TY - JOUR

T1 - Entanglement Renyi Entropy of Two Disjoint Intervals for Large c Liouville Field Theory

AU - Tsujimura, Jun

AU - Nambu, Yasusada

N1 - Funding Information:
The author J.T. was financially supported by JSPS Fellows grant number 22J14390. The author Y.N. was supported in part by JSPS KAKENHI grant number 19K03866.
Publisher Copyright:
© 2022 by the authors.

PY - 2022/12

Y1 - 2022/12

N2 - Entanglement entropy (EE) is a quantitative measure of the effective degrees of freedom and the correlation between the sub-systems of a physical system. Using the replica trick, we can obtain the EE by evaluating the entanglement Renyi entropy (ERE). The ERE is a q-analogue of the EE and expressed by the q replicated partition function. In the semi-classical approximation, it is apparently easy to calculate the EE because the classical action represents the partition function by the saddle point approximation and we do not need to perform the path integral for the evaluation of the partition function. In previous studies, it has been assumed that only the minimal-valued saddle point contributes to the EE. In this paper, we propose that all the saddle points contribute comparably but not necessarily equally to the EE by dealing carefully with the semi-classical limit and then the (Formula presented.) limit. For example, we numerically evaluate the ERE of two disjoint intervals for the large c Liouville field theory with (Formula presented.). We exploit the BPZ equation with the four twist operators, whose solution is given by the Heun function. We determine the ERE by tuning the behavior of the Heun function such that it becomes consistent with the geometry of the replica manifold. We find the same two saddle points as previous studies for (Formula presented.) in the above system. Then, we provide the ERE for the large but finite c and the (Formula presented.) in case that all the saddle points contribute comparably to the ERE. In particular, the ERE is the summation of these two saddle points by the same weight, due to the symmetry of the system. Based on this work, it shall be of interest to reconsider EE in other semi-classical physical systems with multiple saddle points.

AB - Entanglement entropy (EE) is a quantitative measure of the effective degrees of freedom and the correlation between the sub-systems of a physical system. Using the replica trick, we can obtain the EE by evaluating the entanglement Renyi entropy (ERE). The ERE is a q-analogue of the EE and expressed by the q replicated partition function. In the semi-classical approximation, it is apparently easy to calculate the EE because the classical action represents the partition function by the saddle point approximation and we do not need to perform the path integral for the evaluation of the partition function. In previous studies, it has been assumed that only the minimal-valued saddle point contributes to the EE. In this paper, we propose that all the saddle points contribute comparably but not necessarily equally to the EE by dealing carefully with the semi-classical limit and then the (Formula presented.) limit. For example, we numerically evaluate the ERE of two disjoint intervals for the large c Liouville field theory with (Formula presented.). We exploit the BPZ equation with the four twist operators, whose solution is given by the Heun function. We determine the ERE by tuning the behavior of the Heun function such that it becomes consistent with the geometry of the replica manifold. We find the same two saddle points as previous studies for (Formula presented.) in the above system. Then, we provide the ERE for the large but finite c and the (Formula presented.) in case that all the saddle points contribute comparably to the ERE. In particular, the ERE is the summation of these two saddle points by the same weight, due to the symmetry of the system. Based on this work, it shall be of interest to reconsider EE in other semi-classical physical systems with multiple saddle points.

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U2 - 10.3390/e24121758

DO - 10.3390/e24121758

M3 - Article

AN - SCOPUS:85144649324

SN - 1099-4300

VL - 24

JO - Entropy

JF - Entropy

IS - 12

M1 - 1758

ER -