Aspects of Renormalized Entanglement Entropy
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1 Aspects of Renormalized Entanglement Entropy Tatsuma Nishioka (U. Tokyo) based on with Klebanov, Pufu and Safdi with Banerjee and Nakaguchi T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 1 / 52
2 What s the role of entanglement entropy in QFT? An order parameter for various phase transitions Confinement/deconfinement (like Polyakov loop) Quantum phase transition (no symmetry breaking, no classical order parameter) Reconstruction of bulk geometry from entanglement Similarity between MERA and AdS space 1st law of entanglement and linearized Einstein equation of GR T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
3 What s the role of entanglement entropy in QFT? An order parameter for various phase transitions Confinement/deconfinement (like Polyakov loop) Quantum phase transition (no symmetry breaking, no classical order parameter) Reconstruction of bulk geometry from entanglement Similarity between MERA and AdS space 1st law of entanglement and linearized Einstein equation of GR T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
4 What s the role of entanglement entropy in QFT? An order parameter for various phase transitions Confinement/deconfinement (like Polyakov loop) Quantum phase transition (no symmetry breaking, no classical order parameter) Reconstruction of bulk geometry from entanglement Similarity between MERA and AdS space 1st law of entanglement and linearized Einstein equation of GR T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
5 What s the role of entanglement entropy in QFT? An order parameter for various phase transitions Confinement/deconfinement (like Polyakov loop) Quantum phase transition (no symmetry breaking, no classical order parameter) Reconstruction of bulk geometry from entanglement Similarity between MERA and AdS space 1st law of entanglement and linearized Einstein equation of GR T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
6 What s the role of entanglement entropy in QFT? An order parameter for various phase transitions Confinement/deconfinement (like Polyakov loop) Quantum phase transition (no symmetry breaking, no classical order parameter) Reconstruction of bulk geometry from entanglement Similarity between MERA and AdS space 1st law of entanglement and linearized Einstein equation of GR Holography geometrizes the renormalization group (RG) flow T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
7 What s the role of entanglement entropy in QFT? Entanglement entropy as a measure of degrees of freedom c c UV c IR 1/Energy Construct a monotonic function c(energy) of the energy scale Entropic c-theorem in two dimensions F -theorem in three dimensions T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
8 What s the role of entanglement entropy in QFT? Entanglement entropy as a measure of degrees of freedom UV fixed point c RG flow c UV IR fixed point c IR 1/Energy Construct a monotonic function c(energy) of the energy scale Entropic c-theorem in two dimensions F -theorem in three dimensions T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
9 What s the role of entanglement entropy in QFT? Entanglement entropy as a measure of degrees of freedom UV fixed point c RG flow c UV IR fixed point c IR 1/Energy Construct a monotonic function c(energy) of the energy scale Entropic c-theorem in two dimensions F -theorem in three dimensions T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
10 What s the role of entanglement entropy in QFT? Entanglement entropy as a measure of degrees of freedom UV fixed point c RG flow c UV IR fixed point c IR 1/Energy Construct a monotonic function c(energy) of the energy scale Entropic c-theorem in two dimensions F -theorem in three dimensions T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 2 / 52
11 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 3 / 52
12 Definition of entanglement entropy Divide a system to A and B = Ā: H tot = H A H B A B Definition S A = tr A ρ A log ρ A T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 4 / 52
13 Definition of entanglement entropy Divide a system to A and B = Ā: H tot = H A H B A B Definition S A = tr A ρ A log ρ A T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 4 / 52
14 Definition of entanglement entropy Definition S A = tr A ρ A log ρ A Ψ : wave function of a ground state ρ tot = 1 Ψ Ψ Ψ Ψ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 4 / 52
15 Definition of entanglement entropy Definition S A = tr A ρ A log ρ A Ψ : wave function of a ground state Reduced density matrix: ρ tot = 1 Ψ Ψ Ψ Ψ ρ A = tr B ρ tot = i ψ i B ρ tot ψ i B H B = { ψb 1, ψ2 B, } orthonormal basis T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 4 / 52
16 Example: two spin system A B Hilbert spaces: H A = { A, A }, H B = { B, B } T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 5 / 52
17 Example: two spin system Given a ground state ( Ψ Ψ = 1): Ψ = cos θ A B + sin θ A B T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 5 / 52
18 Example: two spin system Given a ground state ( Ψ Ψ = 1): Ψ = cos θ A B + sin θ A B Reduce density matrix: ρ A = B Ψ Ψ B + B Ψ Ψ B = cos 2 θ A A + sin 2 θ A A T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 5 / 52
19 Example: two spin system Reduce density matrix: ρ A = B Ψ Ψ B + B Ψ Ψ B = cos 2 θ A A + sin 2 θ A A Matrix notation: ρ A = ( cos 2 θ 0 0 sin 2 θ ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 5 / 52
20 Example: two spin system Matrix notation: ρ A = ( cos 2 θ 0 0 sin 2 θ ) EE as a function of θ: Ψ = cos θ A B + sin θ A B S A = tr A ρ A log ρ A = cos 2 θ log(cos 2 θ) sin 2 θ log(sin 2 θ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 5 / 52
21 Example: two spin system EE as a function of θ: Ψ = cos θ A B + sin θ A B S A = tr A ρ A log ρ A = cos 2 θ log(cos 2 θ) sin 2 θ log(sin 2 θ) cos 2 θ = 1 2 : Maximally entangled, S A = log 2 cos 2 θ = 0, 1: No entanglement, S A = 0 S A T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 5 / 52
22 Quantum mechanical system Suppose Ψ = d A i=1 db j=1 c ij ψ i A ψj B, d A,B dimh A,B T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 6 / 52
23 Quantum mechanical system Suppose Ψ = d A i=1 db j=1 c ij ψ i A ψj B, d A,B dimh A,B c ij = c A i c B j : pure product state Ψ = Ψ A Ψ B, Ψ A,B i c A,B i ψ i A,B, ρ A = Ψ A Ψ A S A = 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 6 / 52
24 Quantum mechanical system Suppose Ψ = d A i=1 db j=1 c ij ψ i A ψj B, d A,B dimh A,B c ij c A i c B j : entangled state c ij = U ik λ k V kj, U, V : unitary, Ψ = min(d A,d B ) k=1 λ k ψ k A ψ k B, λ k 0, k λ 2 k = 1, S A = k λ 2 k log λ2 k T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 6 / 52
25 Quantum mechanical system Suppose Ψ = d A i=1 db j=1 c ij ψ i A ψj B, d A,B dimh A,B c ij c A i c B j : entangled state c ij = U ik λ k V kj, U, V : unitary, Ψ = min(d A,d B ) k=1 λ k ψ k A ψ k B, λ k 0, k λ 2 k = 1, S A = k λ 2 k log λ2 k = S B T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 6 / 52
26 Quantum mechanical system Suppose Ψ = d A i=1 db j=1 c ij ψ i A ψj B, d A,B dimh A,B c ij c A i c B j : entangled state c ij = U ik λ k V kj, U, V : unitary, Ψ = min(d A,d B ) k=1 λ k ψ k A ψ k B, λ k 0, k λ 2 k = 1, S A = k λ 2 k log λ2 k = S B Maximally entangled state For λ 1 = λ 2 = = 1/ min(d A, d B ), S A = log min(d A, d B ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 6 / 52
27 Properties of entanglement entropy For a pure ground state S A = S Ā T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 7 / 52
28 Properties of entanglement entropy For a pure ground state S A = S Ā Strong subadditivity S A B C + S B S A B + S B C S A + S C S A B + S B C for any three disjoint regions A, B and C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 7 / 52
29 Properties of entanglement entropy For a pure ground state S A = S Ā Strong subadditivity S A B C + S B S A B + S B C S A + S C S A B + S B C for any three disjoint regions A, B and C Mutual information I(A, B) S A + S B S A B 0 for any disjoint two regions A and B T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 7 / 52
30 Rényi entropies n-th Rényi entropy S n (A) = 1 1 n log tr Aρ n A T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 8 / 52
31 Rényi entropies n-th Rényi entropy S n (A) = 1 1 n log tr Aρ n A It reduces to the entanglement entropy in n 1 limit S A = lim n 1 S n (A) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 8 / 52
32 Rényi entropies n-th Rényi entropy S n (A) = 1 1 n log tr Aρ n A It reduces to the entanglement entropy in n 1 limit S A = lim n 1 S n (A) Inequalities n S n 0 ( ) n 1 n n S n 0 n ((n 1)S n ) 0 2 n ((n 1)S n ) 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 8 / 52
33 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 9 / 52
34 QFTs and replica trick Useful trick: S A = n log tr A ρ n A n=1 (tr A ρ A = 1) Z n : partition function on n-covering space tr A ρ n A = Z n (Z 1 ) n T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 10 / 52
35 QFTs and replica trick Useful trick: S A = n log tr A ρ n A n=1 (tr A ρ A = 1) Z n : partition function on n-covering space tr A ρ n A = Z n (Z 1 ) n T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 10 / 52
36 Path integral representation of the wave function t = t t = φ a Ψ = φ a t = 0 Ψ φ b = t = 0 φ b t = t = States φ a,b are the boundary conditions at t = 0 x T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 11 / 52
37 Replica trick and covering space [ρ A ] ab = 1 Z 1 [Dφ B (t = 0, x B)] ( φ A a φ B ) Ψ Ψ ( φ A b φb ), T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 12 / 52
38 Replica trick and covering space [ρ A ] ab = 1 Z 1 [Dφ B (t = 0, x B)] ( φ A a φ B ) Ψ Ψ ( φ A b φb ), = 1 Z 1 [Dφ B (t = 0, x B)] φ B φ A b φ A a φ B φ B φ B t = 0 t = 0 + B A B x T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 12 / 52
39 Replica trick and covering space [ρ A ] ab = 1 Z 1 φ A b φ A a t = 0 + t = 0 B A B x T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 12 / 52
40 Replica trick and covering space φ A 2 φ A 1 φ A 3 tr A ρ n A = 1 (Z 1 ) n φ A 2 n copies Z n φ A 1 φ A n = Z n (Z 1 ) n T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 12 / 52
41 Replica trick and covering space Entanglement entropy S A = ( n 1) log Z n n=1 All we need to know is the partition function Z n on the n-fold cover M n! T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 12 / 52
42 Replica trick and covering space Entanglement entropy S A = ( n 1) log Z n n=1 All we need to know is the partition function Z n on the n-fold cover M n! Comment Regarding β = 2πn as an inverse temperature S A = (β β 1) (βf ) β=2π where βf (β) = log Z n T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 12 / 52
43 Example: A half space Suppose A = {x > 0, t = 0} on M = R 2 M n : ds 2 = dr 2 + r 2 dθ 2 with r 0, θ θ + 2πn log Z n = 1 2 log det( 2 + m 2 ) Mn S A = 1 12 log(m2 ɛ 2 ) ɛ 1 : UV cutoff θ θ + 2πn A φ(x) I = 1 2 d 2 x [ ( µφ) 2 + m 2 φ 2] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 13 / 52
44 Example: A half space Suppose A = {x > 0, t = 0} on M = R 2 M n : ds 2 = dr 2 + r 2 dθ 2 with r 0, θ θ + 2πn log Z n = 1 2 log det( 2 + m 2 ) Mn S A = 1 12 log(m2 ɛ 2 ) ɛ 1 : UV cutoff θ θ + 2πn A φ(x) I = 1 2 d 2 x [ ( µφ) 2 + m 2 φ 2] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 13 / 52
45 Example: A half space Suppose A = {x > 0, t = 0} on M = R 2 M n : ds 2 = dr 2 + r 2 dθ 2 with r 0, θ θ + 2πn log Z n = 1 2 log det( 2 + m 2 ) Mn S A = 1 12 log(m2 ɛ 2 ) ɛ 1 : UV cutoff θ θ + 2πn A φ(x) I = 1 2 d 2 x [ ( µφ) 2 + m 2 φ 2] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 13 / 52
46 Example: A half space Suppose A = {x > 0, t = 0} on M = R 2 M n : ds 2 = dr 2 + r 2 dθ 2 with r 0, θ θ + 2πn log Z n = 1 2 log det( 2 + m 2 ) Mn S A = 1 12 log(m2 ɛ 2 ) ɛ 1 : UV cutoff θ θ + 2πn A φ(x) I = 1 2 d 2 x [ ( µφ) 2 + m 2 φ 2] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 13 / 52
47 UV structures of the partition function The partition function has UV divergences log Z n [g µν ] = C d 2i Λ d 2i R i i=0 M n where Λ 1 is a UV cutoff scale, R is a Ricci scalar The n-fold cover M n differs from M M 1 near the entangling surface Σ A R i n R i # M n M Σ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 14 / 52
48 UV structures of the partition function The partition function has UV divergences log Z n [g µν ] = C d 2i Λ d 2i R i i=0 M n where Λ 1 is a UV cutoff scale, R is a Ricci scalar The n-fold cover M n differs from M M 1 near the entangling surface Σ A R i n R i # M n M Σ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 14 / 52
49 Regularization of M n Σ Cone M n r = 0 Regularized cone M n r = ɛ r θ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 15 / 52
50 Regularization of M n On the regularized geometry M n [Fursaev-Patrushev-Solodukhin 13] 1 = n 1 M n M 1 R = n R + 4π(1 n) 1 + O ( (1 n) 2) M n M 1 Σ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 15 / 52
51 UV structure of entanglement entropy The entropy has UV divergences coming from the correlation near Σ UV structure of entanglement entropy S A = c d 2 Λ d 2 + c d 4 Λ d 4 +, with coefficients schematically written as c d 2i = R l K 2m, K : the extrinsic curvature l+m=i 1 It starts from the area law divergence Σ c d 2 Vol(Σ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 16 / 52
52 UV structure of entanglement entropy The entropy has UV divergences coming from the correlation near Σ UV structure of entanglement entropy S A = c d 2 Λ d 2 + c d 4 Λ d 4 +, with coefficients schematically written as c d 2i = R l K 2m, K : the extrinsic curvature l+m=i 1 It starts from the area law divergence Σ c d 2 Vol(Σ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 16 / 52
53 Summary of UV divergences QFT in d dimensions with UV cutoff length ɛ 1/Λ: In even dimensions S A = c d 2 ɛ d 2 + c d 4 ɛ d c 2 ɛ 2 + c 0 log l A ɛ + c 0 : depends on the central charges of conformal anomalies In odd dimensions S A = c d 2 ɛ d 2 + c d 4 ɛ d c 1 d 1 + ( 1) 2 F ɛ F : scheme independent constant for CFT T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 17 / 52
54 UV finiteness of the mutual information For two disjoint regions A and B the mutual information I(A, B) = S A + S B S A B The UV divergences cancel out! + = 0 Σ(A) Σ(B) Σ(A B) A B The mutual information is scheme independent T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 18 / 52
55 UV finiteness of the mutual information For two disjoint regions A and B the mutual information I(A, B) = S A + S B S A B The UV divergences cancel out! + = 0 Σ(A) Σ(B) Σ(A B) A B The mutual information is scheme independent T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 18 / 52
56 UV finiteness of the mutual information For two disjoint regions A and B the mutual information I(A, B) = S A + S B S A B The UV divergences cancel out! + = 0 Σ(A) Σ(B) Σ(A B) A B The mutual information is scheme independent T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 18 / 52
57 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 19 / 52
58 The AdS geometries Consider the flat (d + 2)-dimensional pseudo Euclidean space defined by ds 2 = dy 2 1 dy dy dy 2 d The AdS d+1 space with the radius L is defined as a submanifold satisfying y 2 1 y y y 2 d = L2 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 20 / 52
59 The AdS geometries Consider the flat (d + 2)-dimensional pseudo Euclidean space defined by ds 2 = dy 2 1 dy dy dy 2 d The AdS d+1 space with the radius L is defined as a submanifold satisfying y 2 1 y y y 2 d = L2 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 20 / 52
60 Poincaré coordinates The coordinate transformations y 1 = L2 t 2 + z 2 + d 1 i=1 x2 i 2z, y d = L2 t 2 + z 2 + d 1 i=1 x2 i 2z y 0 = Lt/z, y i = Lx i /z, (i = 1, d 1) The metric becomes ds 2 = L 2 dz2 dt 2 + d 1 i=1 dx2 i z 2 These coordinates cover half of the whole AdS d+1 space and the boundary at z = 0 is R 1,d 1 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 21 / 52
61 Poincaré coordinates The coordinate transformations y 1 = L2 t 2 + z 2 + d 1 i=1 x2 i 2z, y d = L2 t 2 + z 2 + d 1 i=1 x2 i 2z y 0 = Lt/z, y i = Lx i /z, (i = 1, d 1) The metric becomes ds 2 = L 2 dz2 dt 2 + d 1 i=1 dx2 i z 2 These coordinates cover half of the whole AdS d+1 space and the boundary at z = 0 is R 1,d 1 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 21 / 52
62 Poincaré coordinates The coordinate transformations y 1 = L2 t 2 + z 2 + d 1 i=1 x2 i 2z, y d = L2 t 2 + z 2 + d 1 i=1 x2 i 2z y 0 = Lt/z, y i = Lx i /z, (i = 1, d 1) The metric becomes ds 2 = L 2 dz2 dt 2 + d 1 i=1 dx2 i z 2 These coordinates cover half of the whole AdS d+1 space and the boundary at z = 0 is R 1,d 1 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 21 / 52
63 The AdS/CFT and GKP-W relation The AdS/CFT relates the partition functions GKP-W relation AdS d+1 space e I bulk[b=ads d+1 ] = Z CFT [ B] Consider the Einstein-Hilbert action I bulk [B] = 1 d d+1 x ( g R + 16πG N B ) d(d 1) L 2 R d z = ɛ z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 22 / 52
64 The AdS/CFT and GKP-W relation The AdS/CFT relates the partition functions GKP-W relation AdS d+1 space e I bulk[b=ads d+1 ] = Z CFT [ B] Consider the Einstein-Hilbert action I bulk [B] = 1 d d+1 x ( g R + 16πG N B ) d(d 1) L 2 R d z = ɛ z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 22 / 52
65 Holographic entanglement entropy Following the GKP-W S A = lim n 1 n (I bulk [B n ] n I bulk [B]) AdS space A z = ɛ z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 23 / 52
66 Holographic entanglement entropy Following the GKP-W S A = lim n 1 n (I bulk [B n ] n I bulk [B]) Holographic formula [Ryu-Takayanagi 06, Lewkowycz-Maldacena 13] S A = Area(γ A) 4G N A AdS space γ A z = ɛ z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 23 / 52
67 Holographic entanglement entropy Following the GKP-W S A = lim n 1 n (I bulk [B n ] n I bulk [B]) Holographic formula [Ryu-Takayanagi 06, Lewkowycz-Maldacena 13] S A = Area(γ A) 4G N Reproduce the area law divergence S A = Area( A) ɛ d 2 + ɛ : UV cutoff at z = ɛ A z = ɛ AdS space γ A z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 23 / 52
68 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
69 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] A B C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
70 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] A B C = A B C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
71 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] A A A B = B B C C C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
72 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] A A A S A B C + S B S A B + S B C B = B B C C C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
73 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] A A A S A B C + S B S A B + S B C B = B B C C C A B C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
74 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] A A A S A B C + S B S A B + S B C B = B B C C C A A B = B C C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
75 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] S A B C + S B S A B + S B C A B C = A B C A B C A B C = A B C A B C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
76 Holographic proof of strong subadditivity SSA follows from the minimality [Headrick-Takayanagi 07] A A A S A B C + S B S A B + S B C B = B B C C C S A + S C S A B + S B C A B = A B A B C C C T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 24 / 52
77 Spherical entangling surface in CFT d In the Poincaré patch, Σ = {ρ = R, t = 0} ds 2 = L 2 dz2 + dt 2 + dρ 2 + ρ 2 dω 2 d 2 z 2 The area functional for z = z(ρ) ρ Area(γ A) = L d 1 Vol(S d 2 ) The minimal surface R z(ρ) = R 2 ρ 2 0 dρ ρd ( ρz) z d 1 (ρ) 2 z = ɛ γ A z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 25 / 52
78 Spherical entangling surface in CFT d In the Poincaré patch, Σ = {ρ = R, t = 0} ds 2 = L 2 dz2 + dt 2 + dρ 2 + ρ 2 dω 2 d 2 z 2 The area functional for z = z(ρ) ρ Area(γ A) = L d 1 Vol(S d 2 ) The minimal surface R z(ρ) = R 2 ρ 2 0 dρ ρd ( ρz) z d 1 (ρ) 2 z = ɛ γ A z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 25 / 52
79 Spherical entangling surface in CFT d In the Poincaré patch, Σ = {ρ = R, t = 0} ds 2 = L 2 dz2 + dt 2 + dρ 2 + ρ 2 dω 2 d 2 z 2 ρ γ A The area functional for z = z(ρ) Area(γ A) = L d 1 Vol(S d 2 ) R 0 dρ ρd ( ρz) z d 1 (ρ) 2 R z = ɛ z z The minimal surface z(ρ) = R 2 ρ 2 0 R ρ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 25 / 52
80 Spherical entangling surface in CFT d ρ Holographic EE γ A S A = Ld 1 Vol(S d 2 ) 4G N = Ld 1 Vol(S d 2 ) 4G N 1 ɛ/r dy (1 y2 ) d 3 2 y d 1 [ ] 1 R d 2 d 2 ɛ + d 2 R z = ɛ z z 0 R ρ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 25 / 52
81 Spherical entangling surface in CFT d Holographic EE S A = Ld 1 Vol(S d 2 ) 4G N = Ld 1 Vol(S d 2 ) 4G N In odd dimensions 1 ɛ/r dy (1 y2 ) d 3 2 y d 1 [ ] 1 R d 2 d 2 ɛ + d 2 ρ z = ɛ z γ A z F = Ld 1 4G N π d 2 Γ( d 2 ) R 0 R ρ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 25 / 52
82 Spherical entangling surface in CFT d Holographic EE S A = Ld 1 Vol(S d 2 ) 4G N = Ld 1 Vol(S d 2 ) 4G N 1 ɛ/r dy (1 y2 ) d 3 2 y d 1 [ ] 1 R d 2 d 2 ɛ + d 2 ρ γ A In odd dimensions F = Ld 1 4G N π d 2 Γ( d 2 ) z = ɛ z z In even dimensions R c 0 = ( 1) d 2 +1 A = ( 1) d 2 +1 Ld 2 π d 2 1 2G N Γ( d 2 ) 0 R ρ T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 25 / 52
83 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 26 / 52
84 RG flow and c-function RG transformation: x e t x, dg i dt = βi T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 27 / 52
85 RG flow and c-function RG transformation: x e t x, dg i dt = βi c UV c Zamolodchikov s c-function c (t) = β i c g i = 12G ijβ i β j 0 c IR t T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 27 / 52
86 RG flow and c-function RG transformation: x e t x, dg i dt = βi Zamolodchikov s c-function c (t) = β i c g i = 12G ijβ i β j 0 Monotonic decrease under any RG flow Stationary at fixed points: c g i β i (g ) = 0 g=g Coincide with the central charge c at g = g c(g ) = c T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 27 / 52
87 RG flow and c-function RG transformation: x e t x, dg i dt = βi Zamolodchikov s c-function c (t) = β i c g i = 12G ijβ i β j 0 Monotonic decrease under any RG flow Stationary at fixed points: c g i β i (g ) = 0 g=g Coincide with the central charge c at g = g c(g ) = c c-function can be a measure of degrees of freedom! [Zamolodchikov 86, Cardy 88, Komargodski-Shwimmer 11] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 27 / 52
88 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 28 / 52
89 Entropic c-theorem 2d entropic c-function: Interpolate two fixed points: c(r) 3r ds A(r) dr c(r) c UV (r 0), c(r) c IR (r ) S A = c 3 log(r/ɛ) for CFT 2 SSA + Lorentz invariance monotonicity [Casini-Huerta 04] c (r) 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 29 / 52
90 Entropic c-theorem 2d entropic c-function: Interpolate two fixed points: c(r) 3r ds A(r) dr c(r) c UV (r 0), c(r) c IR (r ) S A = c 3 log(r/ɛ) for CFT 2 SSA + Lorentz invariance monotonicity [Casini-Huerta 04] c (r) 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 29 / 52
91 Entropic c-theorem 2d entropic c-function: Interpolate two fixed points: c(r) 3r ds A(r) dr c(r) c UV (r 0), c(r) c IR (r ) S A = c 3 log(r/ɛ) for CFT 2 SSA + Lorentz invariance monotonicity [Casini-Huerta 04] c (r) 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 29 / 52
92 Proof of entropic c-theorem t A A B R r A B B rr x T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 30 / 52
93 Proof of entropic c-theorem t A A B R r A B B rr x SSA S A + S B S A B + S A B T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 30 / 52
94 Proof of entropic c-theorem t A A B R r A B B rr x SSA 2S( rr) S(R) + S(r) c (r) 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 30 / 52
95 Entanglement and c-theorem in 2d Entropic c-function (not stationary at a fixed point) c(t) = c for CFT, c (t) 0 Zamolodchikov s c-function (stationary at a fixed point) c (t) = 3 2 G ijβ i β j 0, c g i = G ijβ j Thermal c-function F Therm c T 2 Every c-function coincides at a conformal fixed point T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 31 / 52
96 Entanglement and c-theorem in 2d Entropic c-function (not stationary at a fixed point) c(t) = c for CFT, c (t) 0 Zamolodchikov s c-function (stationary at a fixed point) c (t) = 3 2 G ijβ i β j 0, c g i = G ijβ j Thermal c-function F Therm c T 2 Every c-function coincides at a conformal fixed point T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 31 / 52
97 Entanglement and c-theorem in 2d Entropic c-function (not stationary at a fixed point) c(t) = c for CFT, c (t) 0 Zamolodchikov s c-function (stationary at a fixed point) c (t) = 3 2 G ijβ i β j 0, c g i = G ijβ j Thermal c-function F Therm c T 2 Every c-function coincides at a conformal fixed point T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 31 / 52
98 Entanglement and c-theorem in 2d Entropic c-function (not stationary at a fixed point) c(t) = c for CFT, c (t) 0 Zamolodchikov s c-function (stationary at a fixed point) c (t) = 3 2 G ijβ i β j 0, c g i = G ijβ j Thermal c-function F Therm c T 2 Every c-function coincides at a conformal fixed point T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 31 / 52
99 C-theorem in 3d? Thermal c-theorem: C T -theorem: [Petkou 94] F Therm c Therm T 3 C T UV C T IR, T µν (x)t ρσ (0) CFT = C T I µν,ρσ (x) x 6 F -theorem: [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10] F UV (S 3 ) F IR (S 3 ), F = log Z(S 3 ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 32 / 52
100 C-theorem in 3d? Thermal c-theorem: Counter example by [Sachdev 93] C T -theorem: [Petkou 94] F Therm c Therm T 3 C T UV C T IR, T µν (x)t ρσ (0) CFT = C T I µν,ρσ (x) x 6 F -theorem: [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10] F UV (S 3 ) F IR (S 3 ), F = log Z(S 3 ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 32 / 52
101 C-theorem in 3d? Thermal c-theorem: Counter example by [Sachdev 93] C T -theorem: [Petkou 94] F Therm c Therm T 3 C T UV C T IR, T µν (x)t ρσ (0) CFT = C T I µν,ρσ (x) x 6 F -theorem: [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10] F UV (S 3 ) F IR (S 3 ), F = log Z(S 3 ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 32 / 52
102 C-theorem in 3d? Thermal c-theorem: Counter example by [Sachdev 93] F Therm c Therm T 3 C T -theorem: [Petkou 94] Counter example by [TN-Yonekura 13] C T UV C T IR, T µν (x)t ρσ (0) CFT = C T I µν,ρσ (x) x 6 F -theorem: [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10] F UV (S 3 ) F IR (S 3 ), F = log Z(S 3 ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 32 / 52
103 C-theorem in 3d? Thermal c-theorem: Counter example by [Sachdev 93] F Therm c Therm T 3 C T -theorem: [Petkou 94] Counter example by [TN-Yonekura 13] C T UV C T IR, T µν (x)t ρσ (0) CFT = C T I µν,ρσ (x) x 6 F -theorem: [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10] F UV (S 3 ) F IR (S 3 ), F = log Z(S 3 ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 32 / 52
104 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 33 / 52
105 Conformal map t τ ρ φ φ Conformal Map θ = 0 π/2 R 3 S 3 ds 2 = dt 2 + dρ 2 + ρ 2 dφ 2 ds 2 = dθ 2 + sin 2 θdτ 2 + cos 2 θdφ 2 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 34 / 52
106 Conformal map t τ ρ φ φ Conformal Map θ = 0 π/2 R 3 S 3 ds 2 = dt 2 + dρ 2 + ρ 2 dφ 2 ds 2 = dθ 2 + sin 2 θdτ 2 + cos 2 θdφ 2 For CFT 3 Zn[R 3 ] = Z[S 3 n] S 3 n: n-fold cover of S 3 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 34 / 52
107 EE in CFT 3 and F -theorem We use a renormalized partition function in the F -theorem F (S 3 ) log Z (ren) (S 3 ) = finite T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 35 / 52
108 EE in CFT 3 and F -theorem We use a renormalized partition function in the F -theorem F (S 3 ) log Z (ren) (S 3 ) = finite For CFT 3 [Casini-Huerta-Myers 11] S A (R) = log Z[S 3 ] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 35 / 52
109 EE in CFT 3 and F -theorem We use a renormalized partition function in the F -theorem F (S 3 ) log Z (ren) (S 3 ) = finite For CFT 3 [Casini-Huerta-Myers 11] S A (R) = log Z[S 3 ] = α R ɛ F (S3 ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 35 / 52
110 EE in CFT 3 and F -theorem We use a renormalized partition function in the F -theorem F (S 3 ) log Z (ren) (S 3 ) = finite For CFT 3 [Casini-Huerta-Myers 11] S A (R) = log Z[S 3 ] = α R ɛ F (S3 ) Proof of the F -theorem by using entanglement entropy? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 35 / 52
111 Renormalized entanglement entropy Interpolating function between F UV and F IR T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 36 / 52
112 Renormalized entanglement entropy Interpolating function between F UV and F IR Monotonically decreasing under RG flow T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 36 / 52
113 Renormalized entanglement entropy Interpolating function between F UV and F IR Monotonically decreasing under RG flow Renormalized entanglement entropy [Liu-Mezei 12] F(R) (R R 1)S A (R) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 36 / 52
114 Renormalized entanglement entropy Renormalized entanglement entropy [Liu-Mezei 12] F(R) (R R 1)S A (R) For CFT 3 S A (R) = α 2πR ɛ F (S 3 ) F(R) = F (S 3 ) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 36 / 52
115 Renormalized entanglement entropy Renormalized entanglement entropy [Liu-Mezei 12] F(R) (R R 1)S A (R) For CFT 3 S A (R) = α 2πR ɛ Proof of monotonicity [Casini-Huerta 12] F (S 3 ) F(R) = F (S 3 ) SSA + Lorentz invariance F (R) = R S (R) 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 36 / 52
116 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 37 / 52
117 EE in gapped phase Large m expansion: [cf. Grover-Turner-Vishwanath 11] S A (R) = α l Σ ɛ + β m l Σ γ + l=0 c Σ 1 2l m 2l+1 l Σ : length of Σ = A γ: topological entanglement entropy [Kitaev-Preskill 05, Levin-Wen 05] c Σ 1 2l = Σ f(κ, sκ, s 2 κ, ) κ 1/m f: even for κ κ (S A = S ) Ā A T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 38 / 52
118 EE in gapped phase Large m expansion: [cf. Grover-Turner-Vishwanath 11] S A (R) = α l Σ ɛ + β m l Σ γ + l=0 c Σ 1 2l m 2l+1 l Σ : length of Σ = A γ: topological entanglement entropy [Kitaev-Preskill 05, Levin-Wen 05] c Σ 1 2l = Σ f(κ, sκ, s 2 κ, ) κ 1/m f: even for κ κ (S A = S ) Ā A T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 38 / 52
119 Dimensional reduction for free massless fields Dimensional reduction: R 2,1 S 1 R 2,1 [Huerta 11, Klebanov-TN-Pufu-Safdi 12] Entangling surface: Σ S 1 Σ 4d EE from 3d EE S (3+1) Σ S 1 = n Z S (2+1) Σ ( m = 2πn L ) A A L T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 39 / 52
120 Dimensional reduction for free massless fields Dimensional reduction: R 2,1 S 1 R 2,1 [Huerta 11, Klebanov-TN-Pufu-Safdi 12] Entangling surface: Σ S 1 Σ 4d EE from 3d EE S (3+1) Σ S 1 = n Z S (2+1) Σ ( m = 2πn L ) A A L T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 39 / 52
121 Shape dependence of EE in 3d [Klebanov-TN-Pufu-Safdi 12] Log divergence in the large L limit S (3+1) Σ 2 = S (2+1) =Σ S 1 Σ (m n ) n Z 4d EE has a logarithmic divergence S (3+1) log Σ 2 = c ( K a 2π µνk a µν 1 ) 2 (Ka µ µ ) 2 log ɛ, Σ 2 ( ) dκ 2 From 6d anomaly, c Σ 3 = # ds κ 4 + # ds Σ Σ ds T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 40 / 52
122 Shape dependence of EE in 3d [Klebanov-TN-Pufu-Safdi 12] Log divergence in the large L limit S (3+1) Σ 2 =Σ S 1 L L π 1/ɛ 0 dp S (2+1) Σ (m = p) 4d EE has a logarithmic divergence S (3+1) log Σ 2 = c ( K a 2π µνk a µν 1 ) 2 (Ka µ µ ) 2 log ɛ, Σ 2 ( ) dκ 2 From 6d anomaly, c Σ 3 = # ds κ 4 + # ds Σ Σ ds T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 40 / 52
123 Shape dependence of EE in 3d [Klebanov-TN-Pufu-Safdi 12] Log divergence in the large L limit S (3+1) Σ 2 =Σ S 1 L L π 1/ɛ 0 dp S (2+1) Σ (m = p) c Σ 1 log ɛ 4d EE has a logarithmic divergence S (3+1) log Σ 2 = c ( K a 2π µνk a µν 1 ) 2 (Ka µ µ ) 2 log ɛ, Σ 2 ( ) dκ 2 From 6d anomaly, c Σ 3 = # ds κ 4 + # ds Σ Σ ds T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 40 / 52
124 Shape dependence of EE in 3d [Klebanov-TN-Pufu-Safdi 12] Log divergence in the large L limit S (3+1) Σ 2 =Σ S 1 L L π 1/ɛ 0 dp S (2+1) Σ (m = p) c Σ 1 log ɛ 4d EE has a logarithmic divergence S (3+1) log Σ 2 = c ( K a 2π µνk a µν 1 ) 2 (Ka µ µ ) 2 log ɛ, Σ 2 ( ) dκ 2 From 6d anomaly, c Σ 3 = # ds κ 4 + # ds Σ Σ ds T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 40 / 52
125 Shape dependence of EE in 3d [Klebanov-TN-Pufu-Safdi 12] Log divergence in the large L limit S (3+1) Σ 2 =Σ S 1 L L π 1/ɛ 0 dp S (2+1) Σ (m = p) c Σ 1 log ɛ 4d EE has a logarithmic divergence S (3+1) log Σ 2 = c ( K a 2π µνk a µν 1 ) 2 (Ka µ µ ) 2 log ɛ, For free n 0 scalar fields and n 1/2 Dirac fermions c Σ 1 = (n 0 + 3n 1/2 ) ds κ 2 Σ 2 ( ) dκ 2 From 6d anomaly, c Σ 3 = # ds κ 4 + # ds Σ Σ ds T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 40 / 52 Σ
126 Shape dependence of EE in 3d [Klebanov-TN-Pufu-Safdi 12] Log divergence in the large L limit S (3+1) Σ 2 =Σ S 1 L L π 1/ɛ 0 dp S (2+1) Σ (m = p) c Σ 1 log ɛ 4d EE has a logarithmic divergence S (3+1) log Σ 2 = c ( K a 2π µνk a µν 1 ) 2 (Ka µ µ ) 2 log ɛ, For free n 0 scalar fields and n 1/2 Dirac fermions c Σ 1 = (n 0 + 3n 1/2 ) ds κ 2 Σ 2 ( ) dκ 2 From 6d anomaly, c Σ 3 = # ds κ 4 + # ds Σ Σ ds T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 40 / 52 Σ
127 REE of free massive scalar in IR region REE: Σ = a circle of radius R κ = 1 R REE is monotonically decreasing to zero in IR! What happens in small mass region? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 41 / 52
128 REE of free massive scalar in IR region REE: Σ = a circle of radius R κ = 1 R F(R) = REE is monotonically decreasing to zero in IR! What happens in small mass region? π 120mR + O ( 1/(mR) 3) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 41 / 52
129 REE of free massive scalar in IR region REE: Σ = a circle of radius R κ = 1 R F(R) = π 120mR + O ( 1/(mR) 3) REE is monotonically decreasing to zero in IR! What happens in small mass region? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 41 / 52
130 REE of free massive scalar in IR region REE: Σ = a circle of radius R κ = 1 R F(R) = π 120mR + O ( 1/(mR) 3) REE is monotonically decreasing to zero in IR! What happens in small mass region? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 41 / 52
131 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 42 / 52
132 Numerical study of REE for free massive scalar Perturbation around m = 0 doesn t work for this case (will be discussed from the holographic viewpoint) Numerical method [Huerta 11]: F(0) = F UV (S 3 ) F is not stationary at UV fixed point! [Klebanov-TN-Pufu-Safdi 12, TN 14] IR divergence? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 43 / 52
133 Numerical study of REE for free massive scalar Perturbation around m = 0 doesn t work for this case (will be discussed from the holographic viewpoint) Numerical method [Huerta 11]: F(0) = F UV (S 3 ) F is not stationary at UV fixed point! [Klebanov-TN-Pufu-Safdi 12, TN 14] IR divergence? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 43 / 52
134 Numerical study of REE for free massive scalar Perturbation around m = 0 doesn t work for this case (will be discussed from the holographic viewpoint) Numerical method [Huerta 11]: F(0) ' = FUV (S3 ) F is not stationary at UV fixed point! [Klebanov-TN-Pufu-Safdi 12, TN 14] F = FUV 0.13(mR)2 IR divergence? F T. Nishioka (Tokyo) HmRL Oct 15, Chuo, 3rd SiGT 2 43 / 52
135 Numerical study of REE for free massive scalar Perturbation around m = 0 doesn t work for this case (will be discussed from the holographic viewpoint) Numerical method [Huerta 11]: F(0) ' = FUV (S3 ) F is not stationary at UV fixed point! [Klebanov-TN-Pufu-Safdi 12, TN 14] (mr)2 F (mr)2 =0 6= 0 IR divergence? F T. Nishioka (Tokyo) HmRL Oct 15, Chuo, 3rd SiGT 2 43 / 52
136 HEE in confining geometry Cap off in IR of AdS space confining gauge theory Two minimal surfaces: disk type cylinder type A phase transition happens from the disk type to the cylinder type as the radius is increased ( a confinement/deconfinement transition) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 44 / 52
137 HEE in confining geometry Cap off in IR of AdS space confining gauge theory Confining geometry Two minimal surfaces: disk type cylinder type UV boundary R d IR boundary No world! A phase transition happens from the disk type to the cylinder type as the radius is increased ( a confinement/deconfinement transition) z = ɛ z = z 0 z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 44 / 52
138 HEE in confining geometry Cap off in IR of AdS space confining gauge theory Confining geometry Two minimal surfaces: disk type cylinder type UV boundary R d IR boundary No world! A phase transition happens from the disk type to the cylinder type as the radius is increased ( a confinement/deconfinement transition) z = ɛ z = z 0 z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 44 / 52
139 HEE in confining geometry Cap off in IR of AdS space confining gauge theory Two minimal surfaces: disk type cylinder type IR boundary A z = ɛ z = z 0 z A phase transition happens from the disk type to the cylinder type as the radius is increased ( a confinement/deconfinement transition) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 44 / 52
140 HEE in confining geometry Cap off in IR of AdS space confining gauge theory Two minimal surfaces: disk type cylinder type A phase transition happens from the disk type to the cylinder type as the radius is increased ( a confinement/deconfinement transition) IR boundary A z = ɛ z = z 0 z IR boundary A z = ɛ z = z 0 z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 44 / 52
141 HEE in confining geometry Cap off in IR of AdS space confining gauge theory Two minimal surfaces: disk type cylinder type A phase transition happens from the disk type to the cylinder type as the radius is increased ( a confinement/deconfinement transition) IR boundary A z = ɛ z = z 0 z IR boundary A z = ɛ z = z 0 z T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 44 / 52
142 Holographic example: CGLP solution Dual to a gapped (2 + 1)-dim QFT [Cvetic-Gibbons-Lu-Pope 00] Relevant deformation at UV: S = S UV + g d 3 x O(x) [O] = 7 3, [g] = 2 3 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 45 / 52
143 Holographic example: CGLP solution Dual to a gapped (2 + 1)-dim QFT [Cvetic-Gibbons-Lu-Pope 00] Relevant deformation at UV: S = S UV + g d 3 x O(x) [O] = 7 3, [g] = 2 3 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 45 / 52
144 Holographic example: CGLP solution Dual to a gapped (2 + 1)-dim QFT [Cvetic-Gibbons-Lu-Pope 00] Relevant deformation at UV: S = S UV + g d 3 x O(x) [O] = 7 3, [g] = 2 3 REE is stationary at UV fixed point [Klebanov-TN-Pufu-Safdi 12] g F(g 3/2 R) g=0 = 0 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 45 / 52
145 Holographic example: CGLP solution Dual to a gapped (2 + 1)-dim QFT [Cvetic-Gibbons-Lu-Pope 00] Relevant deformation at UV: S = S UV + g d 3 x O(x) [O] = 7 3, [g] = 2 3 REE is stationary at UV fixed point [Klebanov-TN-Pufu-Safdi 12] g F(g 3/2 R) g=0 = 0 When is REE stationary for a relevant perturbation? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 45 / 52
146 Relevant perturbation in AdS/CFT Perturbation of CFT d : S = S CFT + g d d x O Holographically described by a free massive scalar Φ of mass M in AdS d+1 ± = d 2 ± (ML AdS ) 2 + d2 4 Two boundary conditions near z 0 [Klebanov-Witten 99] Φ(z, x) z + [A( x) + ] + z [B( x) + ] = + : A = O, B = g (standard quantization) = : A = g, B = O (alternative quantization) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 46 / 52
147 Relevant perturbation in AdS/CFT Perturbation of CFT d : S = S CFT + g d d x O Holographically described by a free massive scalar Φ of mass M in AdS d+1 ± = d 2 ± (ML AdS ) 2 + d2 4 Two boundary conditions near z 0 [Klebanov-Witten 99] Φ(z, x) z + [A( x) + ] + z [B( x) + ] = + : A = O, B = g (standard quantization) = : A = g, B = O (alternative quantization) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 46 / 52
148 Relevant perturbation in AdS/CFT Perturbation of CFT d : S = S CFT + g d d x O Holographically described by a free massive scalar Φ of mass M in AdS d+1 ± = d 2 ± (ML AdS ) 2 + d2 4 Two boundary conditions near z 0 [Klebanov-Witten 99] Φ(z, x) z + [A( x) + ] + z [B( x) + ] = + : A = O, B = g (standard quantization) = : A = g, B = O (alternative quantization) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 46 / 52
149 Relevant perturbation of HEE [TN 14] The backreacted geometry by the scalar field { #z 2 +, d/2 f(z) = 1 + #z d (log z) 2 +, = d/2 HEE of a disk in the backreacted geometry (t gr d ) { ds #t dt = 2 /(d ) 1 +, d/2 #t log 2 t +, = d/2 Near the UV fixed point (t = 0) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 47 / 52
150 Relevant perturbation of HEE [TN 14] The backreacted geometry by the scalar field { #z 2 +, d/2 f(z) = 1 + #z d (log z) 2 +, = d/2 HEE of a disk in the backreacted geometry (t gr d ) { ds #t dt = 2 /(d ) 1 +, d/2 #t log 2 t +, = d/2 Near the UV fixed point (t = 0) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 47 / 52
151 Relevant perturbation of HEE [TN 14] The backreacted geometry by the scalar field { #z 2 +, d/2 f(z) = 1 + #z d (log z) 2 +, = d/2 HEE of a disk in the backreacted geometry (t gr d ) { ds #t dt = 2 /(d ) 1 +, d/2 #t log 2 t +, = d/2 Near the UV fixed point (t = 0) Classification of EE for a relevant preturbation (1) d/2 < < d : stationary (ds/dt t=0 = 0) (2) d/3 < d/2 : stationary, but the perturbation fails (3) d/2 1 < d/3 : neither stationary nor perturbative T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 47 / 52
152 Comparison to field theory results Free massive scalar in 3d: = 1 Free massive fermion in 2d (an interval): = 1 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 48 / 52
153 Comparison to field theory results Free massive scalar in 3d: = 1 non-perturbative) (3) (non-stationary, Free massive fermion in 2d (an interval): = 1 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 48 / 52
154 Comparison to field theory results Free massive scalar in 3d: = 1 non-perturbative) (3) (non-stationary, S(mR) = S(0) #(mr) 2 + Consistent with the numerical computation of the F-function! [Klebanov-TN-Pufu-Safdi 12, TN 14] Free massive fermion in 2d (an interval): = 1 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 48 / 52
155 Comparison to field theory results Free massive scalar in 3d: = 1 non-perturbative) (3) (non-stationary, S(mR) = S(0) #(mr) 2 + Consistent with the numerical computation of the F-function! [Klebanov-TN-Pufu-Safdi 12, TN 14] Free massive fermion in 2d (an interval): = 1 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 48 / 52
156 Comparison to field theory results Free massive scalar in 3d: = 1 non-perturbative) (3) (non-stationary, S(mR) = S(0) #(mr) 2 + Consistent with the numerical computation of the F-function! [Klebanov-TN-Pufu-Safdi 12, TN 14] Free massive fermion in 2d (an interval): = 1 (2) (stationary, non-perturbative) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 48 / 52
157 Comparison to field theory results Free massive scalar in 3d: = 1 non-perturbative) (3) (non-stationary, S(mR) = S(0) #(mr) 2 + Consistent with the numerical computation of the F-function! [Klebanov-TN-Pufu-Safdi 12, TN 14] Free massive fermion in 2d (an interval): = 1 (2) (stationary, non-perturbative) S(mR) = S(0) #(mr) 2 log 2 (mr) + Agree with the known results! [Casini-Fosco-Huerta 05, Herzog-TN 13] T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 48 / 52
158 1 Basics of entanglement entropy Field theoretic methods Holographic method 2 Renormalization group flow Entropic c-theorem in 2d F -theorem and EE in 3d REE in gapped phase (Non-)Stationarity of REE REE on sphere T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 49 / 52
159 Entanglement across a circle on S 2 To regularize the IR divergence, put a theory on R S 2 Take a cap-like region A = {0 θ θ 0 } EE as a function of the angle θ 0 and the radius R: S A (R, θ 0 ) θ 0 R T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 50 / 52
160 Entanglement across a circle on S 2 To regularize the IR divergence, put a theory on R S 2 Take a cap-like region A = {0 θ θ 0 } EE as a function of the angle θ 0 and the radius R: S A (R, θ 0 ) θ 0 R T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 50 / 52
161 Entanglement across a circle on S 2 To regularize the IR divergence, put a theory on R S 2 Take a cap-like region A = {0 θ θ 0 } EE as a function of the angle θ 0 and the radius R: S A (R, θ 0 ) θ 0 R T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 50 / 52
162 Entanglement across a circle on S 2 To regularize the IR divergence, put a theory on R S 2 θ 0 Take a cap-like region A = {0 θ θ 0 } EE as a function of the angle θ 0 and the radius R: S A (R, θ 0 ) R Can an REE be defined on a cylinder R S 2? T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 50 / 52
163 Renormalized entanglement entropy on cylinder REE on cylinder [Banerjee-TN-Nakaguchi 15] UV finite due to S A R sin θ 0 ɛ + F C (R, θ 0 ) (tan θ 0 θ0 1) S A (R, θ 0 ) Reduce to the REE on flat space in the limit R, θ 0 0 with Rθ 0 fixed Stationary at the UV fixed point of a free massive scalar field! T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 51 / 52
164 Renormalized entanglement entropy on cylinder REE on cylinder [Banerjee-TN-Nakaguchi 15] UV finite due to S A R sin θ 0 ɛ + F C (R, θ 0 ) (tan θ 0 θ0 1) S A (R, θ 0 ) Reduce to the REE on flat space in the limit R, θ 0 0 with Rθ 0 fixed Stationary at the UV fixed point of a free massive scalar field! T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 51 / 52
165 Renormalized entanglement entropy on cylinder REE on cylinder [Banerjee-TN-Nakaguchi 15] UV finite due to S A R sin θ 0 ɛ + F C (R, θ 0 ) (tan θ 0 θ0 1) S A (R, θ 0 ) Reduce to the REE on flat space in the limit R, θ 0 0 with Rθ 0 fixed Stationary at the UV fixed point of a free massive scalar field! T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 51 / 52
166 Renormalized entanglement entropy on cylinder REE on cylinder [Banerjee-TN-Nakaguchi 15] F C (R, θ 0 ) (tan θ 0 θ0 1) S A (R, θ 0 ) UV finite due to S A R sin θ 0 ɛ + Reduce to the REE on flat space in the limit R, θ 0 0 with Rθ 0 fixed Stationary at the UV fixed point of a free massive scalar field! F C (θ 0, mr) θ 0 π /4 (45 ) π /3 (60 ) π /2 (90 ) (mr) 2 T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 51 / 52
167 Summary Construct c-functions with EE in two and three dimensions Examine the REE near UV fixed points and find the non-stationarity possibly due to the IR divergence REE on cylinder may serve as a c-function in Zamolodchikov s sense (monotonicity and stationarity) T. Nishioka (Tokyo) Oct 15, Chuo, 3rd SiGT 52 / 52
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