Gauge/Gravity Duality 2013@Max Plank Inst. , Munich July 29 th – August 2 nd ,2013 Holographic Entanglement Entropy of Excited States Tadashi Takayanagi (YITP, Kyoto Univ.) Based on arXiv:1212.1164 (PRL 110, 091602 (2013)) with Jyotirmoy Bhattacharya (Kavli IPMU), Masahiro Nozaki (YITP), Tomonori Ugajin (Kavli IPMU, YITP) arXiv:1302.5703 (JHEP 05(2013)080) with Masahiro Nozaki (YITP), Tokiro Numasawa (YITP) arXiv:1304.7100 with Masahiro Nozaki (YITP), Tokiro Numasawa (YITP), Andrea Prudenziati (YITP) + work in progress with Jyotirmoy Bhattacharya (Kavli IPMU)
① Introduction What is the entanglement entropy (EE) ? A measure how much a given quantum state is quantum mechanically entangled (or complicated). ~ The amount of `active’ degrees of freedom (or its information) Why interesting and useful ? At present, it seems still difficult to observe EE in real experiments ( → a developing subject). But, recently it is very common to calculate EE in ` numerical experiments ’ of cond-mat systems. e.g. computing central charges, detecting spin liquids
Advantages of EE • EE = A quantum order parameter (~a generalization of `Wilson loops’) Classify quantum phases. • The entanglement entropy (EE) is a helpful bridge between gravity (string) and cond-mat physics. Gravity Entanglement Cond-mat. g Area S systems AdS/CFT A (Holography) • A universal quantity which characterizes the properties of non-equilibrium states.
Quantum Many-body Systems (Cond- mat, QFTs, CFTs, …..) AdS/CFT EE, ES, (Holography) Tensor networks, etc. HEE, BH info. Quantum Quantum gravity Information String theory Theory
Information vs. Energy 1 st law of thermodynamics: T ・ dS = dE Temp. Information Energy ⇒ Can we find an analogous relation in any quantum systems which are far from the equilibrium ? Something like: Tent ・ dS A = dE A ?? Information in A Energy in A What ? = EE Can we observe EE ?? The main motivation of this talk.
Contents ① Introduction ② Basic Facts about the Entanglement Entropy (EE) ③ `The First law’ for the EE of Excited States ④ Entanglement Density and SSA ⑤ Holographic Local Quenches and EE ⑥ What is the Einstein equation for HEE ? ⑦ Conclusions
② Basic Facts about the Entanglement Entropy (EE) (2-1) Definition of Entanglement Entropy Divide a quantum system into . H H H tot A B two subsystems A and B. Example: Spin Chain A B Define the reduced density matrix for A by A taking a trace over the Hilbert space of B . Now the entanglement entropy is defined N : time slice by the von-Neumann entropy: B A A B
(2-2) Basic Properties of EE (i) If is a pure state (i.e. ) and , H H H tot A B then ⇒ EE is not extensive ! (ii) Strong Subadditivity (SSA) [Lieb-Ruskai 73] When for any , , H H H H tot A B C , S S S S A B B C A B C B C B A . S S S S A B B C A C (Actually, these two inequalities are equivalent .)
The strong subadditivity can also be regarded as the concavity of von-Neumann entropy. Indeed, if we assume A,B,C are numbers, then ( ) ( ) ( ) ( ), S A B S B C S A B C S B x y 2 ( ) ( ), S S x S y S(x) 2 2 d ( ) 0 . S x 2 dx (i.e. concave function of x) x
(2-3) Area law [Bombelli-Koul-Lee-Sorkin 86, Srednicki 93] EE in QFTs includes UV divergences. Area Law In a d+1 dim. QFT (d>1) with a UV relativistic fixed point, the leading term of EE at its ground state behaves like Area( A) ~ ( subleading terms), S A 1 d a a where is a UV cutoff (i.e. lattice spacing). [d=1: log div.] Intuitively, this property is understood like: ∂ A A Most strongly entangled
(2-4) Holographic Entanglement Entropy (HEE) [Ryu-TT 06] d 2 2 2 dt dx dz i 2 2 1 i ds R AdS AdS 2 z is the minimal area surface CFT AdS A (codim.=2) such that d 1 d 2 (We omit the time direction. ) and ~ . A A A A A homologous A Note: In time-dependent b.g., B z we need to employ the covariant version [Hubeny-Rangamani-TT 07]. (UV cut off) z
Verification of HEE • Confirmations of basic properties: Area law, Strong subadditivity (SSA), Conformal anomaly,…. • Direct Derivation of HEE from AdS/CFT: (i) Pure AdS, A = a round sphere [Casini-Huerta-Myers 11] (ii) Euclidean AdS/CFT [Lewkowycz-Maldacena 13, cf. Fursaev 06] (iii) Disjoint Subsystems [Headrick 10, Faulkner 13, Hartman 13] (iv) General time-dependent AdS/CFT → Not yet. [But, ∃ confirmations of SSA: Allais-Tonni 11, Callan-He-Headrick 12, Wall 13] • Corrections to HEE beyond the supergravity limit: [Higher derivatives: Hung-Myers-Smolkin 11, de Boer-Kulaxizi-Parnachev 11,….. ] [1/N effect: Barrella-Dong-Hartnoll-Martin 13, Faulkner-Lewkowycz-Maldacena 13] [Higher spin gravity: de Boer-Jottar 13, Ammon-Castro-Iqbal 13]
Holographic Proof of Strong Subadditivity The holographic proof of SSA inequality is very quick ! [Headrick-TT 07] A A A B S S S S = B B A B B C A B C B C C C A A A B B S S S S = B A B B C A C C C C
General Behavior of HEE [Ryu-TT 06] A Area law divergence A universal quantity F which Agrees with conformal anomaly characterizes odd dim. CFT. (central charge) ⇒ A proof of c-theorem in 3 dim. [Calabrese-Cardy 04, Solodukhin 08, (F-theorem). [Casini-Huerta 12, Liu- Hung-Myers-Smolkin 11 …] Mezei 12, Myers- Singh 12, …]
③ `The First law’ for EE of Excited States [Bhattacharya-Nozaki-Ugajin-TT 12] (3-1) Outline Since the EE in a QFT is UV divergent, we would like to focus on the difference between the values of EE. In other words, we will consider excited states and calculate: Ground State . S S S A A A This is always finite and we will compare this entropy with the energy in A: d . E dx T A tt A
(3-2) Holographic Calculation Consider an asymptotically AdS d+2 background (= an excited state in CFT d+1 ): 2 R d 2 2 2 2 ( ) ( ) , ds f z dt g z dz dx i 2 1 i z 1 1 d d ( ) 1 ..., ( ) 1 ..... f z mz g z mz 1 d dR m . T We do not care the details of IR. tt 16 G N energy density AdS bdy z IR UV ??? ∞ 0
Holographic Entanglement Entropy Analysis If we assume a small subsystem A with the size such that l 1 d 1 , ml then we can show A , T S E ent A A Pure AdS d where , . S S S E dx T A A A A tt A The `entanglement temperature’ is given by c T ent . l The constant c is universal in that it only depends on the shape of the subsystem A: 2 d . . when A a round sphere. e g c 2
Holographic Prediction Consider an exited state in a CFT which has an approximate translational and rotational invariance. If the size of the subsystem A (= ) is small enough such that l 1 2 d d / ( ), T l R G O N tt N then the following `1 st law’ like relation is satisfied: c , , T S E T ent A A ent l Info. Energy Note 1: The constant c depends only on the geometry of A. Note 2: For more general critical points with the z dynamical exponent z , we have . T c l ent
Example 1. Excited States in 2d CFT 2 2 2 l ( ) . S h h l E A A 2 3 3 l tot ( , ) conformal dim. [Agrees with Alcaraz-Berganza-Sierra 11] h h the total length of the system l tot Example 2. 3d CFT at finite temp. T 12 ( , ) d E T l ( , ) A BH , T T l 10 ent BH ( , ) d S T l 8 A BH l fixed 6 1 ( 0 ) T c l l 4 ent . T BH =3 2 T BH =2 ( ) T T l ent BH T BH =1 l 0.05 0.10 0.15 0.20 0.25 0.30
④ Entanglement Density [Nozaki-Numasawa-TT 13] We focus on the EE for a pure state in 2d CFTs for simplicity. Let us estimate the EE for the subsystem A (=an interval) by summing all of the EE between two infinitesimal regions: l l 2 1 ( , ) ( , ) ( , ) . S l l dx dy n y x dy n x y 1 2 A l l 1 2 n(y,x) A l 2 y l 1 y x n(x,y)
l l 2 1 ( , ) ( , ) . S dx dy n y x dy n x y A l l 1 2 S l l 1 2 A ( , ) ( , ) ( , ). dy n y l dy n l y dxn x l 2 2 2 l l l 2 1 2 Therefore we find 2 S A 2 ( , ). n l l 1 2 l l 1 2 We will call the entanglement density . ( , ) n l 1 l 2 Clearly this quantity should be non-negative. As we will see, this fact comes from the SSA.
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