Quantum Entanglement and Local Operators Tadashi Takayanagi Yukawa - PowerPoint PPT Presentation

Strings 2014 @ Princeton, June 23-27th, 2014 Quantum Entanglement and Local Operators Tadashi Takayanagi Yukawa Institute for Theoretical Physics (YITP), Kyoto University Based on arXiv:1401.0539 [PRL 112(2014)111602] arXiv:1403.0702, arXiv:1405.5946 (see also arXiv: arXiv:1302.5703 [JHEP05(2013)080]) with Pawel Caputa (YITP), Song He (YITP), Masahiro Nozaki (YITP) Tokiro Numasawa (YITP) and Kento Watanabe (YITP).

① Introduction In QFTs , the entanglement entropy (EE) provides us a universal physical quantity (~order parameter) . For example, we can characterize the degrees of freedom of CFTs (~central charges) from the EE for ground states. (i) 2d CFT [Holzhey-Larsen-Wilczek 94, Calabrese-Cardy 04,..] (ii) 3d CFT [F-th: Jafferis-Klebanov-Pufu-Safdi 11, Entropic proof: Casini-Huerta 12] (iii) 4d CFT [Ryu-TT 06, Solodukhin 08, Sinha-Myers 10, Casini-Huerta-Myers 11 ,…]

  . H H H tot A B A B It is also helpful to look at (n-th) Renyi entanglement entropy (REE) which generalizes the EE : If we know all of , we find all eigenvalues of . (so called entanglement spectrum)

In gravity , we might expect that quantum entanglement gives a quantum bit of spacetime (~ a plank size unit) . (i) BH entropy [Bekenstein 73, Hawking 75,…] (ii) Holographic EE (HEE) (iii) Entanglement/Gravity [Ryu-TT 06, Hubeny-Rangamani- TT 07,…] [Swingle 09, Raamsdonk 09, Myers 12, … ] MERA Entangler  [Vidal 05] A = AdS A B Planck length A

The entanglement entropy is also a useful quantity to characterize excited states . Well-studied examples are quantum quenches: [Calabrese-Cardy 05, 07, …., Liu’s talk] (a) Global quantum quenches m(t)   .   /  S A c t t t* (b) Local quantum quenches   .    2 d log / S c t A Here we want to focus on more elementary excited states: (c) Local operator insertions at a time  ⇒ ? (The main aim of this talk) S A

Consider excited states defined by local operators:  ( ) 0 . O x O(x)     .    We study ( ) ( ) ( ) n n n ( ) 0 S S O x S A A A ( n ) S ~ Loss of information when we assume A that the region B is invisible. ~ ``degrees of freedom’’ of the operator O.

Two limits A B (1) In this case, we find a property analogous to   the first law of thermodynamics:    ( ) n S O E A A [Bhattacharya-Nozaki-Ugajin-TT 12, Blanco-Casini-Hung-Myers 13, Wong-Klich-Pando Zayas-Vaman 13 …, Raamsdonk’s talk] (2) This leads to a very `entropic’ quantity ! ⇒ The main purpose of this talk. [Nozaki-Numasawa-TT 14, He-Numasawa-Watanabe-TT 14, Caputa-Nozaki-TT 14]

Contents ① Introduction ② Replica Calculations of EE for locally excited states ③ Case 1: Free scalar CFTs in any dimensions ④ Case 2: Rational 2d CFTs ⑤ Case 3: Large N CFTs and AdS/CFT ⑥ Conclusions

② Replica Calculations of EE for locally excited states (2-1) Replica method for ground states A basic method to find EE in QFTs is the replica method . In the path-integral formalism, the ground state wave  function can be expressed as follows:   t t t  0     , Path integrate x   

t Then we can express [   t  A ]     0 as follows: Tr ab b A B A x    Glue each boundaries successive ly.    a b  n Tr   A a b  - sheeted n ( ) Z  n .   Riemann surface n ( ) Z n 1 cut sheets n

(2-2) Replica Method for Excited States We want to calculate for        iHt H H iHt ( , ) ( ) 0 0 ( ) t x e e O x O x e e A    ( , ) 0 0 ( , ), O x O x e l       τ ε τ ε ( ), it, it e l  where is the UV regulator for the operator.   1 d Here we consider a 1 dim. CFT on R . d         1 d i  ( , , , , ) R We set . x x x x i re 1 2 1 d

In this way, the Renyi EE can be expressed in terms of correlation functions (2n-point function etc.) on Σ n : 1         ( ) 1 1 n n n  log ( , ) ( , ) ( , ) ( , ) S O r O r O r O r    A l l e e l l e e  1 n n      log ( , ) ( , ) . n O r O r   l l e e  1 Σ n n-sheets

③ Case 1: Free scalar CFTs in any dimensions [Numasawa-Nozaki-TT 14] We focus on the free massless scalar field theory on Σ n           d 1 S d x  and calculate 2n-pt functions using the Green function:   1 / n 1 / n   1 a a      3 d [( , , ); ( , , )] , G r x s y            n 2 1 / 1 / n n 4 ( 1 / ) 2 cos ( ) / n rs a a a a n a rs  where .       2 2 2 2 1 | | a x y r s The operator is chosen as O   k . O : : k

Time evolution in free massless scalar theory      We chose with 10 x l l       ( 2 ) 1 for : : (i.e. 1 ) S A O k        and 0 .  x x 2 d O   i 2 dim. ( : :) e 4 dim. 6 dim.  ( ) n f S A Interested quantities ! t l   2 2 t     ( 2 ) . . log . E g S    ( 4 dim) A 2 2   t l Note ：  ( ) n f is `topologically invariant’ S A under deformations of A.

     ( ) n f k for in d 1 2 dim. S O A Renyi Entropy EE [For a proof: Nozaki 14] EPR state !

Heuristic Explanation First , notice that in free CFTs, there are definite (quasi) particles moving at the speed of light.       . L=A R=B L R left - moving right - moving    k      k j k j vac ( ) ( ) vac C  k j L R 0 j    k  / 2 k 2 . C j k j  k j 0 L R j   1      k ( ) n f nk n Agree with log 2 ( ) , S C   A k j 0 1 j n replica      k  f k log 2 2 log [ ]. S k C C Calculations !  A k j k j 0 j

④ Case 2: Rational 2d CFTs [He-Numasawa-Watanabe-TT 14] (4-1) Free Scalar CFT in 2d Consider following two operators in the free scalar CFT:  O     ( ) i n f (i) : : 0 . e S 1 A     i i ⇒ Direct product state 0 0 O e e L R 1 L R         ( ) n f (ii) i i : : : : log 2 . O e e S 2 A           i i i i 0 0 0 0 O e e e e L R L R 2 L R L R       ⇒ EPR state L R L R

(4-2) General Results for 2d Rational CFTs First, focus on n=2 REE and assume O = a primary op.    We can employ the following conformal map: 2 1    i . z w re It is straightforward to rewrite the n=2 REE in terms of  1  C 4-pt functions on . ( , ) ( , ) ( , ) ( , ) O w w O w w O w w O w w  1 1 2 2 3 3 4 4 2 O     4 | | ( , ). z z G z z 13 24 O          ( ) , ( ) , w i it l w i it l   1 1 z z      12 34          , . z z z z   ( ) , ( ) . w i it l w i it l 2 2 ij i j   z z     2 2 i i 13 24 , . w e w w e w 3 1 4 2

  0 We can show that the limit leads to  t  0 (i) Early time: l     2 2 ( , ) ( ( ), ( )) ( 0 , 0 ). z z O O Chiral Fusion t  l (ii) Late time: Transformation      2 2 z → 1-z ( , ) ( 1 ( ), ( )) ( 1 , 0 ). z z O O O(x) Subsystem A x  l 0 Note: It is straightforward to confirm   ( ) n 0 S at early time (i). A

In terms of conformal block, we find at late time: O O      p ( , ) ( | ) ( | ) G z z C F p z F p z O OO O O p p p O O   ( | ) ( | ) F I z F I z O O ( , ) z z  ( 1 , 0 )        O z  2 2 [ ] ( 1 ) , F O z O , I I [ ] F , O where is so called the fusion matrix, defined by p q     ( | 1 ) [ ] ( | ). F p z F O F q z , O p q O q O O O O p q O O O O

Then the n=2 REE is simply expressed at late time:    ( 2 ) f log [ ]. S F O , A I I In rational 2d CFTs, we can rewrite this in term of the quantum dimension d O S 1   , I O , d O [ ] S F O [Moore-Seiberg 89] , , I I I I   ( 2 ) f log . S d as follows: A O   ( ) n f log S d Actually, more generally we can prove A O for any n.