Short interval expansion of R\'enyi entropy on torus

Bin Chen; Jia-ju Zhang; Jun-Bao Wu

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Short interval expansion of R\'enyi entropy on torus

Bin Chen, Jun-Bao Wu, Jia-ju Zhang

High Energy Physics - Theory · Submitted 17 Jun 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim At both low and high temperature limits, we read the R\'enyi entropies to order $\ell^6$, and find good agreements with holographic results.

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We investigate the short interval expansion of the R\'enyi entropy for two-dimensional conformal field theory (CFT) on a torus. We require the length of the interval $\ell$ to be small with respect to the spatial and temporal sizes of the torus. The operator product expansion of the twist operators allows us to compute the short interval expansion of the R\'enyi entropy at any temperature. In particular, we pay special attention to the large $c$ CFTs dual to the AdS$_3$ gravity and its cousins. At both low and high temperature limits, we read the R\'enyi entropies to order $\ell^6$, and find good agreements with holographic results. Moreover, the expansion allows us to read $1/c$ contribution, which is hard to get by expanding the thermal density matrix. We generalize the study to the case with the chemical potential as well.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the short interval expansion of the R\'enyi entropy for two-dimensional conformal field theory (CFT) on a torus.	contribution 0.97	0.90	0.08	asserted 0.85
s1	We require the length of the interval $\ell$ to be small with respect to the spatial and temporal sizes of the torus.	method 0.63	0.43	0.03	asserted 0.91
s2	The operator product expansion of the twist operators allows us to compute the short interval expansion of the R\'enyi entropy at any temperature.	method 0.99	0.84	0.16	asserted 0.98
s3	In particular, we pay special attention to the large $c$ CFTs dual to the AdS$_3$ gravity and its cousins.	contribution 0.84	0.50	0.06	asserted 0.87
s4	At both low and high temperature limits, we read the R\'enyi entropies to order $\ell^6$, and find good agreements with holographic results.	result 1.00	0.95	0.34	asserted 0.99
s5	Moreover, the expansion allows us to read $1/c$ contribution, which is hard to get by expanding the thermal density matrix.	contribution 0.42	0.87	0.18	asserted 0.98
s6	We generalize the study to the case with the chemical potential as well.	contribution 1.00	0.92	0.05	asserted 0.99

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Original abstract

We investigate the short interval expansion of the R\'enyi entropy for two-dimensional conformal field theory (CFT) on a torus. We require the length of the interval $\ell$ to be small with respect to the spatial and temporal sizes of the torus. The operator product expansion of the twist operators allows us to compute the short interval expansion of the R\'enyi entropy at any temperature. In particular, we pay special attention to the large $c$ CFTs dual to the AdS$_3$ gravity and its cousins. At both low and high temperature limits, we read the R\'enyi entropies to order $\ell^6$, and find good agreements with holographic results. Moreover, the expansion allows us to read $1/c$ contribution, which is hard to get by expanding the thermal density matrix. We generalize the study to the case with the chemical potential as well.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Subsystem quantum R\'enyi-Bhattacharyya distance in 2D CFTs and its holographic applications
hep-th 2026-10 accept novelty 7.0 of 10

A systematic calculation of the subsystem Bhattacharyya distance in 2D CFTs reveals that distinguishability scales as the central charge when classical data differ and as its inverse when they agree.
On thermality of CFT eigenstates
hep-th 2017-05 conditional novelty 6.0 of 10

In 2D conformal field theories, the trace-square distance between a heavy eigenstate's reduced density matrix and the thermal one scales as (subsystem fraction)^8 and vanishes as 1/c^2 at large central charge.

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20 arXiv papers cite this in 63 sentences

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Papers citing it · 34 sentences without a confident label

Uses its method or tool 7 papers

Using the method in appendix B of [ref] , the conformal transformations of $,,$ in [ref] , as well as the structure constants in [ref] and ([ref]), we get the one-point functions _(L,q) = 23452 π^8 c/59535 L^8 +36608 π^8 (1008 c-1) q^2/3969 L^8 +18304 π^8 (13608 c+335) q^3/1323 L^8 _(L,q) = +36608 π^8 (54096 c+5795) q^4/1323L^8 +O(q^5), _(L,q) = -13 π^8 c (5 c+22) (465 c-127)/10125(105 c+11)L^8 +1664 π^8 (5 c+22) (945 c^2+2184 c-10) q^2/675 (105 c+11)L^8 _(L,q) = +416 π^8 (5 c+22) (11340 c^2+31323 c-1517) q^3/225 (105 c+11)L^8 _(L,q) = +1664 π^8 (64575 c^3+334935 c^2+226879 c+26048) q^4/225 (105 c+11)L^8 +O(q^5), _(L,q) = π^8 c (2 c-1) (3 c+46) (5 c+3) (5 c+22) (7 c+68)/1296 (1050 c^2+3305c-251) L^8 ( 1+3264 q^2/c+13536 q^3/c _(L,q) = +576 (325 c+4814) q^4/c (5c+22)+O(q^5) ).

Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis · OPE of twist operators · 0.81
For the short interval we use the OPE of twist operators [ref] to calculate Rényi entropy, and the result can be written as a sum of the products of one-point functions [ref] .

R\'enyi entropy at large energy density in 2D CFT · Microcanonical ensemble state · 0.99
Following [ref] , we can use the OPE of twist operators to obtain the short interval expansion of the Rényi entropy S_A^(n) = n+112nlogℓ - (n+1)ℓ^26n ( T + 14 ^2 ) + O(ℓ^4), where the expectation values on the torus read [ref] T = - 2π^2 qL^2 _q Z(q)Z(q), = πL η()^2Z(q).

R\'enyi entropy and subsystem distances in finite size and thermal states in critical XY chains · CFT results · 0.95

Uses its data or values 6 papers

Using the results of $b_K$ in [ref] , we get the relevant results of $a_K$ a_T = -1/6, a_TT = -1/30 c, a_TTT = -4/315 c^2, a_ = -1/126 c (5 c+22), a_TT = 1/315 c^2, a_TTTT = -c+8/630 c^3.

Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis · The $n→1$ limit · 0.98
The result of $T _ϕ$ is given in ([ref]), and the result for $T _$ in high temperature is given in [ref] , i.e., it is

Thermality and excited state R\'enyi entropy in two-dimensional CFT · Check ETH · 0.87
Below we list some of the $b_K$ coefficients that we will require [ref] [equation] Collecting the leading order terms in the SIE for the $n$-replicated torus partition function, we have the following, [equation] where contributions from the primaries and their descendants are contained in the $ϕ$ summation and the `$…$' come with higher powers of $ℓ/L$.

Monstrous entanglement · Short interval expansion for entropy · 0.84

Compares with it 2 papers

We will then review the results in [ref] for latter comparisons.

Thermality and excited state R\'enyi entropy in two-dimensional CFT · Rényi entropy in 2D CFT: formulation and review · 0.86
This is consistent with ([ref]) and the results in [ref] .

Corrections to holographic entanglement plateau · Contributions from the vacuum module · 0.87

Cites it as background 6 papers

We will first review the result for the Rényi entropy of one interval $A=[0,ℓ]$ on a torus in the 2D free massless Majorana fermion theory [ref] , and then we will recompute it using twist operators [ref] and their operator product expansion (OPE) [ref] .

R\'enyi entropy and subsystem distances in finite size and thermal states in critical XY chains · von Neumann and Rényi entropies · 0.80
There are many investigations of holographic Rényi entropy in AdS$_3$/CFT$_2$ correspondence [ref] .

Holographic R\'enyi entropy for two-dimensional $\mathcal{N}$=(2,2) superconformal field theory · Introduction · 0.99
The results will then depend on the details of the CFT, e.g., on the spectrum and structure constants [ref] .

Thermality and excited state R\'enyi entropy in two-dimensional CFT · Rényi entropy in 2D CFT: formulation and review · 0.84

What it cites · 55 sentences without a confident label

Uses data or values from 2 papers

The needed OPE coefficients $d_K$ have been calculated in [ref] d_T=n^2-1/12n^2, d_=(n^2-1)^2/288 n^4, d_=-(n^2-1)^2 (2 n^2(35 c+61)-93)/10368 n^6(70 c+29), d_=(n^2-1)^3/10368 n^6, d_TT^j_1j_2=18n^4c1s^4_j_1j_2+(n^2-1)^2144n^4, d_T^j_1j_2=n^2-196n^6c1s^4_j_1j_2+(n^2-1)^33456n^6, d_TTT^j_1j_2j_3=-18n^6c^21s^2_j_1j_2s^2_j_2j_3s^2_j_3j_1 +n^2-196n^6c ( 1s^4_j_1j_2+1s^4_j_2j_3+1s^4_j_3j_1 ) +(n^2-1)^31728n^6, with s_j_1j_2=sin(j_1-j_2)πn.

On short interval expansion of R\'enyi entropy · OPE of twist operators · 0.95
The needed OPE coefficients $d_K$ have been calculated in [ref] d_T=n^2-1/12n^2, d_=(n^2-1)^2/288 n^4, d_=-(n^2-1)^2 (2 n^2(35 c+61)-93)/10368 n^6(70 c+29), d_=(n^2-1)^3/10368 n^6, d_TT^j_1j_2=18n^4c1s^4_j_1j_2+(n^2-1)^2144n^4, d_T^j_1j_2=n^2-196n^6c1s^4_j_1j_2+(n^2-1)^33456n^6, d_TTT^j_1j_2j_3=-18n^6c^21s^2_j_1j_2s^2_j_2j_3s^2_j_3j_1 +n^2-196n^6c ( 1s^4_j_1j_2+1s^4_j_2j_3+1s^4_j_3j_1 ) +(n^2-1)^31728n^6, with s_j_1j_2=sin(j_1-j_2)πn.

Holographic R\'enyi entropy for CFT with $W$ symmetry · OPE of twist operators · 0.95

Compares with 2 papers

We would like to compute the entropy at both the low and high temperature limits, in order to compare with the results got in [ref] .

Single Interval R\'enyi Entropy At Low Temperature · Vacuum conformal family · 1.00
We would like to compute the entropy at both the low and high temperature limits, in order to compare with the results got in [ref] .

Holographic R\'enyi Entropy of Single Interval on Torus: with W symmetry · Vacuum conformal family · 1.00

Cites as background 22 papers

In the previous studies of the Rényi entropy in a large $c$ CFT on the torus [ref] , there is no restriction on the length of the interval except that it should not be comparable with the size the torus , but one has to take the low or high temperature limit in order to expand the thermal density matrix appropriately.

Single Interval R\'enyi Entropy At Low Temperature · Conclusion · 0.91
In the previous studies of the Rényi entropy in a large $c$ CFT on the torus [ref] , there is no restriction on the length of the interval except that it should not be comparable with the size the torus , but one has to take the low or high temperature limit in order to expand the thermal density matrix appropriately.

Holographic R\'enyi Entropy of Single Interval on Torus: with W symmetry · Conclusion · 0.91
In the previous studies of the Rényi entropy in a large $c$ CFT on the torus [ref] , there is no restriction on the length of the interval except that it should not be comparable with the size the torus , but one has to take the low or high temperature limit in order to expand the thermal density matrix appropriately.

Holographic Calculation for Large Interval R\'enyi Entropy at High Temperature · Conclusion · 0.91

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