Localized shocks

TL;DR

The paper investigates how interior Einstein-Rosen bridges of two-sided AdS black holes are encoded in boundary dynamics for spatially localized perturbations. By combining chaotic spin-chain numerics and holographic shock-wave analysis, it establishes a precise correspondence between the growth of localized precursors and a minimal tensor-network representation that mirrors the ERB geometry on large scales. It shows that a localized precursor expands ballistically with radius r[W_x(t_w)] ≈ v_B (t_w − t_*), and that the tensor network for products of precursors organizes into a folded-time, cone-like structure whose holographic dual reproduces the ERB patches. This work extends the TN–ERB correspondence from spatially homogeneous shocks to localized perturbations, suggesting a universal geometrical- informational description of interior growth across lattice and holographic systems, with finite-coupling and stringy corrections suggested for future study.

Abstract

We study products of precursors of spatially local operators, $W_{x_{n}}(t_{n}) ... W_{x_1}(t_1)$, where $W_x(t) = e^{-iHt} W_x e^{iHt}$. Using chaotic spin-chain numerics and gauge/gravity duality, we show that a single precursor fills a spatial region that grows linearly in $t$. In a lattice system, products of such operators can be represented using tensor networks. In gauge/gravity duality, they are related to Einstein-Rosen bridges supported by localized shock waves. We find a geometrical correspondence between these two descriptions, generalizing earlier work in the spatially homogeneous case.

Figures (12)

• Figure 1:
• Figure 2: Ballistic growth of the operator $Z_1(t_w)$, evolved with the chaotic $g =-1.05$, $h = 0.5$ Hamiltonian (solid) and the integrable $g = 1$, $h = 0$ Hamiltonian (dotted). Left:$c_k(t)$ is the sum of the squares of the coefficients of Pauli strings of length $k$ in $Z_1(t_w)$. Notice that the integrable and chaotic behavior is rather similar until the strings grow to reach the end of the chain ($n=8$ spins). Right: for both types of evolution, the size grows linearly until it approaches the size of the system. After this point, the chaotically-evolving operator saturates, while the integrably-evolving operator begins to shrink. The blue "staircase" curves show the size $s[Z_1(t_w)]$. The smooth black curves show $s_2[Z_1(t_w)] \propto \sum_k k \ c_k(t_w)$.
• Figure 3: The tensor network description of the identity operator (left) and operators $e^{-iHt}$ with successively larger $t$.
• Figure 4: Left: the naive tensor network describing a precursor operator $e^{-iHt_w}We^{iHt_w}$. The red network represents backwards time evolution, the black dot represents the local operator $W$, and the green network represents forwards time evolution. Shading indicates the region affected by the linearly growing $W$ insertion. In the unshaded region, the forwards and backwards evolutions cancel. Right: the network after removing tensors that cancel. The dotted lines indicate contractions; their endpoints should be identified.
• Figure 5: Fibering the position-dependent time fold over the $x$ space gives the minimalized TN. The geometry at right is equivalent to the right panel of Fig. \ref{['fig-tensor-network']}.