Causal Evolutions of Bulk Local Excitations from CFT

TL;DR

This work probes bulk locality in AdS/CFT by constructing boundary states $| abla_\alpha\rangle$ dual to bulk-localized excitations and evaluating boundary two-point functions $\langle O O\rangle_{\Psi_\alpha}$ in 2D CFTs. The authors formulate the calculation via four-point functions with heavy and light operators, relate the bulk-localized states to HKLL bulk reconstruction, and analyze both holographic CFTs and free fermion cases. They find causal, light-like propagation when the bulk scalar is near the BF bound in holographic CFTs, while such propagation is absent in free fermion CFTs; the short-distance behavior obeys a universal first-law-like relation involving $\Delta_{\Psi_\alpha}$ and $\Delta_O$. Geometrically, the exterior region aligns with BTZ physics for mass $\Delta_{\Psi_\alpha}$, whereas interior regions require solving the Einstein-scalar system, illustrating how bulk causal structure and entanglement properties emerge from CFT data and providing a framework to study interior bulk dynamics.

Abstract

Bulk localized excited states in an AdS spacetime can be constructed from Ishibashi states with respect to the global conformal symmetry in the dual CFT. We study boundary two point functions of primary operators in the presence of bulk localized excitations in two dimensional CFTs. From two point functions in holographic CFTs, we observe causal propagations of radiations when the mass of dual bulk scalar field is close to the BF bound. This behavior for holographic CFTs is consistent with the locality and causality in classical gravity duals. We also show that this cannot be seen in free fermion CFTs. Moreover, we find that the short distance behavior of two point functions is universal and obeys the relation which generalizes the first law of entanglement entropy.

Figures (7)

• Figure 1: The plot of the ratio ${\langle \Psi_\alpha |O(1)O(e^{-i\sigma})| \Psi_\alpha\rangle\over \langle 0|O(1)O(e^{-i\sigma})|0\rangle}$ at the time $t=0$ (left graph) and $t=\pi/2$ (right graph) as a function of $\sigma$. At $t=0$, the excitation is localized at the center of AdS. While at $t=\pi/2$, the excitation is spread throughout AdS. We chose $h_{\alpha}=1/2$ and $h_{O}={c\over 24}$ and made order $n=4$ approximation. The blue, red and green curve corresponds to the value $\epsilon=0.35, 1$, and $\infty$. Note that when $\epsilon=\infty$ the left and right graphs do coincide, because the corresponding BTZ solution is static. We can also see the two point function increases under the time evolution when $0\leq t\leq \pi/2$ for finite $\epsilon$.
• Figure 2: The sketches of time evolution of the local excitation $|\Psi_\alpha\rangle$ in a global AdS$_3$. The gray regions describe the bulk scalar field excitation. In the white region we can ignore matter field contributions and the spacetime can be approximated by the BTZ black hole solution with the correct mass. The left picture corresponds to the light excitation $h_\alpha\simeq 1/2$. Since the dual bulk scalar field is light, we expect light like propagation of excitations. The right picture describes the geometry for $h_\alpha \gg 1$. Since the scalar field is very massive, the excitations does not reach the AdS boundary. Also the primary state includes in $|\Psi_\alpha\rangle$ gives a back reacted geometry (deficit angle spacetime) localized at the center of AdS.
• Figure 3: The plot of the ratio $R(t)\equiv {\langle \Psi_\alpha(t) |O(1)O(e^{-i\sigma})| \Psi_\alpha(t)\rangle\over \langle 0|O(1)O(e^{-i\sigma})|0\rangle }$ (left graph) and the difference $R(t)-R(0)$ (right graph) as a function of $\sigma$ for $t=0,\pi/6,\pi/3,\pi/2,2\pi/3,5\pi/6,\pi$ (from the bottom to the top graph). At $t=0$, the excitation is localized at the center of AdS. While at $t=\pi/2$, the excitation is spread throughout AdS. We chose $h_{\alpha}=1/2$ and $h_{O}={c\over 24}$ for $\epsilon=0.35$. We employed $n=4$ approximation. In the right graph, we marked points $\sigma=0,\pi/6,\pi/3,\pi/2,2\pi/3,5\pi/6$ with blue dots, corresponding to the wave fronts as computed in (\ref{['ww']}).
• Figure 4: The two point function ratios for various $\alpha_O=\sqrt{1-24h_{O}/c}$. The left graph shows $R(t)\equiv{\langle \Psi_\alpha(t) |O(1)O(e^{-i\sigma})| \Psi_\alpha(t)\rangle\over \langle 0|O(1)O(e^{-i\sigma})|0\rangle }$ as a function of $\sigma$ for $t=0,\pi/6,\pi/3, \pi/2,2\pi/3,5\pi/6,\pi$ from the bottom to top. The right graph describes the difference $R(t)-R(0)$ as a function of $\sigma$ for $t=0,\pi/6,\pi/3,\pi/2,2\pi/3,5\pi/6,\pi$. At $t=0$, the excitation is localized at the center of AdS. While at $t=\pi/2$, the excitation is spread throughout AdS. We chose $h_{\alpha}=1/2$ and employed $n=4$ approximation. In the right graph, we marked points $\sigma=0,\pi/6,\pi/3,\pi/2,2\pi/3,5\pi/6$ with blue dots, corresponding to the light-like wavefront as computed in (\ref{['ww']}).
• Figure 5: The two point function ratios for various $h_\alpha$. The left graph shows $R(t)\equiv{\langle \Psi_\alpha(t) |O(1)O(e^{-i\sigma})| \Psi_\alpha(t)\rangle\over \langle 0|O(1)O(e^{-i\sigma})|0\rangle }$ as a function of $\sigma$ for $t=0,\pi/6,\pi/3,\pi/2,2\pi/3,5\pi/6,\pi$ from the bottom to top. The right graph describes the difference $R(t)-R(0)$ as a function of $\sigma$ for $t=0,\pi/6,\pi/3,\pi/2,2\pi/3,5\pi/6,\pi$ from the bottom to top. We chose $\alpha=1/2$ and employed $n=4$ approximation. In the case $h_\alpha = 1$, we find approximate light-like wavefront as in $h_\alpha = 1/2$. At $h_{\alpha}=10$, we cannot observe light-like wavefront, because the bulk scalar field is too heavy.