Researchers P. C. van de Graaf and H. T. C. Stoof of Utrecht University have demonstrated that entanglement entropy in an acoustic black hole scales linearly with size, a “volume law”, challenging the expectation that black hole entropy is determined solely by the area of its event horizon. Their work introduces a method to numerically compute this entanglement, revealing that this scaling originates from “non-separable long-distance correlations” created by phonon pairs at the black hole’s horizon. The team found the system to be locally thermal, with the volume-scaling portion of entanglement entropy closely matching the thermal entropy of outgoing Hawking radiation.

Acoustic Black Hole Setup in Bose-Einstein Condensates

This volume law scaling indicates that the entanglement is not simply determined by the event horizon’s area, but rather extends into the bulk of the simulated system, suggesting a more complex relationship between entanglement and geometry than previously understood.

Further investigation into the acoustic black hole setup involved a detailed review of the system’s properties, including the normalization conditions required for the mode functions describing the sound waves. The equations of motion governing these modes, known as the Bogoliubov-de Gennes equations, were central to the simulations.

The team acknowledges the concurrent work by Chandran and Fisher, who independently found a volume law for entanglement negativity in a similar system using different methods. The work was supported by preliminary investigations from master students Hendrik Snijder, Ken de Ruiter, Mike Hakvoort and Pablo Vis, and benefited from discussions with Remco Bus.

Recati’s Model and Acoustic Black Hole Construction

Constructing an acoustic black hole that accurately models theoretical predictions requires careful consideration of boundary conditions and potential instabilities; infinite system size is often demanded, a practical impossibility for simulation. This approach allows for precise control over divergences in entanglement entropy through the implementation of both infrared and ultraviolet regulators, essential for accurate calculations.

These correlations are demonstrably non-separable, indicating the presence of entangled degrees of freedom arising from the pair production process at the horizon. Detailed analysis of the φ-φ correlation function, for example, provides a visual representation of these entangled correlations and their spatial distribution within the acoustic black hole.

Correlator Method for Entanglement Entropy Calculation

The correlator method offers a distinct advantage over standard entanglement entropy calculations, allowing researchers to analyze arbitrary Gaussian states that are important for modeling complex systems like acoustic black holes. 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.

Regularization of infrared and ultraviolet divergences was achieved through a tight-binding approximation and the introduction of a mass term into the equations of motion, enabling the application of the correlator method. The team’s work demonstrates a transition in the prefactor α governing logarithmic scaling with subsystem size, shifting from 1/3 for systems where the healing length is much smaller than the lattice spacing, to 1/2 when the healing length is significantly larger.

This change highlights the sensitivity of entanglement to the underlying parameters defining the system’s structure and the importance of accurately modeling these parameters in simulations. The presence of the black-hole horizon itself is the primary driver of the observed volume-law scaling, manifesting far from the event horizon.

They speculate that the anomalous φ-φ correlator could potentially represent a two-particle wave function, offering a pathway to compute entanglement between Hawking pairs and compare it to the total entanglement entropy determined in this study. “We suspect it might be possible to interpret the anomalous φ-φ correlator as a two-particle wave function,” said van de Graaf, indicating a direction for future research.

The team introduced a method to numerically compute the entanglement entropy of an acoustic black hole. The work ultimately provides a framework for understanding how entanglement arises in analogue gravity systems and its connection to the fundamental properties of black holes.

Volume Law Scaling of Entanglement Entropy Confirmed

This scaling is not universal; it appears only in transsonic configurations, disappearing entirely when the flow is entirely subsonic, as detailed in the research. This correspondence allows for a numerical approximation of the entanglement entropy using the scattering amplitudes, a method detailed in the paper’s appendix, and supports the idea that volume scaling arises from local thermal effects. Subleading to this volume term, the simulations also reveal the expected logarithmic scaling due to short-distance correlations across the boundaries of the examined subregion, a standard feature of entanglement in quantum field theories.

The work builds upon established understanding of black hole thermodynamics, noting the similarities between black holes and thermodynamical systems, and extends it by demonstrating how entanglement contributes to the overall entropy in analogue gravity systems. The team hopes that future work will explicitly determine the wave function of the Hawking pairs and directly examine the entanglement between them, potentially unlocking further insights into the fundamental relationship between entanglement and gravity.

Local Thermal Entropy of Hawking Radiation

The scaling of entanglement entropy in simulated black holes deviates from established theory, with recent work demonstrating a linear relationship between entropy and system size rather than the expected area law. Simulations at Utrecht University reveal that entanglement entropy exhibits this “volume law” scaling, indicating that long-range correlations contribute significantly to the overall entropy of the system.

This challenges the conventional understanding of black hole entropy, which typically associates it solely with the area of the event horizon. The researchers note that the observed behavior is highly dependent on these correlations, highlighting their important role in determining the overall entanglement structure.

Bogoliubov Theory Benchmarking of Entanglement Entropy

The scaling behavior was observed using Bogoliubov theory, a framework employed to model quantum fields in curved spacetime. Specifically, the simulations demonstrate that the logarithmic scaling coefficient of the entanglement entropy, denoted as α, converges to a value of 1/3 for low values of the ratio between the healing length and step size (ξ/a). This value aligns with predictions from conformal field theory, validating the model’s accuracy in certain regimes.

Increasing the gap parameter, ω₀, however, shifts the scaling from logarithmic to constant, indicating a transition in the entanglement structure. Analysis of the homogeneous Bogoliubov theory revealed an empirical equation for entanglement entropy: $S_{E}$ = α(ξ/a) log(l/a) + c′, where α is a function of system parameters, l is the subsystem size, and c′ is a non-universal constant.

This ansatz, tested against exact results from conformal field theory, confirms the logarithmic scaling expected for short length scales, though with a varying prefactor dependent on system parameters. The simulations also show that for values of ξ/a less than one, the logarithmic scaling coefficient approaches the conformal field theory value of 1/3, further solidifying the connection between the acoustic black hole model and established theoretical predictions.

The logarithmic scaling coefficient α also varies with the dimensionless gap parameter ω₀, defined as ω₀a/c_r, where $c_{r}$ is the sound velocity. Simulations demonstrate that as ω₀ increases, the entanglement entropy transitions from logarithmic to constant behavior, indicating a change in the dominant contribution to entanglement. This behavior is particularly evident in finite-size systems with N = 50 points and ξ = a, where the transition is clearly visible in the plotted data.

👉 More information
🗞 Entanglement entropy of an acoustic black hole
✍️ P. C. van de Graaf and H. T. C. Stoof
🧠 DOI: http://link.aps.org/doi/10.1103/zk1n-x14g

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