Computing entanglement entropy in 2d CFTs

Brief review

We mostly follow the review by Cardy and Calabrese.

Rényi and von Neumann entropies

The Rényi entropy $S_{\alpha }$ depends on $\alpha \in \mathbb {C}$. The von Neumann entropy is

$S=\lim _{\alpha \to 1}S_{\alpha }\ .$

Then entanglement entropy is the von Neumann entropy of the reduced density matrix of a subsystem. By extension, the term is also applied to the Rényi entropy of the subsystem.

In dimension $2=1+1$, the subsystem can be chosen as a union of $N$ disjoint intervals,

$A=(u_{1},v_{1})\cup (u_{2},v_{2})\cup \cdots (u_{N},v_{N})\ .$

The corresponding reduced density matrix coincides with the partition function of a Euclidean 2d CFT on a surface with open cuts on $A$.

Replicas and Riemann surfaces

For $\alpha =n\in \{2,3,4,\dots \}$, the Rényi entropy $S_{n}[A]$ then coincides with the partition function on the Riemann surface that is obtained by joining $n$ replicas of the sphere along the open cuts. The resulting Riemann surface has the genus

$g=(n-1)(N-1)\ .$

The moduli of the Riemann surface are given in terms of the positions of the interval endpoints. In general, the number of moduli is larger than $2N-3$ positions (modulo global conformal transformations), so we obtain a Riemann surface of special type, sometimes called a Rényi surface. For example, for $(n,N)=(3,2)$, Rényi surfaces are a $1$-dimensional subspace of the $3$-dimensional space of genus two Riemann surfaces.

Orbifold CFT

Computing partition functions on higher-genus Riemann surfaces is not easy, even for CFTs that are fully solved. The standard manipulation is, in string theory language, to replicate the target space rather than the worldsheet. The partition function on an $n$-sheeted sphere is identical to a sphere correlation function of $2N$ twist fields in the orbifold CFT

${\text{CFT}}_{n}={\frac {{\text{CFT}}^{n}}{\mathbb {Z} _{n}}}\ .$

Here the group $\mathbb {Z} _{n}$ performs cyclic permutations of the sheets. Explicitly,

$S_{n}[A]_{\text{CFT}}=\left\langle \prod _{i=1}^{N}\tau (u_{i}){\tilde {\tau }}(v_{i})\right\rangle _{{\text{CFT}}_{n}}$

This might be viewed as a special case of a correlation function in the symmetric product orbifold CFT ${\frac {{\text{CFT}}^{n}}{S_{n}}}$. Work on the symmetric product orbifold however prioritizes low values of the genus $g$ and/or the large $n$ limit, so it is of limited relevance.

The twist fields $\tau ,{\tilde {\tau }}$ should be viewed as belonging to the identity sector. They are mapped to the identity field by a singular conformal transformation. Their left and right conformal dimensions are

$\Delta _{n}={\frac {c}{24}}\left(n-{\frac {1}{n}}\right)\ .$

The singular conformal transformation maps the null vector condition $L_{-1}{\text{Id}}=0$ to $L_{-{\frac {1}{n}}}\tau =0$, from which the conformal dimension of $\tau$ can be deduced. Twist fields are therefore degenerate fields of the orbifold CFT.

In the case $N=1$, the Rényi entropy reduces to a two-point function of twist fields, and can therefore be computed up to a constant prefactor.

Motivations

CFT motivations

The Rényi entropy is closely related to the partition functions on higher genus Riemann surfaces: these are geometric quantities that do not depend on a choice of fields. Unlike correlation functions, these quantities can therefore be compared between different CFTs.

Thanks to its representation in terms of an orbifold CFT, the Rényi entropy can be a bit easier to compute than a higher genus partition function. However, it still remains difficult to compute, even in minimal models. On the other hand, the entanglement entropy is intractable, even for compactified free bosons.

Extrinsic motivations

It is easy to compute the entanglement entropy in numerical simulations, and to deduce the central charge.

Exact results

Rényi entropy of two intervals for a compactified free boson

Let $\eta$ be the radius of compactification, with $\eta =1$ the self-dual radius. Let $x$ be the cross-ratio of the $N=2$ intervals' endpoints, and let $\tau$ be such that $x={\frac {\theta _{2}(\tau )^{4}}{\theta _{3}(\tau )^{4}}}$, with $\theta _{i}$ are Jacobi theta functions. In the case $n=2$, the Rényi entropy is

$S_{2}[A]={\frac {\theta _{3}(\eta \tau )\theta _{3}(\eta ^{-1}\tau )}{\theta _{3}(\tau )^{2}}}$

The generalization is

$S_{n}[A]={\frac {\Theta (\eta \Gamma )\Theta (\eta ^{-1}\Gamma )}{\Theta (\Gamma )^{2}}}$

where $\Gamma$ is a square matrix of size $n-1$ with the elements

$\Gamma _{rs}={\frac {2i}{n}}\sum _{\xi \in (0,1)\cap {\frac {1}{n}}\mathbb {Z} }\sin(\pi \xi )\cos \left(2\pi (r-s)\xi \right){\frac {_{2}F_{1}(\xi ,1-\xi ,1,1-x)}{_{2}F_{1}(\xi ,1-\xi ,1,x)}}$

and

$\Theta (\Gamma )=\sum _{m\in \mathbb {Z} ^{n-1}}e^{i\pi (m\Gamma m)}$

It is not known how to analytically continue $S_{n}[A]$ to $n=1$ in order to obtain the entanglement entropy.

Rényi entropy in the Ising model

The Rényi entropy of two intervals in the Ising model has been determined analytically, It is quite similar to the case of the free boson, and again the continuation to $n=1$ is not known.

Rényi entropy in minimal models

The null vector equation $L_{-{\frac {1}{n}}}\tau =0$ cannot be enough for determining Rényi entropies, otherwise we would obtain results that would not depend on the CFT, only on its central charge. In minimal models, additional null vectors lead to more equations, and in some cases the Rényi entropy can be determined.

Bootstrap

The Rényi entropy for two disjoint intervals can be decomposed into Virasoro conformal blocks, and continued to $n=1$. The difficulty with this approach is that the Virasoro symmetry is much smaller than the orbifold symmetry, so that the number of Virasoro primary fields is large. If one used the orbifold symmetry itself, the difficulty would be continuing the results to $n=1$, as the symmetry only makes sense at integer $n$.