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1 CFT OTO ) PTEP 2016 (2016) no.11, 113B06 (arxiv [hep-th]) Collaboration with P. Caputa(YITP) and A.Veliz-Osorio(Queen Mary U.)
2
3 i i +1 H = X i i z z i+1 + h X i i x i z = i x =
4 "" "i ## #i!!!i i h z i zi j h zih i zi j =0 h i z j zi e i j h i z j zi i j
5 i i +1 " " " # # " " " # # " " " # # " " " # # " " " # # " " " # #
6 !(k) k =
7
8 : C(t) =hvw(t)v W (t)i V,W :operator, W (t) e iht We iht [Larkin-Ovchinikov,64] [Kitaev,14] h[v,w(t)][v,w(t)] i = hvw(t)w (t) V i + hw (t)vv W (t) i hvw(t)v W (t) i hw (t)vw(t) V i [q(t),p(0)] {q(t),p(0)} P.B = i~e Lt = ie L (t t ) t = 1 L log ~ 1 q e Lt q
9 [cf:stanford-shenker 13]
10 hw (t)v (0)W (t)v (0)i ( C VW (t) # e L (t t ) (t apple t ) [cf:robert-stanford 14, ] W (t), V! 0 (t>t ) L = 2, t = 1 L log c C VW 1 e 2 (t t ) t = 2 log c t 0
11 C(t) =hvw(t)v W (t)i I z x h (t, 0) (0,x) (t, 0) (0,x)i h ih i t x =2, =1
12 h i h ih i =0 h i h ih i = 1 h i h ih i =1
13 V = O i (t, x), W = O j (t, x) C ij (t) = ho i (t, 0)O j (0,x)O i(t, 0)O j (0,x)i ho i (t, 0)O i(t, 0)ihO j (0,x)O j(0,x)i lim t!1 C ij (t) = S ij S 00 S 00 = S ij 1 S 00 S 0i S 0j S 00 d i d j S ij : modular S matrix d i = S 0i S 00 [Caputa- -Osorio,16] [Gu-Qi,16] S ab = ha : out b : ini
14 S ij = S 00 S 0 S 0 S 0 S S S 0 S S 1 A = p p p 1 p 1 2 A h i h ih i =0 h i h ih i = 1 = = S S 00 S 00 S 0 S 00 S 0 S S 00 S 00 S 0 S 00 S 0 h i h ih i =1 = S S 00 S 00 S 0 S 00 S 0
15 C ij (t) W S ij S 00 S 00 S 0i S 00 S 0j x V x t t
16 t!1
17 (T 2 ) N /Z N [Caputa-Kusuki-Takayanagi-Watanabe, 17] C N /S N [Belin, 17]
18
19
20 Non-integrability seems to be important in BH physics... black hole formation = thermalization in dual CFT black hole non-formation in c=1 matrix model(integrable) [Karczmarec-Maldacena-Strominger,04] black hole is thermal object in AdS/CFT, :spectrum of primary state Black hole with temp. field with mass Eigenstate represent thermal state (ETH) related to quantum chaos [Srednicki,94]
21 Furthermore, BH show initial state sensitivity: BH metric: ds 2 = l2 dudv + R 2 (1 uv) 2 d 2 (1 + uv) 2 [Stanford-Shenker,13] [Caputa-Simon-Stikonas -Takayanagi-Watanabe,15] Proper Energy at t=0 slice is Because of Blue Shift, tiny perturbation back reacts the geoemtry ds 2 = l2 dudv + R 2 [1 u(v + (u))] 2 d 2 [1 + u(v + (u))] 2 E M ert w/l 2
22 tiny perturbation change the structure of entanglement TFD : entanglement is special W (t w ) make more typical entanglemet Intuitively, Entanglement is scrambled!! (we can see this using mutual information) is related to C(t) =hvw(t)v W (t)i OTO (Out of-time-oerder) cor. (we will see in section 3)
23 l s l p R AdS l p N R AdS l s l p R AdS l s
24 [Belavin-Polyakov-Zamolodchikov, 84]
25 [Gabadiel-Gopakumar, 11] [Castro-Gopakumar-Hartman-Maloney-Volpato, 11]
26 [Heemskerk-Penedones-Polchinski-Sully,11] t R AdS l s l p
27 V L W L W R V R htfd V L W R W l V R TFDi En e 2 E n i 1 E n i 2 TFDi = X n E E 0 e 2 t TFDi
28 { i } 4 i=1 4 > 2 > 3 > W V W V hw (i 4 )W (i 3 )V (i 2 )V (i 1 )i 4! 4 it 3! 3 it V V W (t) W (t) hw (t + i 4 )W (t + i 3 )V (i 2 )V (i 1 )i
29 S 1 V V W W V V W (t) W (t)
30 ho i O j O io j i = z 12 2 i z 34 2 j G(z, z) F ii jj(a z) G(z, z) = X a F ii jj(a z) F ii jj(a z) z = z 12z 34 z 13 z 24,z ij = z i z j
31 z z O i (0) O i (0) O j O j O j O j O i (t) O i (t) x x z t SL(2,C ) ( ) O i (t) O i (0) O j (1) O j (1)
32 t j 0 i 0
33 z z O j O i (0) O i (0) O j Oj O i (t) O i (t) O j x x z t O i (0) O i (t) O j (1) O j (1)
34 t j 0 i 0
35 M 00 = S ij S 00 1 d i d j [Moore-Seiberg,89]
36 t O j j 0 i O i (t) 3D TQFT picture O j O i (t) 0 lim t!1 C ij(t) = hlink(w i,w j )i hw i ihw j i = S ij S 00 1 d i d j [Witten, 88]
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