Black-Hole Signatures in the Finite-Temperature Critical Ising Chain

Abstract

We demonstrate that the finite-temperature critical transverse-field Ising chain exhibits quantitative signatures of black-hole physics in its dual gravitational description within the AdS/CFT correspondence. Its finite-temperature dynamics and thermodynamics are consistently captured by a mixed thermal-AdS/BTZ black hole saddle, leading to three mutually compatible observations. First, antipodal excitation transport collapses onto a universal temperature-dependent curve determined by the relative AdS and BTZ contributions to the gravitational partition function, reflecting horizon absorption. Second, in the high-temperature regime, the retarded response exhibits exponential relaxation governed by the lowest quasi-normal mode of the dual black hole. Third, the temperature derivative of the von Neumann entropy develops a pronounced minimum at a temperature consistent with the Hawking-Page transition. These results identify critical quantum spin chains as minimal and experimentally accessible platforms for probing dynamical and thermodynamic aspects of quantum black holes in controllable many-body systems.

Figures (5)

• Figure 1: (a) Schematic illustration of the finite-temperature critical spin chain and its dual gravitational description, comprising thermal AdS spacetime and a BTZ black hole. A localized perturbation generates excitations that correspond to particles propagating from the boundary into the bulk. They either reach the antipodal point through the AdS geometry (red arrow) or fall into the black hole (dashed purple arrow). (b) Temperature dependence of antipodal excitation transport for various system sizes $L$ and angular momenta $M$. All numerical data collapse onto a universal curve determined by the relative partition weights $Z_{\mathrm{AdS}}(T)/Z_{\mathrm{grav}}(T)$, shown as the red line, with effective parameters $\ell_{\mathrm{eff}}=1.28$ and $G_{\mathrm{eff}}=1.33$. See text for details.
• Figure 2: Manifestation of the quasi-normal mode of the BTZ black hole in the critical spin chain. The time dependence of the spatially summed retarded response function $R(t)$ for the scalar perturbation in the high-temperature regime is computed for a critical spin chain of length $L=10^{3}$. The magnitude $|R(t)|$ exhibits a clear exponential decay governed by the lowest quasi-normal mode of the BTZ black hole, ($\exp(-2\pi T\Delta t)$ with $\Delta=1$), over an extended time window. Beyond the leading exponential behavior, $|R(t)|$ develops a temperature-dependent offset given by $\exp(-\pi^{2}T/2)/2$. See text for details.
• Figure 3: Entropy and its temperature derivative for the critical Ising spin chain and its dual gravitational system. The entropy of the spin chain, $S(T)$, is computed for system size $L=10^{3}$, while the entropy of the dual gravitational system, $S_{\mathrm{grav}}(T)$, is obtained directly from its partition function Eq. (\ref{['eq:Partition_function_gravity_system']}). At low temperatures, the spin-chain entropy remains close to zero, consistent with the gravitational result $S_{\mathrm{grav}}\approx S_{\mathrm{AdS}}(T)=0$. At high temperatures, $S(T)$ exhibits a linear dependence on $T$, in agreement with the BTZ black hole--dominated scaling in the bulk description, $S_{\mathrm{grav}}(T)\approx S_{\mathrm{BTZ}}(T)\propto T$. In the intermediate temperature regime, $dS(T)/dT$ develops a pronounced minimum at $T=0.16\pm0.01$, in good agreement with the corresponding minimum of the gravitational entropy derivative, $dS_{\mathrm{grav}}(T)/dT$, which occurs at the Hawking--Page transition temperature $T_{\mathrm{HP}}=1/2\pi\approx0.16$. See text for details.
• Figure S1: Null geodesics in BTZ spacetime. The plots show representative null trajectories in the $(t,r)$ plane, where $r\equiv\rho/\sqrt{\rho^{2}+\ell^{2}}$ is a compactified radial coordinate. A null ray sent from the boundary falls into the horizon and does not return to the boundary, illustrating why BTZ saddles do not contribute to the transported excitation. (a) and (b) correspond to horizon radii $\rho_{h}=0.3$ and $\rho_{h}=0.1$, respectively; in both cases we take $\Omega=5$, $\ell=1$, and $\rho\in(\rho_{h},2]$.
• Figure S2: Relation between $G_{\mathrm{eff}}$ and $\ell_{\mathrm{eff}}$ for the higher-curvature theory of Ref. Emparan_2020_JHEP. The red curve shows the analytic relation in Eq. (\ref{['eq:quBTZ_relation']}). The color scale indicates the sum of squared errors (SSE) between the effective-theory prediction and the numerical fit to the transported excitation; darker colors correspond to smaller SSE. The curve passes through the region of minimal SSE, demonstrating consistency between the fitted $(\ell_{\mathrm{eff}},G_{\mathrm{eff}})$ and the higher-curvature correction. Here we fix $\ell_{\mathrm{classical}}=1$ and $G_{\mathrm{classical}}=3$, as in the main text.