Entanglement and apparent thermality in simulated black holes

The gist: The apparent thermality of Hawking radiation in simulated black holes is found to be present only when bipartitions are made across the event horizon, indicating that genuine thermalization requires interactions deep in the black hole interior <ref:2509.22774#pg11>.

Model and Setup

The study utilizes a chiral spin-chain simulator to model fermions propagating on a black hole space-time in the continuum limit, which describes non-interacting Dirac fermions in curved space-time geometry <ref:2509.22774#pg14>. The mean-field Hamiltonian of the chiral spin-chain is transformed into a free fermion Hamiltonian, which can be diagonalized to obtain a single-particle Hamiltonian with a dispersion relation E(p) = v sin(acp) ± 2u cos (acp/2) <ref:2509.22774#pg17>. In the continuum and low-energy limit, this results in an action that is the action of a massless Dirac spinor on a (1+1)-dimensional Riemann-Cartan space-time <ref:2509.22774#pg4>. This line element is described by ds2 = 1 − v 2/u squared dt squared − 2v/u squared dtdx − 1/u squared dx squared.

Entropy and Thermality Analysis

The derivation of the fermion zero-mode entropy in the interior and exterior regions relies on the assumption that the state is approximately thermal with Hawking temperature, leading to an equivalence between thermodynamic entropy and entanglement entropy across horizon bipartitions. The density of states for zero-modes in a sub-region of length s enclosed in a sub-region of the black hole’s interior can be expressed as N(0) = NF / 2πħ ∫ s ac dx Z R dp δ(E(p)). Under the assumption of thermality, the thermodynamic entropy S(T) is found to be S(T) = π 2/3 N(0)T. When assuming the temperature T equals the Hawking temperature TH = ħκ / 2π, where κ is the surface gravity of the Gullstrand-Painlevé metric, the fermion zero-mode entropy of the black hole’s interior region is S(TH) = NF/12 ln 2xh/ac + S0.

Entanglement Entropy and Breakdown

Numerical simulations of the chiral spin-chain's entanglement entropy are compared against analytic predictions for fermion zero-mode entropy to identify regimes where thermality holds. The paper demonstrates that while the equivalence between fermion zero-mode entropy and entanglement entropy is expected when the ground state is thermal, this equality breaks down for arbitrary partitions away from the horizon. This breakdown of thermality for arbitrary partitions indicates that no genuine thermalization occurs in the free-theory regime, implying that the simulated black hole’s information is not entirely erased.

Mode Occupation and Unruh Effect

The mode occupation expectation value ⟨0Mc†kck0M⟩ of the mean-field Hamiltonian is investigated to probe thermality. When partitioned at the event horizon (∆n = 0), this mode occupation follows a Fermi-Dirac distribution f(EGP,k, T) = 1 / e EGP,k/T + 1, indicating that the reduced state is indeed thermal. However, as the partition is shifted away from the horizon (∆n ≠ 0), deviations from a Fermi-Dirac distribution appear and grow with increasing ∆n. This confirms that thermality holds only for partitions near the event horizon. Furthermore, by modifying the mean-field Hamiltonian to approximate an entanglement Hamiltonian through a coordinate transformation to Rindler space-time, the effective temperature extracted from mode occupation can be reliably identified as the Hawking temperature TH = κ/2π.

Conclusion on Free Theory

The analysis confirms that thermality, and thus the equivalence between statistical and entanglement entropy, holds only for partitions near the event horizon. The breakdown of thermality for arbitrary partitions signals that no genuine thermalization occurs in the free-theory regime. This finding suggests that strong interactions deep in the black hole interior are necessary to induce genuine thermalization, resulting in a thermal distribution even for sufficiently small, but otherwise arbitrary partitions. The paper concludes by showing that this breakdown of thermality is verified by studying the subsystem’s mode occupation, which reveals a Fermi-Dirac distribution only when the lattice is partitioned near the event horizon.

How it works

The method involves calculating analytic expressions for the fermion zero-mode entropy in both interior and exterior regions based on assumptions of thermality at Hawking temperature. This is achieved by using a non-linear dispersion relation to find a finite density of states of the fermion zero-modes, which is then used with the assumption of thermality to obtain their thermal energy and hence, thermodynamic entropy. The density of states N(0) for the interior region is derived using a Dirac delta identity and considering sub-luminal mean-field dispersion relation. The resulting entropy S(TH) is then calculated assuming T = TH, leading to the expression S(TH) = NF/6 ln 2xh/ac + N˜F/12 1/2 sxh + sxh 1/2 + ln 2x1/2 hac! + ln s1/2 - x1/2 h.

Verification of Thermality

The thermality condition is verified by studying the mode occupation expectation value, ⟨0Mc†kck0M⟩, of the mean-field Hamiltonian with respect to the ground state. For a horizon partition (∆n = 0), this mode occupation follows a Fermi-Dirac distribution f(EGP,k, T) = 1 / e EGP,k/T + 1. As the partition is shifted away from the horizon (∆n ≠ 0), deviations from a Fermi-Dirac distribution appear and grow with increasing ∆n. This confirms that the reduced state deviates progressively from thermal behaviour as the partition deviates from the horizon.

Extraction of Hawking Temperature

The effective Hawking temperature is extracted by transforming the mean-field Hamiltonian into a form with only nearest-neighbour interactions and a linear coupling profile, which approximates both the entanglement Hamiltonian and the Dirac Hamiltonian near the event horizon. This transformation corresponds to a change of coordinates that preserves the curvature of the underlying space-time, thus preserving Hawking temperature invariance. In this regime, taking v(x) to be zero and making u(x) position dependent corresponds to a coordinate transformation from Gullstrand–Painlevé to Rindler space-time, which approximates Schwarzschild space-time near the black hole’s horizon.

Role of Interactions

The paper explicitly states that the breakdown of thermality for arbitrary partitions indicates that no genuine thermalisation occurs in the free-theory regime. This strongly suggests that strong interactions inside the simulated black hole are required to induce genuine thermalization, resulting in a thermal distribution even for sufficiently small, but otherwise arbitrary partitions. The study points towards future research on this strong interaction regime as a potential avenue for exploring genuine thermalization.

Final Results Summary

The key finding is that thermality holds only when the partition is located at the event horizon due to unique physics emerging there. This is supported by the observation that mode occupation reveals a Fermi-Dirac distribution only for horizon partitions. The breakdown of thermality for arbitrary partitions confirms that no genuine thermalisation occurs in free theories, meaning information is not entirely erased.

How it works

The extraction of the Hawking temperature is achieved by mapping the black hole geometry to a mean-field Hamiltonian that approximates an entanglement Hamiltonian, which yields a Fermi-Dirac distribution from which TH can be reliably extracted. This procedure fails when the chirality term v(x) is non-zero because the mean-field Hamiltonian no longer approximates the corresponding entanglement Hamiltonian. The analysis shows that near the event horizon, this approximation holds, and by mapping to Rindler space-time, thermality is confirmed with a temperature equal to TH = κ/2π.

Appendix C Details

The derivation of fermion zero-mode entropy for both interior and exterior regions involves determining the density of states for both regions using the non-linear dispersion relation. The density of states N(0) is given by a complex expression involving integrals over space, which is solved in the limit 0 < ac << xh to yield N(0) = NF xh/πħ 2 ln 2xh/ac + ln(2) − 7/2 + N˜F xh/πħ 1/2 sxh + sxh 1/2 + ln s1/2 - x1/2 h.

1. Improve simulation fidelity for non-horizon partitions by incorporating interaction effects. The paper states, Beyond the mean-field limit, we expect strong interactions inside the simulated black hole to induce genuine thermalisation, suggesting that AI models trained on or using this framework can predict thermal behavior for arbitrary partitions where mean-field theory fails.

2. Develop a mechanism to extract Hawking temperature from non-horizon partitions by modifying the Hamiltonian to approximate the entanglement Hamiltonian with only a linear nearest-neighbour coupling. This allows the system to probe the effective temperature extracted from the mode occupation’s Fermi-Dirac distribution when v(x) is set to zero and the curvature needs to be encoded in a linear coupling u(x).

3. Implement an automated check for thermality by comparing analytic predictions with numerical results across all bipartitions. The paper notes, we can identify the regimes where this equality and, by extension, the assumption of thermality is valid and where it breaks down, allowing AI to flag partitions that deviate from the horizon partition's unique physics.

4. Develop a generalized model for fermion zero-mode entropy that accounts for both interior and exterior regions using a combined density of states expression. This allows the system to calculate S(TH) = NF/6 ln 2xh/ac + N˜F/12 1/2 sxh + sxh 1/2 + ln 2x h / (ac!) + ln s 1/2 - x h / (s 1/2 - x h!) for arbitrary regions.

5. Integrate the Bogoliubov transformation results into a predictive model for Unruh effect temperature extraction. The AI can use the derived Bogoliubov transformations and the fact that AˆK0M⟩ = BˆK0M⟩ = 0 to calculate vacuum expectation values, confirming that the reduced state deviates progressively from thermal behaviour as the partition deviates from the horizon.

We investigate the apparent thermality of Hawking radiation in the semi-classical limit of quantum black holes using the mean-field limit of a chiral spin-chain simulator, which models fermions propagating on a black hole space-time in the continuum. In this free-theory regime, no genuine thermalisation occurs. Nevertheless, we show that a bipartition across the event horizon yields a reduced density matrix whose mode occupations follow an apparent thermal Fermi-Dirac distribution. In contrast, partitions away from the horizon do not exhibit such thermal distributions, reflecting the absence of thermal behaviour. Our results demonstrate that Hawking radiation appears thermal only with respect to horizon bipartitions in free theories, with genuine thermal behaviour expected to emerge only in the presence of strong interactions deep in the black hole interior, a study of which we leave for future work.