operator method i etaglemet etropy Noburo hiba YITP Kyoto U. Luch semiar at YITP, Jauary 14, 2015 N.hiba, JEP 1412 2014 152 [arxiv:1408.1637] YITP 2015/1/14
Cotets operator method = ew computatioal method of etaglemet etropy EE Its applicatio to Etaglemet Etropy of Disjoit Regios i Locally Excited tates
1.Itroductio Etaglemet etropy EE is the quatity which measures the degree of quatum etaglemet. EE is a useful tool to study global properties of QFTs. I the light of d/cft correspodece, the geometries of gravitatioal spacetimes ca be ecoded i the quatum etaglemet of dual CFTs.
The defiitio ad basic properties of Reyi etaglemet etropy We decompose the total ilbert space ito subsystems ad. tot We trace out the degrees of freedom of ad cosider the reduced desity matrix of. Tr tot EE is defied as vo Neuma etropy. : tr log The Reyi EE is the geeralizatio of EE ad defied as 1 1 : logtr lim 1
EE i QFT geometric etropy Geeral properties 1. If a composite system is i a pure state, the 2.If, the ad Total system: d dimesioal space-like maifold N ubsystem: d dimesioal domai I d+1 dimesioal QFT, we defie the subsystem geometrically. Mutual Reyi iformatio I :, :, I 0
2.ome computatioal methods of EE Euclidea time path itegral method: Whe the subsystem is disjoit, this method is ot useful. Real time method: This method is applicable oly for the Gaussia desity matrix operator method : we ca use the geeral properties of the operator to compute systematically the Reyi etropy for a arbitrary state ad this method is useful whe the subsystem is disjoit.
operator method i EE We cosider the geeral scalar field i d+1 dimesioal spacetime ad do ot specify its amiltoia. We cosider copies of the scalar fields ad the j-th copy of j the scalar field is deoted by { }. Thus the total ilbert space,, is the tesor product of the copies of the ilbert space, where is the ilbert space of oe scalar field. We defie the desity matrix i as where is a arbitrary desity matrix i. We ca express as Tr Tr Tr E
Tr Tr E x d [ x, y] i x y x where is a cojugate mometa of, ad J j x K j x ad exist oly i ad 1 J J 1
Geeral properties of E 1 ymmetry: 0 2 Locality: whe ad E E E 3 For arbitrary operators j = 1, 2,..., o, F j where F j Tr c F j 4 The cyclic property: 5 The relatio betwee ad for pure states: E E where ad are arbitrary pure states. j j c
E 5 The relatio betwee ad for pure states: E where ad are arbitrary pure states. j j c This is the geeralizatio of for a pure state Tr Tr c
EE of Disjoit Regios i Locally Excited tates r C 1 R 2 R We cosider the mutual Reyi iformatio of disjoit compact spatial regios ad i the locally excited states. :, I where
I, i the geeral QFT which has a mass gap We ca reproduce these results from the quatum mechaics.
I, i the free massless scalar field theory we impose the coditio that uder the sig chagig trasformatio,, the operators O is trasformed as where O =0 or 1. i The case O i O i ' ii The case O ad O i O i ' j O j '
Coclusio We developed the computatioal method of EE based o the idea that Tr is writte as the expectatio value of the local operator at. Tr Tr E The advatages of this methods are as follows: 1 we ca use ordiary techique i QFT such as OPE ad the cluster decompositio property 2 we ca use the geeral properties of the operator to compute systematically the Reyi etropy for a arbitrary state. We could apply the operator method to perturbative calculatio i a iteractig field theory.