglement entropy of the subregion [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The entanglement entropy of the region [103 a, 103 a + la], as function of l/a for different values of the speed of sound in the supersonic region cl. The speed of sound in the subsonic region is given by cr/|v| = 3/2, the healing length ξl = 2a and gap ω0a/cl = 5 × 10−6 . occupation numbers as Sth = X i [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The entanglement entropy per lattice site [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The ϕ − ϕ correlation function of the acoustic black hole, with a gap ω0a/cl = 5 × 10−3 . The flow profile has cl/|v| = 1/2 and cr/|v| = 3/2, with healing length ξl = a [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
read the original abstract

We introduce a method to numerically compute the entanglement entropy of an acoustic black hole. It is shown that the entanglement entropy of sufficiently large subregions scales linearly with size and thus shows a volume law instead of an area law. The origin of this scaling can be traced back to the non-separable long-distance correlations due to the production of phonon pairs at the horizon. The system is shown to be locally thermal, such that the part of the entanglement entropy scaling with volume is well approximated by the thermal entropy of the outgoing Hawking radiation.

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 / 1 minor

Summary. The manuscript introduces a numerical method to compute the entanglement entropy (EE) of an acoustic black hole. It claims that EE for sufficiently large subregions exhibits linear (volume-law) scaling rather than area-law scaling, with the origin traced to non-separable long-distance correlations generated by phonon-pair production at the horizon. The system is reported to be locally thermal, allowing the volume-scaling portion of the EE to be approximated by the thermal entropy of the outgoing Hawking radiation.

Significance. If the numerical results are free of artifacts, the work would provide a concrete demonstration that horizon-induced correlations in an analog gravity system can produce volume-law entanglement, offering a controllable setting to study modifications to standard area-law expectations. This could serve as a benchmark for theoretical models connecting Hawking radiation to quantum information measures.

major comments (2)
  1. [Numerical method (around the description of the covariance matrix and Bogoliubov modes)] The numerical method section provides insufficient detail on discretization scheme, covariance-matrix construction for the Gaussian state, and convergence tests with system size (while holding the horizon region fixed in physical units). Without explicit demonstration that the linear coefficient in the EE scaling converges and is insensitive to periodic-boundary or mode-truncation artifacts, the central claim that the volume law originates from physical long-distance phonon-pair correlations cannot be substantiated.
  2. [Results section discussing local thermality and thermal-entropy approximation] The assertion that the volume-scaling part of the EE is 'well approximated' by the thermal entropy of outgoing Hawking radiation requires a quantitative comparison (e.g., explicit subtraction or ratio plot) that is independent of the local-thermality definition used to identify the radiation. If local thermality is inferred from the same correlation functions that enter the EE, the approximation risks circularity and must be shown to hold under controlled variations of the cutoff or dispersion relation.
minor comments (1)
  1. [Abstract and introduction] The qualifier 'sufficiently large subregions' in the abstract and main text should be accompanied by explicit bounds relative to the computational domain size and the horizon scale to allow readers to assess whether the reported regime remains free of finite-size contamination.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate additional details and quantitative comparisons as requested.

read point-by-point responses

  1. Referee: [Numerical method (around the description of the covariance matrix and Bogoliubov modes)] The numerical method section provides insufficient detail on discretization scheme, covariance-matrix construction for the Gaussian state, and convergence tests with system size (while holding the horizon region fixed in physical units). Without explicit demonstration that the linear coefficient in the EE scaling converges and is insensitive to periodic-boundary or mode-truncation artifacts, the central claim that the volume law originates from physical long-distance phonon-pair correlations cannot be substantiated.

    Authors: We agree that the numerical method section would benefit from expanded detail. In the revised manuscript we will add: (i) a precise description of the spatial discretization of the acoustic metric and the associated Bogoliubov mode equations, (ii) the explicit construction of the covariance matrix from the mode functions for the Gaussian state, and (iii) systematic convergence tests in which the total system size is increased while the physical horizon region is held fixed. These tests will include explicit checks of the linear coefficient in the entanglement entropy against periodic-boundary conditions and mode truncation, thereby confirming that the reported volume-law scaling is insensitive to these numerical choices and originates from the physical long-range correlations. revision: yes

  2. Referee: [Results section discussing local thermality and thermal-entropy approximation] The assertion that the volume-scaling part of the EE is 'well approximated' by the thermal entropy of outgoing Hawking radiation requires a quantitative comparison (e.g., explicit subtraction or ratio plot) that is independent of the local-thermality definition used to identify the radiation. If local thermality is inferred from the same correlation functions that enter the EE, the approximation risks circularity and must be shown to hold under controlled variations of the cutoff or dispersion relation.

    Authors: We acknowledge the need for an explicit, non-circular quantitative comparison. In the revision we will include a new figure or subsection that directly compares the volume-law coefficient extracted from the entanglement entropy with the thermal entropy of the outgoing Hawking radiation (via subtraction or ratio plots). We will also report results under controlled variations of the ultraviolet cutoff and dispersion relation to demonstrate that the approximation remains valid independently of the specific local-thermality criterion employed. revision: yes

Circularity Check

0 steps flagged

Numerical computation yields volume-law EE without definitional reduction to inputs

full rationale

The paper introduces a numerical method for computing entanglement entropy in an acoustic black hole and reports a volume-law scaling for large subregions, attributing it to non-separable phonon-pair correlations at the horizon. It separately observes local thermality and notes that the volume-scaling portion approximates the thermal entropy of outgoing Hawking radiation. No quoted step reduces the central claims by construction to fitted parameters, self-citations, or ansatzes; the results emerge from the simulation rather than being tautological with the method's definition. The derivation chain is self-contained against external benchmarks of the acoustic metric and covariance-matrix EE formula.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Since only the abstract is reviewed, the ledger is based on standard assumptions in the field of analog gravity. No specific free parameters or new entities are mentioned.

axioms (1)
  • domain assumption The acoustic black hole model accurately reproduces the quantum field theory behavior near the horizon for phonon modes.
    This is implicit in using the analog system to study black hole effects.

pith-pipeline@v0.9.1-grok · 5609 in / 1337 out tokens · 32823 ms · 2026-06-28T23:45:16.998991+00:00 · methodology

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Reference graph

Works this paper leans on

47 extracted references · 13 canonical work pages · 3 internal anchors

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