Entanglement entropy of two disjoint intervals in c=1 theories

TL;DR

The paper addresses how Renyi entropies for two disjoint intervals reveal rich operator content in $c=1$ theories, deriving an analytic $F_2(x)$ for a free boson on an orbifold tied to the Ashkin–Teller model and validating it with Monte Carlo simulations and Tree Tensor Network methods on the XXZ spin chain. By combining cluster Monte Carlo for the classical AT model with TTN-based quantum simulations, the work demonstrates quantitative agreement with CFT predictions once finite-size corrections are properly accounted, highlighting scaling forms $F_n^{\rm lat}(x)=F_n^{\rm CFT}(x)+\ell^{-2\omega/n}f_n(x)$ and related exponents $K_L$. The results show that disjoint-interval entanglement encodes compactification radii and symmetries beyond central charge, while providing practical, symmetry-aware numerical strategies for testing CFT data in strongly interacting lattice systems. Overall, the study advances the understanding of disjoint-interval entanglement in $c=1$ theories and provides robust methodologies for extracting universal behavior from finite-size data in both classical and quantum models.

Abstract

We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.

Figures (23)

• Figure 1: Survey of $c=1$ theories corresponding to a free boson compactified on a circle (horizontal axis) and on an orbifold (vertical axis) as reported e.g. in Refs. book. For some values of $r_{\rm circle}$ and $r_{\rm orb}$, the corresponding statistical mechanical models are reported. The XXZ spin chain in zero magnetic field lies on the horizontal axis in the interval $r_{\rm circle}\in[0,1/\sqrt2]$. The self-dual line of the Ashkin-Teller model lies on the vertical axis in the interval $r_{\rm orb}\in[\sqrt{2/3},\sqrt{2}]$.
• Figure 2: $F_2(x)$ for the Ashkin-Teller model on the self-dual line for some values of $\eta$. Inset: $F_2(x)-1$ in log-log scale to highlight the small $x$ behavior. The black-dashed line is $\sim x^{1/4}$.
• Figure 3: Phase diagram of the 2D symmetric Ashkin-Teller model defined by the Hamiltonian (\ref{['sat']}). The red ABC line is the self dual line. The point $B$ at $K=0$ corresponds to two uncoupled Ising models. The point $C$ is the critical four-state Potts model at $K=J=(\log 3)/4$. At $J=0$ there are two critical Ising points at $K=\pm(\log(1+\sqrt{2}))/2$, one (Is) ferromagnetic and the other (AFIs) antiferromagnetic. For $K\rightarrow\infty$ there is another critical Ising point at $J=(\log(1+\sqrt{2}))/2$. All continuous lines are critical. The blue lines $C-Is$ and the one starting at $AFIs$ are in the Ising universality class. The red line is critical with continuously varying critical exponents. The region denoted by I corresponds to a ferromagnetic phase for all the variables. In the region II, $\sigma$, $\tau$, and $\sigma\tau$ are paramagnetic. In the region III only $\sigma\tau$ is ferromagnetic and in region IV $\sigma\tau$ exhibits antiferromagnetic order while $\sigma$ and $\tau$ are paramagnetic.
• Figure 4: A typical cluster configuration on a $12\times 12$ lattice. Green lines are $\sigma$-clusters and red dashed lines are $\tau$-clusters. Links in blue are double links. Periodic boundary conditions on both directions are used.
• Figure 5: ${\rm Tr}\,\rho_A^2$ for a single interval of length $\ell$ in a finite system of length $L=120$. Data have been obtained by Monte Carlo simulations using the embedded algorithm. The orange points correspond to the SUSY model and the green ones to the $Z_4$ parafermions. The black crosses at $\ell=10$ are data obtained using the direct algorithm. Inset: behavior of the statistical error of ${\rm Tr}\,\rho_A^2$ vs $\ell$ for the SUSY model. The blue-dashed line is the expected form $A+B\ell^{1/2}$.