Phase transition in the Rényi-Shannon entropy of Luttinger liquids

The Rényi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge $c=1$, we show that it exhibits a phase transition at some value $n_c$ of the Rényi parameter $n$ which depends on the Luttinger parameter $R$. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and $J_1-J_2$ spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for $n<n_c$. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.

Figure 1

Height shift $\delta h=\pi r/2$ for the semi-infinite strip with bottom boundary condition $|i_{\max }\rangle$.

Figure 2

Constant term in the RSE of periodic XXZ and $J_1-J_2$ chains for different values of $\Delta$ and at the critical point $J_2/J_1=0.2411$. Each point comes from fitting the data for $L=20,22,24,26$ and 28 to $aL+b+c/L+d/L^2$. Fat lines: theoretical prediction [Eq. (15)]. Eq. (3) is also plotted above $n_c$ (dashed lines) for comparison.

Figure 3

Logarithmic term in the RSE of open XXZ and $J_1-J_2$ chains, extracted using $S_n\simeq aL+b\ln L+c+d/L$ with four consecutive system sizes ($L,...,L-6$) with $L=16$ and 28. Bold lines: Eq. (15). Inset: scaling close to $n_c(\Delta =0)=4$.