Anomalies and Entanglement Entropy

Anomalies and Entanglement Entropy Tatsuma Nishioka (University of Tokyo) based on a work with A. Yarom (Technion) (to appear) T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 1 / 35

Roles of entanglement entropy in QFT An order parameter for various phase transitions Confinement/deconfinement (like Polyakov loop) [TN-Takayanagi 06, Klebanov-Kutasov-Murugan 07, ] Classification of phases [Calabrese-Cardy 04, Kitaev-Preskill 06, Levin-Wen 06, ] Reconstruction of bulk geometry from entanglement Similarity between MERA and AdS space [Swingle 09, Nozaki-Ryu-Takayanagi 12, ] 1st law of entanglement and linearized Einstein equation of GR [Bhattacharya-Nozaki-Takayanagi-Ugajin 12, Blanco-Casini-Hung-Myers 13, Nozaki-Numasawa-Prudenziati-Takayanagi 13, Lashkari-McDermott-Raamsdonk 13, Faulkner-Guica-Hartman-Myers-Raamsdonk 13, ] T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 2 / 35

Renormalization group and entanglement entropy Entanglement entropy as a measure of degrees of freedom Construct a monotonic function c(energy) of the energy scale Entropic c-theorem in two dimensions [Casini-Huerta 04] F -theorem in three dimensions [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10, Liu-Mezei 12, Casini-Huerta 12] T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 3 / 35

Renormalization group and entanglement entropy Entanglement entropy as a measure of degrees of freedom UV fixed point RG flow IR fixed point Construct a monotonic function c(energy) of the energy scale Entropic c-theorem in two dimensions [Casini-Huerta 04] F -theorem in three dimensions [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10, Liu-Mezei 12, Casini-Huerta 12] T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 3 / 35

Renormalization group and entanglement entropy 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 [Casini-Huerta 04] F -theorem in three dimensions [Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinha 10, Liu-Mezei 12, Casini-Huerta 12] T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 3 / 35

Universal terms Conformal anomalies in even d dimensions: [cf. Myers-Sinha 10] S A = c d 2 ɛ d 2 + + c 0 log ɛ +, c 0 central charges Relation to sphere partition function F ( 1) d 1 2 log Z[S d ]: [Casini-Huerta-Myers 11] S A = c d 2 d 1 + + ( 1) 2 ɛd 2 F T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 4 / 35

More universal terms? Gravitational anomaly in CFT 2 with c L c R [Wall 11, Castro-Detournay-Iqbal-Perlmutter 14] (Holography: [Guo-Miao 15, Azeyanagi-Loganayagam-Ng 15]) T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 5 / 35

More universal terms? Anomalies are robust against perturbation, which may be detected through entanglement entropy Gravitational anomaly in CFT 2 with c L c R [Wall 11, Castro-Detournay-Iqbal-Perlmutter 14] (Holography: [Guo-Miao 15, Azeyanagi-Loganayagam-Ng 15]) T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 5 / 35

Outline 1 Entanglement entropy in QFT 2 Weyl anomalies 3 Consistent gravitational anomalies 4 Other anomalies T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 6 / 35

Outline 1 Entanglement entropy in QFT 2 Weyl anomalies 3 Consistent gravitational anomalies 4 Other anomalies T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 7 / 35

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) Sep 10, 2015 @ Tohoku 8 / 35

Replica trick Rewrite the definition 1 S A = lim n 1 1 n log tr Aρ n A φ A 2 φ A 1 tr A ρ n A = 1 (Z 1 ) n φ A 3 φ A 2 n copies Z n (Z 1 ) n φ A 1 φ A n T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 9 / 35

Replica trick Entanglement entropy S A = lim n 1 W n nw 1 n 1 W n = log Z[M n ]: the Euclidean partition function M n : the n-fold cover with a surplus angle 2π(n 1) around the entangling surface Σ A T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 10 / 35

The n-fold cover M n Suppose A is a semi-infinite line in two dimensions A = {0 r <, τ = 0} τ τ + 2πn M n = r = 0 A r = T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 11 / 35

Strategy Use a partition function W reproducing flavor and (mixed-)gravitational anomalies Evaluate W on the n-fold cover M n that is an S 1 fibration over an entangling region A τ A Calculate the variation of the entanglement entropy with W T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 12 / 35

Outline 1 Entanglement entropy in QFT 2 Weyl anomalies 3 Consistent gravitational anomalies 4 Other anomalies T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 13 / 35

Weyl anomaly CFT is classically invariant under the Weyl rescaling δ σ g µν = 2σg µν because of T µ µ = 0: δ σ W = 1 d d x g σ T µ µ = 0 2 The Weyl anomalies in even d dimensions: T µ µ = 2( 1) d/2 a E i c i I i E: the Euler density, I i : Weyl invariants in d dimensions a, c i : central charges T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 14 / 35

Weyl anomaly in entanglement entropy The Weyl rescaling of entanglement entropy: δ σ S A = lim n 1 δ σ W n nδ σ W 1 n 1 The contribution localizes to the entangling surface Σ A: [Fursaev-Solodukhin 95, Fursaev-Patrushev-Solodukhin 13] Cone r = 0 Regularized cone r = δ τ r T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 15 / 35

Weyl anomaly in entanglement entropy The Weyl rescaling of entanglement entropy: δ σ S A = lim n 1 δ σ W n nδ σ W 1 n 1 The contribution localizes to the entangling surface Σ A: [Fursaev-Solodukhin 95, Fursaev-Patrushev-Solodukhin 13] δ σ W n nδ σ W 1 = 1 σ [ T µ µ Mn T µ µ M1 ] 2 M n = (n 1)σ f(a, c i, g µν ) + O ( (n 1) 2) because locally M n M 1 away from Σ Σ T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 15 / 35

Weyl anomaly in entanglement entropy In even dimension: δ σ S A = σ c 0, c 0 = Σ f(a, c i, g µν ) It gives the logarithmic divergence: [Holzhey-Larsen-Wilczek 94, Ryu-Takayanagi 06, Solodukhin 08, Myers-Sinha 10] S A = + c 0 log(l/ɛ) L: typical size of A scaling as L e σ L T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 16 / 35

Outline 1 Entanglement entropy in QFT 2 Weyl anomalies 3 Consistent gravitational anomalies 4 Other anomalies T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 17 / 35

Gravitational anomaly CFT 2 with central charges c L c R has a gravitational anomaly µ T µν = (c L c R )X ν 0 The boost (rotation) of the line A is not a symmetry r = 0 A Rotation by θ r = 0 A θ T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 18 / 35

Covariant and consistent gravitational anomalies Covariant anomaly: X ν ɛ νρ ρ R The stress tensor is covariant Consistent anomaly: ] X ν g νρ λ [ɛ αβ α Γ λ ρβ Satisfies the Wess-Zumino consistency condition T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 19 / 35

Anomaly polynomials and WZ consistency condition The variation of W is given by descent relation δw = Ω 2k 2k-form Ω 2k is fixed by 2k + 2-form anomaly polynomials P 2k+2 P 2k+2 = di 2k+1, δi 2k+1 = dω 2k I 2k+1 : Chern-Simons 2k + 1-form T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 20 / 35

Anomaly inflow mechanism Decompose W into gauge-invariant and anomalous parts W = W gauge-inv + W anom The consistent anomaly for W anom can be fixed by the anomaly inflow mechanism [Callan-Harvey 85, Jensen-Loganayagam-Yarom 12] δw anom = i δ I CS 1 M Given anomaly polynomials, one can write down (a representative of) W anom! T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 21 / 35

Anomalous partition function in 2d Anomaly polynomial in 2d P 2d = c g tr(r R) Chern-Simons form, P 2d = di CS : I CS = c g tr [Γ dγ + 23 ] Γ Γ Γ Variation of the partition function: δw n = i c g ( µ ξ ν )dγ µ ν M n (δg µν = (L ξ g) µν, Γ µ ν Γ µ νλdx λ) T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 22 / 35

Replica space for a semi-infinite line For a semi-infinite line M n : ds 2 = dr 2 + r 2 dτ 2, τ τ + 2πn Rotation by θ on M 1 = Rotation by nθ on M n τ τ + 2πn A r = 0 Rotation by θ r = 0 A nθ M n M n T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 23 / 35

Einstein anomaly For a diffeomorphism δ ξ g µν = L ξ g µν δ ξ S A = lim n 1 δ ξ W n nδ ξ W 1 n 1 where δ ξ W n = i M n c g ( µ ξ ν )dγ µ ν Rotation τ τ + θ, ξ τ = θ Gravitational anomaly for a rotation by angle θ δ θ S A = 4πi c g θ T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 24 / 35

Anomalies in 4d Anomaly polynomial: P 4d = c m F tr(r R) Chern-Simons form: I CS = A [(1 α)c m tr(r R)] + αc m F j GCS (Γ) Ambiguity in CS form proportional to α T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 25 / 35

Variation of W Suppose a product space M n = C n N C n : two-dimensional cone, N : a (compact) two-manifold The variation under the diffeomorphism: δ ξ W n = i αc m Φ ( µ ξ ν )dγ µ ν with Φ N F C n Same as the 2d gravitational anomaly with c g = αc m Φ T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 26 / 35

Mixed anomaly Take a plane entangling surface M n = τ τ + 2πn A B T 2 Magnetic field B F xy through the entangling surface Mixed anomaly for a rotation by angle θ δ θ S A = 4πi θ α c m B vol(t 2 ) T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 27 / 35

Anomalies in 6d Six-dimensional theories have two types of gravitational anomalies and a mixed gauge-gravitational anomaly: P 6d = c m F 2 tr ( R 2) + c a tr ( R 2) 2 + cb tr ( R 4) Question: Is it always possible to extract coefficients c m, c a and c b from the entanglement entropy? T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 28 / 35

Extract c m and c a Assume a product manifold M n = C n N The factorization of the anomaly polynomial: P 6d = tr ( R 2) [ c m F 2 + c a tr ( R 2)] Gravitational and mixed anomalies under the boost δ θ S A = 4πi θ ( αc m Φ 24π 2 c a τ[σ] ) ( Φ = N F 2, τ[σ] = 1 24π 2 N tr ( R 2)) T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 29 / 35

Extract c b? On a general non-product manifold ds 2 = dr 2 + r 2 ( dτ + ω 1 (y 1 )dy 2 + ω 2 (y 3 )dy 4) 2 + 4 (dy i ) 2 i=1 the coefficient c b appears in the variation of entanglement entropy But there seems to be no corresponding entangling surface for the manifold... A rotating black hole possibly plays a role? T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 30 / 35

Higher dimensions Anomaly polynomial in higher dimensions: P d = c i F k i tr ( R 2n) m i n i n=1 where for each i, 2k i + n 4nmi n = d + 2 Taking a product manifold M n = C n N N = R 2k j K3 m1n 1 (HP n ) mj n n=2 one can read off the coefficient c j T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 31 / 35

Outline 1 Entanglement entropy in QFT 2 Weyl anomalies 3 Consistent gravitational anomalies 4 Other anomalies T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 32 / 35

Gauge anomalies U(1) polygon anomaly in d = 2m dimensions P = cf m For A A + dλ, δ λ W n = ic λ F m M n The external gauge field A satisfying the Z n replica symmetry A(τ = 0) = A(τ = 2π) = = A(τ = 2πn) which yields M n F m = n M 1 F m and δ λ S A = 0 T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 33 / 35

Covariant gravitational anomalies Repeating similar calculations for covariant gravitational anomaly δ ξ W n (cov) = i g cg ( µ ξ ν )ɛ µν ν R M n One gets the variation of entanglement entropy twice as large as the consistent anomaly T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 34 / 35

Summary Systematic treatment of anomalies in entanglement entropy using the anomaly inflow mechanism Our method is valid for any QFT Can be applied as well in higher even dimensions (including mixed anomalies), but how about other anomalies such as parity anomaly in odd dimensions? T. Nishioka (Tokyo) Sep 10, 2015 @ Tohoku 35 / 35