Entanglement entropy and conformal field theory

TL;DR

This paper surveys how 1+1D conformal field theory provides a comprehensive framework for entanglement entropy and related spectra in extended quantum systems. It develops the replica trick and twist-field formalism to compute Tr ρ_A^n on Riemann surfaces, yielding universal scaling with central charge c for a single interval and intricate dependence on geometry (finite size, temperature, boundaries, interfaces, disjoint intervals). It also extends the static CFT results to non-critical and non-equilibrium settings, including massive perturbations and quantum quenches, with clear physical interpretations in terms of quasiparticle pictures and boundary effects, and discusses how the entanglement spectrum and experimental probes via quantum noise можуть be connected. Overall, the work highlights the power of CFT in predicting universal entanglement signatures and provides a toolkit for analyzing real-time dynamics and finite-size systems in 1D quantum critical and near-critical models.

Abstract

We review the conformal field theory approach to entanglement entropy. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries, and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.

Figures (6)

• Figure 1: From density matrix to reduced density matrix. Left: Path integral representation of $\rho(\phi|\phi')$. Center: The partition function $Z$ is obtained by sewing together the edges along $\tau=0$ and $\tau=\beta$ to form a cylinder of circumference $\beta$. Right: The reduced density matrix $\rho_A$ is obtained by sewing together only those points $x$ which are not in $A$.
• Figure 2: A representation of the Riemann surface ${\cal R}_{3,1}$. Reprinted with permission from ccd-07.
• Figure 3: Uniformizing transformation for ${\cal R}_{n,1}$. $w\to\zeta=(w-u)/(w-v)$ maps the branch points to $(0,\infty)$. This is uniformized by the mapping $\zeta\to z=\zeta^{1/n}$.
• Figure 4: The mutual information for fixed four-point ratio $x$ as function of $\eta$ in the gapless phase of the XXZ model. The horizontal lines stand for $Z_{\mathcal{R}_{n,2}}^W$ of Ref. cc-04. Left: mutual information of the von Neumann entropy. Right: mutual information of the Rényi entropy for $n=2$, compared with the compactified boson prediction Eq. (\ref{['F2']}). Reprinted with permission from fps-08
• Figure 6: Left: Space-imaginary time regions for the density matrix in (\ref{['dm0']}). Right: The reduced density matrix $\rho_A$ is obtained by sewing together along $\tau=0$ only those parts of the $x$-axis corresponding to points in $B$ (right part in this plot).