Entropy and modular Hamiltonian for a free chiral scalar in two intervals

We calculate the analytic form of the vacuum modular Hamiltonian for a two interval region and the algebra of a current $j(x)=\partial \varphi (x)$ corresponding to a chiral free scalar $\varphi$ in $d=2$. We also compute explicitly the mutual information between the intervals. This model shows a failure of Haag duality for two intervals that translates into a loss of a symmetry property for the mutual information usually associated with modular invariance. Contrary to the case of a free massless fermion, the modular Hamiltonian turns out to be completely nonlocal. The calculation is done diagonalizing the density matrix by computing the eigensystem of a correlator kernel operator. These eigenvectors are obtained by a novel method that involves solving an equivalent problem for a holomorphic function in the complex plane where multiplicative boundary conditions are imposed on the intervals. Using the same technique we also rederive the free fermion modular Hamiltonian in a more transparent way.

Figure 1

The mutual information $I(A_1,A_2)$ as a function of the cross ratio $\eta$. The continuous solid line corresponds to the mutual information obtained by numerical integration of the analytic expression (5.74) and (5.75). The red points correspond to the simulation on the lattice as it is explained in Sec. 7. The dashed line is the chiral free fermion mutual information $-(1/6)\log (1-\eta )$.

Figure 2

The function $U(\eta )$ as a function of the cross ratio $\eta$. This function is negative because the model of the current is a subalgebra of the free fermion model. It does not have the $\eta \leftrightarrow (1-\eta )$ symmetry expected for the case where the entropy for two intervals is equal to the entropy of its complement. $U(\eta )$ has a $-1/2\log (-\log (1-\eta ))$ divergence for $\eta \to 1$.

Figure 3

The Renyi mutual information $I_n(\eta )$ as a function of the cross ratio $\eta$ for different values of $n$. From top to bottom $n=1/3$, $1/2$, 1, 2, $\infty$. The mutual information $I_1(\eta )=I(\eta )$ is shown with the dashed line.

Figure 4

The function ${\widehat{\alpha }}_r(z)$ for different values of $\eta$, blue $\eta =1/3$, green $\eta =1/2$, and red $\eta =2/3$.

Figure 5

A contour plot of the kernel $N(x^{\prime },y^{\prime })$ giving the nonlocal part of the modular Hamiltonian for two intervals for $\eta =9/10$. $N(x^{\prime },y^{\prime })$ has a logarithmic divergence along the diagonal.