One-loop universality of holographic codes

TL;DR

This work demonstrates that the universal, flat entanglement-spectrum structure of holographic quantum codes persists to one-loop order, despite higher-derivative bulk corrections and infrared bulk fluctuations. By reframing the Lewkowycz-Maldacena analysis as a Hamilton-Jacobi variation with respect to conical defect angles and extending it to arbitrary perturbative higher-derivative terms, the authors show that states of fixed geometric entropy continue to yield density matrices on boundary factors that are projectors, and that the code subspace remains invariant under modular flow. They further prove that density-matrix multiplication within the code preserves the code subspace, forcing the bulk χ_{R_α^2} operators to be projectors and rendering the Ryu-Takayanagi-based universality robust at O(1). The results strengthen the operator JLMS relation on the code subspace and underscore the compatibility of simple tensor-network holographic models with full holographic CFT behavior beyond leading order, while highlighting delicate cancellations required in the bulk RG flow.

Abstract

Recent work showed holographic error correcting codes to have simple universal features at $O(1/G)$. In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order ($O(1)$ corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in $G$, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.

Figures (1)

• Figure 1: After cutting open the $n=1$ bulk saddle $g_1$ (left), three copies may be glued together to construct the $n=3$ bulk saddle $g_{3}$ (right). The black dot in the center is the HRT surface $\gamma_R$.