Entanglement of low-energy excitations in Conformal Field Theory

TL;DR

It is shown that the nth Rényi entropy is related to a 2n-point correlator of primary fields in CFT and this result uncovers a new link between quantum information theory and CFT.

Abstract

In a quantum critical chain, the scaling regime of the energy and momentum of the ground state and low lying excitations are described by conformal field theory (CFT). The same holds true for the von Neumann and Renyi entropies of the ground state, which display a universal logarithmic behaviour depending on the central charge. In this letter we generalize this result to those excited states of the chain that correspond to primary fields in CFT. It is shown that the n-th Renyi entropy is related to a 2n-point correlator of primary fields. We verify this statement for the critical XX and XXZ chains. This result uncovers a new link between quantum information theory and CFT.

Figures (3)

• Figure 1: Riemann surfaces describing the past events in $\zeta$ and $z$. The distinguished point in $z$ is the infinite past $\zeta_\infty= -i \infty$.
• Figure 2: An illustration of the law $F_{\Upsilon_1[2,1]}^{(2)}(x)=F_{\Upsilon_1[2,1]}^{(3)}(x)=1$ for the $XX$ model with $N=100$, $n_F=50$, and for the $XXZ$ with $\Delta=-1/2$ and $N=30$, $n_F=14$ (16 for the excited state). The entropy of ground and excited states coincide and follows the law (\ref{['HLWlaw']}) (continuous lines), up to oscillations Calabrese2010.
• Figure 3: The quantity $F_{\Upsilon_2}^{(2)}$ for the $XX$ model at different filling fractions and for the $XXZ$ ($\Delta=-1/2$, $N=30$, $n_F=14$) model, vs. the CFT prediction (\ref{['3rdF2']}). For $n_F=250$ the oscillations around (\ref{['3rdF2']}) are so small that both curves are indistinguishable. In the inset we show $S_2$ for the ground and excited states ($n_F=250$, $N=500$). The upper inset is a zoom of the region selected by the small rectangle over the curve in the main figure.