Lectures on entanglement entropy in field theory and holography

TL;DR

This work surveys entanglement entropies in quantum field theories and their holographic duals, starting from classical and quantum information concepts and building up to field-theoretic and geometric computations. It explains how the replica trick, modular Hamiltonians, and conformal symmetry govern subsystem entropies in 1+1d and higher dimensions, and then shows how holography encodes these entropies as bulk geometric areas via the Ryu-Takayanagi prescription. The notes detail key results for simple geometries (half-lines, intervals) and finite-temperature or gapped settings, and they discuss important properties such as area laws, strong subadditivity, and the special holographic inequality known as MMI. They culminate in a set of holographic checks and generalizations (HRT, FLM, higher-derivative terms) that illustrate how EE probes bulk spacetime, RG flows, and phase transitions in strongly coupled quantum systems. The overall significance lies in connecting quantum information quantities to geometric and gravitational data, offering a powerful framework for understanding entanglement in QFT and the emergence of spacetime in holography.

Abstract

These notes, based on lectures given at various schools over the last few years, aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how the Ryu-Takayanagi formula geometrically realizes general features of field-theory entanglement, while revealing special properties of holographic theories. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.

Figures (18)

• Figure 1: Schematic illustration of the mutual information and conditional entropy in a classical bipartite system: Two correlated systems $A$ and $B$ can be encoded into $S(A)$ and $S(B)$ bits respectively, such that $I(A:B)$ bits of each are perfectly correlated, $H(A|B)$ bits of $A$ are uncorrelated with those of $B$, and $H(B|A)$ bits of $B$ are uncorrelated with those of $A$. (Factors of $\ln2$ have been dropped for clarity.)
• Figure 2: Page curve: The EE $S(A)$ in a random pure state of a large Hilbert space $\mathcal{H}_{AB}$ as a function of the fractional size $x$ of the $A$ subsystem, defined as $\ln\dim\mathcal{H}_A/\ln\dim\mathcal{H}_{AB}$.
• Figure 3: A region $A$ of size $L$ in a lattice system with lattice spacing $\epsilon$.
• Figure 4: A region $A$ of a Cauchy slice $\Sigma$ and a region $A'$ of another Cauchy slice $\Sigma'$ with the same causal domain (shown in gray), $D(A')=D(A)$.
• Figure 5: The half-line $A$\ref{['half-line']}, its complement $A^c$, and its causal domain $D(A)$, the Rindler wedge \ref{['Rindler']} (shaded), in $(1+1)$-dimensional Minkowski space. Also shown are some lines of constant $\chi$ in the coordinates $(\chi,r)$ of \ref{['Rindler metric']} (purple), each of which is a Cauchy slice for the Rindler wedge, as well as a trajectory of constant $r$ (red).