^{*}

^{3}.

The Rényi entanglement entropy (REE) of the states excited by local operators in two-dimensional irrational conformal field theories (CFTs), especially in Liouville field theory (LFT) and

One can define some observables to detect the property of the vacuum or excited states in a local quantum field theory. For example, entanglement entropy (EE) and the Rényi entanglement entropy (REE) are helpful quantities to use when studying global or nonlocal structures in QFTs. For a subsystem, the entanglement entropy and Rényi entropy, both of them are defined as a function of the reduced density matrix

One might wonder whether there is a kind of topological contribution to the entanglement entropy even for gapless theories, e.g., conformal field theories (CFTs) (e.g., computing topological contributions in entanglement entropy called topological entanglement entropy

The

In 2D rational CFTs, it was found

In this research, the previous study

In this paper, we evaluate the second REE in a compact

The outline of this paper is as follows. In Sec.

An excited state is defined by an operator

This figure is to show our basic setup in the two-dimensional plane

To calculate variation of the

More precisely, the replica method for the local operator excited states in field theory has been explicitly shown in Sec. 2.2 of

Finally, the

We will explain the details at the end of this section.

Firstly, we study

For

We will follow the standard procedure of the analytical continuation of Euclidean theory into its Lorentzian version. The most important and subtle point is that we should treat

To get REE, we solely focus on the conformal cross ratio

It is useful to note the relationship

We are interested in the two limits (i)

The four-point function on

The two-point function looks like

In the late time limit (ii), we finally find that the ratio in

In rational CFTs, we can calculate

Extra care should be taken when we generalize

The replica method for the local operator excited states in field theory has been explained explicitly in Sec. 2.2 of

In other words,

The existence of the translation invariant normalizable vacuum is not self-consistent with the classical equation of motion of Liouville field theory, which has been shown in

The main reason to choose reference and target states from the same class in LFT and SLFT is to calculate a well-defined quantity to restore the quasiparticle picture.

To begin with the analysis of REE in special irrational CFTs, the compact free boson theory with generic radius is a simple irrational theory to see the time evolution of the second REE.

The authors of

For

For

We can analyze the two light cone singularities of the second Reyni entropy of this theory explicitly, via the torus partition function [30]

Then, the late time limit of the second REE in

Finally, the variation of the second REE between the early time and the late time is

The compact free boson theory on generic radius as “nearly rational” theory and the entanglement entropy has quasiparticle behavior, which is consistent with the criterion of a quasiparticle picture

We are mainly interested in the second REE here, which is associated with the four-point function in terms of Eq.

In our setup of REE, the four-point Green function does not involve any discrete terms.

reviewed in AppendicesFirst, let us calculate the REE in the early time limit. One can make use of the

Once we take the early time limit of Eq.

In terms of Eq.

Following standard regularization from Eq. (5.13) in

Then

Generally speaking, if

In the remaining part of this paper, one can refer to the divergent piece of

Generally speaking, for the four-point function

Here we have chosen

Since the factor

Now we will calculate the late time limit

The prime of integration over intermediate momentum

For external Liouville momentum

In the late time limit, the leading contributions to the REE will be as follows:

In this limit, the ratio becomes

Then the second REE in the late time limit reads

Additionally, we have to consider the marginal case:

Since

In the late time limit, the ratio for the second REE is

In this section, we would like to consider the states excited by local operators in super-Liouville field theory. For the sake of consistent notation, we review the corresponding contents of SLFT in Appendix

We start with the external super-Liouville momentum

The two-point Green function for the primary operator in the NS sector is as follows:

Then

This case is similar to that mentioned above, except that

The ratio for the second REE in the early time limit is

Secondly, we consider the second REE in SLFT in the late time limit. For convenience in the late time limit, we have to use a conformal bootstrap equation to express the four-point function

Then the ratio associated with the second REE in super-Liouville field theory can be defined as follows:

This case is similar to that mentioned above, except that

For

In the previous section, we computed the second REE of the local excited states. In this subsection, we use the

In the late time limit, the dominant contribution from the intermediate channel in Eq.

Refer to Eqs.

The

We normalize the two-point function

For the sake of simplicity, we have omitted the normalization factor associated with

Then we get

The

In the early time limit

If we can regard the

We can estimate the difference in the late time limit as follows:

Thus we can estimate the

Then

Then the variation of the difference

This is the main difference between LFT and rational CFTs. In Appendices

To close this section, we would like to comment on how to extend the calculation of

In the previous sections, we computed the

The conformal transformation for the descendant operators can be derived from the energy-momentum tensor and the conformal transformation from the

The two-point function for

In the late time limit,

Due to the fusion rule in LFT or SLFT, [

In Secs.

Here the

Precisely, here we have neglected the normalization factors and associated DOZZ factors.

Finally, we present the main results with the more generic descendent operators:

We do not repeat the calculation in detail here as this has been analyzed in

As we see from Eq.

In this paper, the time evolution of the difference

To answer this question, we calculate the second REE in a compact

To understand these properties, the second REE

Equivalently,

How can one understand the different divergence behaviors of

Finally, one can apply these techniques to calculate the out-of-time-ordered correlation function (OTOC) to check whether the superintegrability of LFT is consistent with the chaotic proposal

We would like to thank Konstantin Aleshkin, Vladimir Belavin, X. Cao, Bin Chen, Harald Dorn, M. R. Gaberdiel, Wu-Zhong Guo, George Jorjadze, Li Li, Zhu-Xi Luo, Tokiro Numasawa, Hao-Yu Sun, J. Teschner, Kento Watanabe, and Jie-qiang Wu for their discussions and suggestions during various stages of the project. We appreciate Harald Dorn, Xing Huang, Axel Kleinschmidt, Hermann Nicolai, and Tadashi Takayanagi, who commented on the draft. We especially thank Xing Huang and Stefan Theisen for intensive discussions during the whole project. S. H. appreciates Axel Kleinschmidt, Hermann Nicolai, Stefan Theisen, and Tadashi Takayanagi for their encouragement and support. S. H. is supported by the Max-Planck fellowship in Germany, the German-Israeli Foundation for Scientific Research and Development, and the National Natural Science Foundation of China (Grant No. 11305235).

The full Liouville action (see a review in

The stress tensor is

The three-point function of the primary operator in LFT is

The

The theory has

The NS-NS primary fields

The dimension of the R-R operator is

To consider both NS and R sectors, we will need various functions defined differently for each sector. Here we use the notations

The four-point function for the NS-NS operator is

Following

The four-point function for R-R operators,

The function

In terms of the functional relation in Eq.

In terms of

The asymptotics behavior of

We define the following functions:

In the paper we used the following:

Reflection properties:

Locations of zeros and poles can be obtained from Eq.

Basic residue:

Following the Appendix about the LFT in

The integrand of

The similar structure constant in SLFT is given in the above Appendix

If

Using the

In this paper, we will use asymptotic form of the conformal block as

Once

In this section, we would like to show some details about how to do the early time integral appearing in the four-point function in LFT. The early time limit

For

But we have to take into account that

The fact that the author writes

For

The integral has again an asymptotics as above Eq.

To calculate the late time limit of

In this subsection, we will see how to associate with the quantum dimension defined in LFT. For the late time limit of the second REE, the fusion matrix element

Just to follow the convention in

The factor

By comparing the definition of the quantum dimension

Similar to Appendix

Alternatively, it can be shown that for

In the second line of