R\'enyi entropy at large energy density in 2D CFT

We investigate the R\'enyi entropy and entanglement entropy of an interval with an arbitrary length in the canonical ensemble, microcanonical ensemble and primary excited states at large energy density in the thermodynamic limit of a large central charge $c$ two-dimensional conformal field theory. The main purpose is to use these results to see whether we can distinguish these various kinds of large energy density states by the R\'enyi entropies of an interval at different size scales, namely, short, medium and long. In evaluating these R\'enyi entropies by the method of twist operators, we find some corrections to the recent results for the holographic R\'enyi entropy of a medium size interval. Based on our derived R\'enyi entropies of the three interval scales, we find that R\'enyi entropy cannot distinguish the canonical and microcanonical ensemble states for a short interval, but can do the job for both medium and long intervals. At the leading order of large $c$ the entanglement entropy cannot distinguish the canonical and microcanonical ensemble states for all interval lengths, but the difference of entanglement entropy for a long interval between the two states would appear with $1/c$ corrections. We also discuss R\'enyi entropy and entanglement entropy differences between the thermal states and primary excited state.


Introduction
Eigenstate thermalization hypothesis (ETH) [1][2][3][4][5] states that in a chaotic system local operators cannot distinguish a highly excited energy eigenstate from a proper thermal state. Then, the natural and complementary questions are whether some nonlocal operators can distinguish the excited and thermal states, and how nonlocal they should be. A natural set of nonlocal operators are the entanglement entropy S A and Rényi entropy S (n) A of a subsystem A of volume V A in a system of volume V and state with density matrix ρ. The reduced density matrix of A is obtained by tracing out the degrees of freedom of its complementĀ, i.e. ρ A = trĀρ. The Rényi entropy is defined as and in n → 1 limit it becomes the entanglement entropy Based on these and other nonlocal quantities the subsystem ETH was proposed [6][7][8]. Moreover, the distinguishability of a thermal state from its microstates is related to the black hole information loss paradox [9,10] through gauge/gravity duality [11][12][13].
The above scheme of distinguishability can be extended to the states with finite energy density, i.e. states of energy E with E/V fixed and finite in the thermodynamic limit V → ∞. For these states the Rényi entropy is expected to follow the volume law [14]. A related question is whether entanglement entropy or Rényi entropy can distinguish canonical and microcanonical ensemble states even in the thermodynamic limit. As local operators cannot distinguish the canonical and microcanonical ensemble states [7,8,15,16], neither can the short interval entanglement entropy or Rényi entropy. It was proposed in [7] that the Rényi entropy of an interval with a length that is comparable to the length of its complement can distinguish the canonical and microcanonical ensemble states, while the entanglement entropy cannot. Recently, in this context it was shown by Dong in [17] (which is motivated by [14] and [7,18]) that the holographic Rényi entropy of a medium size subsystem, i.e.
with V A /V fixed and finite, can distinguish the canonical and microcanonical ensemble states of large energy density, i.e. E/V ∝ c with c being the large central charge. Holographically, one can evaluate the entanglement entropy by Ryu-Takayanagi formula [19][20][21][22][23], and the Rényi entropy by relating it to the refined Rényi entropy [24]S (n) which can be evaluated by the area of a bulk codimension-two cosmic brane.
In this paper we investigate Rényi entropy in the two-dimensional (2D) CFTs of large central charge c for the states of large energy density in the thermodynamic limit. The length of the circle where the CFT lives is L, and the length of the subsystem A is . For comparison, we consider three types of large density states: (i) the canonical ensemble state at inverse temperature β; (ii) the microcanonical ensemble state of energy E = πcL 6λ 2 with λ being a constant; and (iii) a primary excited state of conformal weights h =h = c 24 ( L 2 µ 2 + 1) or energy E = πcL 6µ 2 with constant µ. Furthermore, for each kind of states, we will consider three different scales of . For the canonical ensemble state, we have (S) short interval with 0 < β; (M) medium interval with β L − β; and (L) long interval with L − β L. In the above we have used " " to indicate that we cannot find the sharp regime boundaries. It is similar for the microcanonical ensemble state. For the primary excited state, we have (S) short interval with 0 < µ; (M) medium interval with µ < L/2; and (L) long interval with L/2 < L.
The medium interval regime for canonical and microcanonical ensemble states has been investigated in [17] for holographic CFTs in general dimensions, and the 2D results for holographic Rényi entropy and refined Rényi entropy in this regime are In the above equation we have used "CE" to denote the canonical ensemble state and "ME" to denote the microcanonical ensemble state, and in this paper we will also use "PE" to denote the primary excited state. It was argued in [14,17] that the above results for microcanonical ensemble state also apply to the primary excited state as long as < L/2. Note that, the primary excited state is a pure state so that S PE (L − ) relating the short and long interval ones, and the short interval expansion has been obtained in [6,[25][26][27]. Rényi entropy in canonical ensemble state for all three scales of have been obtained in [28,29] by field theory method. However, Rényi entropy in microcanonical ensemble haven't been explored before. This is one of the concrete results of our paper. Combining all the results, we obtain the piecewise Rényi entropies in canonical ensemble, microcanonical ensemble and primary excited states for arbitrary subsystem size.
Using these piecewise Rényi entropies, we find that the short interval Rényi entropy cannot distinguish a canonical ensemble state from a microcanonical ensemble one as expected, but the medium and long interval ones can. In contrast, at the leading order of c the entanglement entropy of arbitrary cannot distinguish the canonical and microcanonical ensemble states. With the 1/c corrections, however, the entanglement entropy of long interval regime can distinguish these two ensembles. Our findings are consistent with the holographic results in [17]. Furthermore, we also generalize the holographic results of medium interval of [17], i.e. (1.4), to arbitrary interval length as well as to all orders of 1/c.

Canonical ensemble state
We first consider the canonical ensemble state in 2D CFT at inverse temperature β. Rényi entropy of a length interval in the thermodynamic limit is well-known [28] with being the UV cutoff. Note that this formula holds as long as L − β, and was obtained by using the method of twist operators [28,30]. The long interval Rényi entropy has also been investigated in [36,37] and [29], and we just adopt the result in [29] that was obtained using conformal transformations. In the thermodynamic limit, the result is This formula holds as long as β. In the above, we have introduced I n (x) with 0 < x < 1, which is the Rényi mutual information of two disjoint intervals on a complex plane with cross ratio x, see a brief review in Appendix A. Note that I n (x) satisfies the property [31] I n (x) = c(n + 1) 6n log For x 1 it has been calculated up to order x 8 [34,[38][39][40][41]. In this paper we use the small x expansion up to x 8 as the approximation of I n (x) for 0 < x < 1/2. When approaches L − , the long interval Rényi entropy approaches the Rényi entropy of the entire system , we find an extra term, i.e. the first term on the RHS of (2.6). Holographically, the refined Rényi entropy is given by the area of a codimension-2 cosmic brane homologous to the interval A in a backreacted bulk geometry, which is denoted by B n (β, A) in [17]. The second term on the RHS of (2.6), i.e. the result in [17], is the contribution from the part of the cosmic brane parallel to the black hole horizon, and the first term is the part extending along the radial direction of the bulk geometry.
Due to the aforementioned regions of validity for the short and long interval formulas (2.1) and (2.2), we can infer that the medium interval formula (2.2) should also exits a validity region, say  Note that the above CE 2 is just the critical point for the minimal surface of the holographic entanglement entropy [42][43][44].
In summary, the piecewise Rényi entropy for the canonical ensemble state is Note that in the above we have put back the UV cutoff . A typical piecewise Rényi entropy for a canonical ensemble state is shown in Figure 1. We also plot the short interval, medium interval, long   We then consider a large energy density microcanonical ensemble state with energy E = πcL 6λ 2 . For the short interval we use the OPE of twist operators [31][32][33][34] to calculate Rényi entropy, and the result can be written as a sum of the products of one-point functions [25][26][27]35]. It was recently shown in [16] that in the thermodynamic limit and with the identification β = λ, the canonical and microcanonical ensemble states have the same one-point functions so that the resultant short interval Rényi entropies are the same as long as the expansion converges. We thus get the short interval Rényi entropy in microcanonical ensemble state Note that it is only valid for < λ.
For a long interval, we can still use the OPE of twist operators [36,37], and we give the calculation details in Appendix C. The long interval Rényi entropy in microcanonical ensemble state is It is only valid for L − λ < L − . As approaches L − , it approaches the Rényi entropy of the total system Unlike the canonical ensemble case, the RHS of (3.3) does not depend on the Rényi index n.
The Rényi entropy of a medium interval can be calculated holographically as in [17]. The backreacted geometry B n (β n , A) caused by the cosmic brane for a microcanonical ensemble state of dual CFT is approximately the same as B n (β, A) for a canonical ensemble state with the parameter β n given by [17] Using this fact and the result for a canonical ensemble state (2.6), we can get the medium interval refined Rényi entropy for the corresponding microcanonical ensemble statẽ The medium interval Rényi entropy in microcanonical ensemble state can thus be obtained from (3.5), Again, comparing with the results in [17], i.e.S Similar to the canonical ensemble case, due to the limited regimes of validity for the short and long interval formulas (3.1) and (3.2), the medium interval formula (3.6) should also have a validity region, say ME 1 ME 2 for some critical lengths ME 1,2 . Thus, the piecewise Rényi entropy for a microcanonical ensemble state is However, we do not know how to obtain the precise forms of the critical lengths ME 1,2 . Given n and λ, we can get the approximate value of ME 1 by requiring S ME,M ( ME 1 ), and the approximate ME,L ( ME 2 ). As expected, both ME 1 and L − ME 2 are the same order of λ. In fact, by setting λ = β, ME 1,2 are, respectively, close to CE 1,2 . The piecewise Rényi entropy for a typical microcanonical ensemble state is shown in Figure 1. We also plot the short interval, medium interval,

Primary excited state
We finally consider a large energy density primary excited state with conformal weights h =h = c 24 ( L 2 µ 2 + 1) and energy E = πcL 6µ 2 . The short interval Rényi entropy for a primary excited state was calculated in [6,[25][26][27], and in the thermodynamic limit it is No closed form of the short interval Rényi entropy in primary excited state is known, and we use the above expansion as an approximation. The expansion breaks down rapidly as approaches µ.
It was argued in [14,17] that the medium interval Rényi entropy in the primary excited state is the same as that in the microcanonical ensemble state as long as < L/2. If so, this would lead to the medium interval Rényi entropy in primary excited state We use the symbol " ? =" to remind the reader that there is no rigorous justification. We do not know whether the conjecture is true at the leading order of c, but in the next section we will show that even if it is true at the leading order of c there must be some corrections at order O(c 0 ). In [17] the author commented that the conjecture may likely fail for 2D CFTs, which are special compared to their higher dimensional cousins because of the infinite number of commuting conserved quantum Kortewegde Vries charges [45,46]. In fact, a different result of the Rényi entropy in the primary excited state from (4.2) was obtained in [15], and one can see also an earlier proposal in [7]. The medium interval Rényi entropy in primary excited state is an open question, as emphasized in both [15] and [17], however we will plot the figures using the conjecture (4.2).
Again, there should exist a critical length PE so that the short interval formula (4.1) holds only for < PE and the medium interval formula (4.2) holds only for PE < L/2. Given n, µ, we PE,M ( PE ). By setting λ = β = µ, we find PE is close to CE 1 and ME 1 . For the long interval regime L/2 < < L − , we can use S The piecewise Rényi entropy for a typical primary excited state is shown in Figure 1. We also plot the short interval, medium interval, and piecewise Rényi entropies in the primary excited state (4.1), Figure 6 of Appendix B.

Distinguishabilities of various states
As stated in [17], to investigate ETH, it is interesting to use the Rényi entropy and entanglement entropy to distinguish various large energy density states. We use (2.8), (3.7), and (4.3) to calculate the Rényi entropy and entanglement entropy differences among the canonical ensemble, microcanonical ensemble, and primary excited states, and plot the results in Figure 2. Note that since we cannot calculate the precise various critical lengthes, i.e. CE 1,2 , ME 1,2 , PE , the figures around these critical lengthes are just suggestive. We also remind the reader that the medium interval Rényi entropy in the primary excited state (4.2) is a conjecture, and so the corresponding Rényi entropy and entanglement entropy differences are also conjectures and thus suggestive.
The leading order c part of the entanglement entropy for the primary excited state with length 0 < < L/2 was calculated in [47,48], and it is the same as the entanglement entropy in canonical ensemble state with the identification β = µ No closed form of the 1/c corrections is known, and it was calculated by short interval expansion to 8 in [26,27] and to 12 in [49]. Let us denote the reduced density matrices of the interval A for the canonical ensemble and primary excited states by ρ CE ( ) and ρ PE ( ), respectively. Using the fact that in the thermodynamic limit the modular Hamiltonian of ρ CE ( ) is a local integral of the energy density [50,51], the relative entropy will be reduced to the difference of the entanglement entropies, We summarize the distinguishabilities of the canonical ensemble, microcanonical ensemble, and primary excited states in Table 1. Especially, we find that Rényi entropy cannot distinguish the canonical and microcanonical ensemble states for a short interval, but can distinguish the two states for a medium interval and a long interval at the leading order of c. At the leading order of c the entanglement entropy cannot distinguish the canonical and microcanonical ensemble states for any length interval, but the difference of entanglement entropy between the two states would appear with 1/c corrections for a long interval. Both Rényi entropy and entanglement entropy can easily distinguish the thermal states and the primary excited state for a long interval. The Rényi entropy is more powerful than the entanglement entropy to distinguish different states. Our findings are consistent with and generalize the holographic medium interval results in [17], and are also consistent with the conjecture in [7].  Table 1: The distinguishabilities of the canonical ensemble, microcanonical ensemble, and primary excited states for a short, medium, and long interval in terms of the entanglement entropy and Rényi entropy. We mark " " for distinguishable states, and mark "×" otherwise. For some cases we give the references where the results were firstly derived or could be easily inferred from. Note that for "canonical VS primary" and "microcanonical VS primary", we refer to µ < L/2 for a medium interval and refer to L/2 < < L for a long interval. The medium interval Rényi entropy in the primary excited state (4.2) is a conjecture, and we mark "(?)" for the corresponding cases. The other cases are derived in this paper.
In Table 1, there is a puzzle, and it is related to that the medium interval Rényi entropy for the primary excited state (4.2) is a conjecture in [14,17]. Generally, the states become more distinguishable as the length of the interval increases. When two states can be distinguished for a short interval, one may expect that they can also be distinguished for a medium interval. In Table 1 there is one case that this rule does not apply, which is using the leading order Rényi entropy to distinguish the microcanonical ensemble and primary excited states. Generally, there is no theorem to guarantee that the Rényi entropy difference must be nondecreasing with respect to . As we have discussed above, the conjecture (4.2) is not true at least at the order of O(c 0 ), but we cannot conclude whether it is true at the leading order of c.
In the thermodynamics limit, using directly the definition of the relative entropy and the density matrices of the entire system, with β = λ we get the vanishing relative entropy of the entire system in the microcanonical and canonical ensemble states, i.e., S(ρ ME (L) ρ CE (L)) = 0. This is consistent with the results in [17] and in this paper. With β = µ, it is also easy to get the relative entropy of the entire system in the primary excited and canonical ensemble states This is consistent with the result in this paper. We cannot extract more additional useful information from the relative entropy.

Discussion
If ETH works, it should apply not only to the primary states but also the descendant states in the 2D large c CFT. The short interval entanglement entropy for some special descendant states was recently studied in [49,53], and they generally behave differently from the canonical ensemble, microcanonical

Acknowledgments
We would like to thank Anatoly Dymarsky for helpful discussions. WZG is supported by the National is defined as It is a function of the cross ratio , and one can write it as I n (x). For the 2D large c CFT, we only include the contributions from the vacuum conformal family. The Rényi mutual information satisfies the property [31] I n (x) = c(n + 1) 6n log No closed form of I n (x) is known, and for a small x 1 it has been calculated up to order x 8 [34,[38][39][40][41].
One can see the result in [41]. We plot it in Figure 3.

B Plots of Rényi entropies in various states
We plot the short interval, medium interval, long interval, and piecewise Rényi entropies and entan-

C Derivation of long interval Rényi entropy in microcanonical ensemble state
We derive the long interval Rényi entropy (3.2) for the high energy density microcanonical ensemble state with energy E = πcL 6λ 2 in the thermodynamic limit in the 2D large c CFT. The density matrix of the entire system of the microcanonical ensemble state and the partition function are     , and (4.3) to plot the differences of Rényi entropies and entanglement entropies for the canonical ensemble, microcanonical ensemble, and primary excited states. Since we cannot determine the precise various critical lengthes, i.e. CE 1,2 , ME 1,2 , PE , the figures around these critical lengthes cannot be trusted. We also remind the reader that the medium interval Rényi entropy for the primary excited state (4.2) is a conjecture. Especially, the conjecture is not true at least at order O(c 0 ), and the medium interval part of entanglement entropy difference S CE ( ) − S PE ( ) (middle of the 4th row) should be nonvanishing. To draw the figures we have set c = 30, L = 100.
The density matrix of canonical ensemble state with inverse temperature β and the canonical ensemble partition function are There is the relation We have a short interval A with length and its complementĀ, and a long interval with length L − .
Note that S A . From OPE of twist operators [31][32][33][34], for the short interval reduced density matrix ρ A (E) = trĀρ(E) we get [25][26][27]35] with h σ = c(n 2 −1) 24n and the one-point function of a general quasiprimary operator X in the 2D CFT being defined as (C.5) The coefficients b X 1 ···X k were defined in [35] and are related to the OPE coefficients of the twist operators, and their explicit forms are not important to us in this paper. In the thermodynamic limit the one-point function w.r.t. microcanonical ensemble state is the same as the expectation value w.r.t.