Entanglement and boundary critical phenomena

We investigate boundary critical phenomena from a quantum-information perspective. Bipartite entanglement in the ground state of one-dimensional quantum systems is quantified using the Rényi entropy $S_{\alpha }$, which includes the von Neumann entropy $(\alpha \to 1)$ and the single-copy entanglement $(\alpha \to \infty )$ as special cases. We identify the contribution of the boundaries to the Rényi entropy, and show that there is an entanglement loss along boundary renormalization group (RG) flows. This property, which is intimately related to the Affleck-Ludwig $g$ theorem, is a consequence of majorization relations between the spectra of the reduced density matrix along the boundary RG flows. We also point out that the bulk contribution to the single-copy entanglement is half of that to the von Neumann entropy, whereas the boundary contribution is the same.

Figure 1

Bipartite entanglement for the bulk critical transverse Ising model, quantified by the von Neumann entropy $S_1$ of the reduced density matrix. The block size $\ell$ is given as $T\left(\ell \right)$ (see text) which allows for a direct check of the CFT predictions (1, 2). The fit yields the predicted boundary entropies.

Figure 2

Eigenvalue distributions for the reduced density matrix of a block of size $\ell =128a$ in the bulk critical transverse Ising model (3) for various values of the boundary magnetic field $h_b$. The quasiexact DMRG results confirm the majorization relation along the boundary field RG flow (see text). The inset displays the first three eigenvalues and shows that the crossing of the spectra occurs at $i^{*}=1$.