Entanglement spectrum in one-dimensional systems

Abstract

We derive the distribution of eigenvalues of the reduced density matrix of a block of length l in a one-dimensional system in the scaling regime. The resulting "entanglement spectrum" is described by a universal scaling function depending only on the central charge of the underlying conformal field theory. This prediction is checked against exact results for the XX chain. We also show how the entanglement gap closes when l is large.

Figures (3)

• Figure 1: Sum of the first $M$ eigenvalues of the XX model up to $M=100$: $1-s(M)$ as function of $M$ for $\ell=10,100,1000,10000$ (black dots). The red line is the conformal field theory prediction Eq. (\ref{['srformula']}), in which ${\lambda_{\rm max}}$ has been fixed to the maximum eigenvalue obtained numerically.
• Figure 2: Inverse function of $\lambda_i$ (value of the $i$-th eigenvalue of the reduced density matrix) for $\ell=10,100,1000$ for the XX chain. The plot is shown in terms of the scaling variable $2\sqrt{b\ln ({\lambda_{\rm max}}/\lambda)}$. The full line is the CFT prediction.
• Figure 3: The same as in Fig. \ref{['rhoeigcut']} for all $10\leq\ell\leq28$. Even (odd) $\ell$ are shown on the left (right) panel.