Entanglement and apparent thermality in simulated black holes

Abstract

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, whilst genuine thermal behaviour emerges only in the presence of interactions deep in the black hole interior.

Figures (7)

• Figure 1: Schematic of the lattice corresponding to the chiral spin-chain model Hamiltonian of Eq. \ref{['Chiral Spin-Chain Hamiltonian']}, where the lattice sites alternate between two sub-lattices, $A$ and $B$, and a two-site unit cell has been introduced.
• Figure 2: Blue and orange curves show the positive and negative mean-field dispersion relations of Eq. \ref{['Equation: MFT Hamiltonian Dispersion Relation']}, respectively, for fixed $u=1$ and various coupling strengths $v$ (from left to right, $v=0, \ 0.5, \ 1, \ 2$). As the coupling $v$ increases, the Dirac cone about the Fermi point $p_0$ tilts. At the critical point $u^2=v^2$, which corresponds to the event horizon, the cone over-tilts, resulting in the emergence of two further Fermi points $p_\pm$.
• Figure 3: $(a)$ Schematic of spin-chain bipartition at lattice site $n_s$ located a distance $\Delta n$ from $n_h$. $(b)$ Red data points show entanglement entropy $S_{\mathcal{A}}(n_s)$ of $\mathcal{A}$ as a function of the partition site $n_s$ for a system with $n_h=1000$. Blue and orange curves show the fermion zero-mode entropies of Eqs. \ref{['Equation: Entropy of (1+1)D Black Hole Fermion Zero-Modes (Interior)']} and \ref{['Equation: Entropy of (1+1)D Black Hole Fermion Zero-Modes (Exterior)']} for the black hole space-time's interior and exterior regions, respectively, with $N_F=\tilde{N}_F=1$ and the non-universal constant of the exterior region chosen to match that obtained for the horizon partition. (c), (d), and (e) Red data points show the entanglement entropy $S_{\mathcal{A}}(n_s)$ of $\mathcal{A}$ as a function of the partition's location, where the system is partitioned at $n_s=n_h-\Delta n$ (interior), the event horizon $n_h$, and $n_s=n_h+\Delta n$ (exterior), respectively, with $\Delta n=250$. Blue curves show data interpolated using Eqs.\ref{['Equation: Entropy of (1+1)D Black Hole Fermion Zero-Modes (Interior)']}, \ref{['Equation: Entropy of (1+1)D Black Hole Fermion Zero-Modes at Horizon']}, and \ref{['Equation: Entropy of (1+1)D Black Hole Fermion Zero-Modes (Exterior)']} for the fermion zero-modes' entropy for the interior, horizon, and exterior regions of the black hole space-time, respectively. To avoid partitioning the two-site unit cell of the continuum limit, only even $n_h$ were considered. For all figures, systems of $N=10000$ lattice sites with couplings $v(n)=\sqrt{n_h/n}$ and $u=1$ were taken.
• Figure 4: Data points show the interpolated value for the number of fermionic fields $N_F$ as a function of the distance $\Delta n$ of the partition $n_s=n_h-\Delta n$ from the event horizon $n_h$, for a system of $N=10000$ lattice sites with couplings $v(n)=\sqrt{n_h/n}$ and $u=1$. Each interpolated value of $N_F$ is determined by fitting Eq. \ref{['Equation: Entropy of (1+1)D Black Hole Fermion Zero-Modes (Interior)']} to numerical data for the entanglement entropy for a fixed $\Delta n$ as $n_h$ is increased. The deviation from $N_F\approx 1$ as $\Delta n$ increases indicates the breakdown of Eq. \ref{['Equation: Entropy of (1+1)D Black Hole Fermion Zero-Modes (Interior)']} for the fermion zero-mode entropy of the black hole's interior in describing the behaviour of the entanglement entropy as the partition deviates from the event horizon. Inset shows the derivative of the entanglement entropy $S_\mathcal{A}(n_h)$ with respect to $\ln(2n_h)$ following a horizon partition, $\Delta n=0$, for increasing system sizes, indicating the regime where $N_F\approx 1$. As system size increases, the domain for which the gradient plateaus around 1 increases, suggesting that $N_F\rightarrow 1$ in the thermodynamic limit, $N\rightarrow\infty$.
• Figure 5: Red data points show the mode occupation expectation value $\langle 0_M| c_k^\dagger c_k|0_M\rangle$ of the mean-field Hamiltonian that simulates the black hole space-time (Eq.\ref{['Equation: Mean-Field Hamiltonian']} with the Gullstrand-Painlevé couplings $v(x)=\sqrt{x_h/x}$ and $u=1$, defined only on $\mathcal{B}$) with respect to the ground state $|0_M\rangle$ of the Minkowski Hamiltonian (Eq. \ref{['Equation: Mean-Field Hamiltonian']} with couplings $u=1$ and $v=0$, defined over the entire lattice) for a horizon partition ($\Delta n=0$) and interior partition ($\Delta n=50$), respectively. The abscissa shows the energy eigenvalues $E_{GP,k}$ of the single-particle Hamiltonian $h_{GP}$ associated with $H_{MF}$ for the Gullstrand-Painlevé couplings; for finite $N$ the spectrum is bounded due to the lattice bandwidth. Blue curve shows data interpolated using the Fermi-Dirac distribution of Eq. \ref{['Equation: Fermi-Dirac Distribution']} with $T=0.3271$ and $T=1.9237$, respectively. System sizes of $N=2000$ lattice sites, with the horizon located at $n_h=700$, were considered.