TY - JOUR

T1 - Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT

AU - He, Song

AU - Lin, Feng Li

AU - Zhang, Jia ju

N1 - Funding Information:
at http://people.brandeis.edu/~headrick/Mathematica/index.html. JJZ would like to thank the organisers of The String Theory Universe Conference 2017 held in Milan, Italy on 20-24 February 2017 for being given the opportunity to present some of the results as gongshow and poster, and thank the participants, especially Geoffrey Compère, Debajyoti Sarkar and Marika Taylor, for stimulating discussions. SH is supported by Max-Planck fellowship in Germany and the National Natural Science Foundation of China Grant No. 11305235. FLL is supported by Taiwan Ministry of Science and Technology through Grant No. 103-2112-M-003-001-MY3 and No. 103-2811-M-003-024. JJZ is supported by the ERC Starting Grant 637844-HBQFTNCER.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - We investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval Rényi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of Rényi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central charge expansion. We then evaluate the relative entropy of the reduced density matrices to measure the difference between the heavy primary state and thermal state of canonical ensemble, and find that the aforementioned nontrivial eighth order results make the relative entropy unsuppressed in the large central charge limit. By using Pinsker’s and Fannes-Audenaert inequalities, we can exploit the results of relative entropy to yield the lower and upper bounds on trace distance of the excited-state and thermal-state reduced density matrices. Our results are consistent with subsystem weak ETH, which requires the above trace distance is of power-law suppression by the large central charge. However, we are unable to pin down the exponent of power-law suppression. As a byproduct we also calculate the relative entropy to measure the difference between the reduced density matrices of two different heavy primary states.

AB - We investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval Rényi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of Rényi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central charge expansion. We then evaluate the relative entropy of the reduced density matrices to measure the difference between the heavy primary state and thermal state of canonical ensemble, and find that the aforementioned nontrivial eighth order results make the relative entropy unsuppressed in the large central charge limit. By using Pinsker’s and Fannes-Audenaert inequalities, we can exploit the results of relative entropy to yield the lower and upper bounds on trace distance of the excited-state and thermal-state reduced density matrices. Our results are consistent with subsystem weak ETH, which requires the above trace distance is of power-law suppression by the large central charge. However, we are unable to pin down the exponent of power-law suppression. As a byproduct we also calculate the relative entropy to measure the difference between the reduced density matrices of two different heavy primary states.

KW - AdS-CFT Correspondence

KW - Conformal Field Theory

KW - Holography and condensed matter physics (AdS/CMT)

UR - http://www.scopus.com/inward/record.url?scp=85028637869&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028637869&partnerID=8YFLogxK

U2 - 10.1007/JHEP08(2017)126

DO - 10.1007/JHEP08(2017)126

M3 - Article

AN - SCOPUS:85028637869

VL - 2017

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 8

M1 - 126

ER -