Universal entanglement of non-smooth surfaces

Transcription

1 Universal entanglement of non-smooth surfaces Pablo Bueno IFT, Madrid Based on P.B., R. C. Myers and W. Witczak-Krempa, PRL 115, (2015); P.B. and R. C. Myers arxiv: ; + work in progress with the gentlemen above Gravity - New perspectives from strings and higher dimensions Benasque July 16 th, 2015 Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

2 Introduction Outline 1 Introduction 2 Holographic calculations 3 QFT comparison 4 A universal ratio 5 Generalizations Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

3 Introduction Corner contribution to entanglement entropy Let us consider a 3d CFT. If V has a corner of opening angle Ω, the entanglement entropy is given by S EE = c 1 δ q(ω) log ( ) H 2πc 0, δ where q(ω) is a positive even function of Ω constrained to behave as Casini, Huerta; Myers, Singh q(ω 0) = κ/ω, q(ω π) = σ (Ω π) 2. κ and σ encode well-defined information about the CFT Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

4 Introduction Stress tensor 2-point function The stress tensor 2-point function defines a central charge for CFT s in any spacetime dimension. In the vacuum, functional form completely fixed by conformal symmetry and energy conservation Erdmenger, Osborn, Petkou where T ab (x) T cd (0) = C T x 2d I ab,cd(x), I ab,cd (x) 1 2 (I ac(x) I db (x) + I ad (x) I cb (x)) 1 d δ ab δ cd, I ab (x) δ ab 2 x a x b x 2. In 2d CFT s, standard definition of the central charge c: C T = c. In 4d CFT s, C T = 40 c/π 4 where c is the coefficient of the Weylsquared term in the trace anomaly. Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

5 Introduction Conjecture P.B., Myers, Witczak-Krempa Based on strong evidence, we conjecture that σ and C T equal up to a numerical factor for general 3d CFT s are σ C T = π2 24 Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

6 Holographic calculations Holographic entanglement entropy Ryu-Takayanagi prescription for the EE of CFT s dual to Einstein gravity Ryu, Takayanagi S EE (V ) = ext m V [ ] A(m). 4G We extremize the area functional A(m) over all possible bulk surfaces m whose boundary coincides with V. Result for a corner region: Hirata, Takayanagi S EE = L 2 ( ) H H 2G δ q E (Ω) log + O(δ 0 ), δ q E (Ω) = L 2 2G h0 Ω(h 0) = dh 0 [ dy h 0(Ω) 2 (1 + y 2 ) 2 + h 0(Ω) 2 (1 + y 2 ) 2h h h 2 (h0 2 h2 )(h0 2 + (1 + h2 0 )h2 ) ], 8πG. = σ E = L 2 Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

7 Holographic calculations Holographic entanglement entropy L 2 /G L 2 /l 2 Planck # d.o.f. For Einstein gravity this ratio appears everywhere, so we cannot distinguish among charges higher-derivative terms, which introduce new dimensionless quantities λ i. Higher-derivative terms Ryu-Takayanagi no longer valid. Replace by Hung, Myers, Smolkin; Dong; Fursaev, Patrushev, Solodukhin; Sarkar, Wall... S EE (V ) = ext m V S grav(m). Not Wald! Corner contribution for the higher-order theory I = 1 d 4 x [ 6 g 16πG L + R + ( ) 2 L2 λ 1 R 2 + λ 2 R µν R µν + λ GB X 4 +L 4 ( λ 3,0 R 3 + λ 1,1 RX 4 ) + L 6 ( λ 4,0 R 4 + λ 2,1 R 2 X 4 + λ 0,2 X 2 4 ) ]. The final expressions take the form P.B., Myers q(ω) = α q E (Ω) σ = α σ E, α = 1 24λ 1 6λ λ 3,0 + 24λ 1,1 6912λ 4,0 576λ 2,1 + O(λ 2 ). Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

8 Holographic calculations Stress tensor 2-point function In holography, the stress tensor is dual to the normalizable mode of the metric Witten; Gubser, Klebanov, Polyakov metric fluctuations in AdS 4. α [ + 2 L ] h µν λ 2L 2 [ + 2 L ] 2 h µν = 8πGT µν where again P.B., Myers α = 1 24λ 1 6λ λ 3,0 + 24λ 1,1 6912λ 4,0 576λ 2,1 + O(λ 2 ). C T = αc T,E = α 3 π 3 L 2 G. Then, for all the holographic theories considered This is not the case for other charges, e.g., σ = σ E = π2 C T C T,E 24. S disk EE = c 1 δ F σ F = ( 1 2λ GB 24λ 1, λ 2,1 + 96λ 0,2 + O(λ 2 ) ) σ E F E, s th. = c s T 2 σ c s = ( 1 16λ 0,2 + O(λ 2 ) ) σ E c s,e. Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

9 QFT comparison Comparison with QFT calculations Holographic calculations strongly coupled 3d CFT s dual to our bulk theories. Remarkably, q(ω) known for a free scalar and a free Dirac fermion with certain precision. Casini, Huerta C T C fermion T known exactly since long ago in those cases: Osborn, Petkou = 2C scalar T = 3/(16π 2 ). Notably, results for q(π/2)/c T also known for the N = 1, 2, 3, O(N) Wilson-Fisher CFT s. Kallin, Stoudenmire, Frendley, Singh, Melko Given the different nature of the theories, even a qualitative agreement would look surprising... Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

10 QFT comparison Comparison with QFT calculations P.B., Myers, Witczak-Krempa qct AdS/CFT Free scalar Free fermion Ising (N=1) XY (N=2) Heisenberg (N=3) Swingle qct qctholo Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

11 QFT comparison Comparison with QFT calculations Good agreement among all results. q(ω) and C T seem to have the same kind of emergent scaling with the number of d.o.f. In particular, q O(N) (π/2) N q Ising (π/2) and C T O(N) (π/2) NC T Ising (π/2). All curves/data points lie above the holographic curve. Does the holographic q(ω)/c T represent some kind of universal bound? A striking feature occurs as Ω π, where [q(ω)/c T ]/[q(ω)/c T ] AdS/CFT seems to tend to 1 both for the free scalar and the free fermion. This suggests that [σ/c T ]/[σ/c T ] AdS/CFT = 1. Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

12 QFT comparison A universal ratio σ computed approximately for the free fermion and the free scalar by Casini and Huerta: σ fermion , σ scalar , Using the values for C T, one finds σ fermion C fermion , T σ scalar C scalar T For all our holographic theories we had found All the results agree within %! σ = σ E = π2 C T C T,E Given the extremely different nature of the theories and the calculations, we are led to conjecture that this is a universal quantity for general 3d CFT s. Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

13 QFT comparison A universal ratio The EE of a region with an almost smooth corner would be fully determined by the two point function of the stress tensor. We can use our conjecture to predict the exact values of σ scalar and σ fermion σ scalar = 1 256, σ fermion = We are using holography to improve free field theory results! But we can do even better... Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

14 QFT comparison A universal ratio Let us revisit the original free field computations... Casini, Huerta. + + σ scalar = 2π dm db µ H a(1 a) m sech 2 (πb), 1/ [ σ fermion = 4π dm db µ H a(1 a) F ] m cosech 2 (πb), 1/2 0 4π H c 2h X1T 1 ( 2c X2T πa(a 1), h 2 a(a 1) + m 2 ) sin 2 (πa) m 2 cos(2πa) + cos (π 1 4m 2)), ( ( 2 2a 1 πa(1 a) sec π2 2a + 1 4m 2)) Γ( 32 a + 1 ) 2 1 4m 2 c ( m Γ(2 a) 2 Γ a ), 1 4m 2 [ Γ( a) π sinh ( πµ X 1 ( 2 2a+1 µ Γ(a + 1) Γ X 2 X 1 with a replaced by (1 a), T F F1 ( ( ), F 1 a 2 8πc 2 m F 2 ( ) + a 8πc 2 m 2 X 1T 8πhm 2 X 2T + ch F 2 8c h ( a 2 a + m 2) 2 (2a 1)µ ) ( 2 ψ (0)( a + iµ ) ψ (0)( a iµ ))] ) ( ), a iµ Γ a + iµ (cos(2πa) + cosh(πµ)) h(a 2 a + (h + 1)m 2 ),, µ 4m 2 1, ( ) ) ) X 1T + 8πh m X 2T ch 16πa 3 T (c 2 X 1 + hx 2 ) + 8πa 4 T (c 2 X 1 + hx 2 ch(h + 1)m 2, { i b + 1 a 2 for the scalar, i b for the fermion, Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

15 QFT comparison A universal ratio We can improve the accuracy of the numerical results: These agree with our predictions σ scalar (5), σ fermion (7), σ scalar = = , σ fermion = = , with a precision of 1 part in (the precision at which we computed the integrals). Recently, the integrals have been evaluated analitically Helvang, Hadjiantonis The results agree with our expectations. Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

16 Comments and summary of the 3d EE results q(ω)/c T is an almost universal ratio for a broad class of 3d CFTs. Is the holographic curve q(ω)/c T a universal lower bound (reminiscent of η/s = 1/(4π))? Conjecture: σ/c T is a universal quantity for general 3d CFTs: very different theories and procedures (holographic vs field theoretical methods), same result. This result allowed us to improve the free field theory results for σ. Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

17 Generalizations to come soon... Higher-dimensions (cones) Rényi entropies Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

18 Universal holographic cones P.B., Myers Sharpening spheres: cone coefficients σ (d) for general holographic theories in higher-dimensions (Using Mezei s formula) ( 1) d+2 2 d even σ (d) = C (d) π d 1 (d 1)Γ[ d 1 2 ]2 T Γ[ d 2 2 ]Γ[ d 2 ]Γ[d + 2] Cones with non-spherical cross-sections? π/4 d = 3 ( 1) d+1 2 π d = 5 Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

19 Corner Rényi entropies P.B., Myers, Witczak-Krempa Rényi entropies: S n (V ) = 1 1 n log Tr ρn V. Analogous corner coefficients σ n : S n = B n H δ q n(ω) log(h/δ) + c n q n (Ω π) = σ n (π Ω) 2. Twist operators τ n (V ): line operators (3d) extending over V, τ n n = Tr ρ n V. Its scaling dimension h n, defined by the coefficient of the leading divergence in T µν τ n n, e.g., T ab τ n n = h n 2π δ ab y 3. h n independent of the geometry of the entangling surface. Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

20 Corner Rényi entropies P.B., Myers, Witczak-Krempa Generalized conjecture: σ n = h n π(n 1) It reduces to our previous conjecture for n = 1. It works n for a free scalar and a free fermion! Some interesting consequences, e.g., Π 2 σ n 0 = 1 24 C T c s 12π 3 n 2, s thermal = c s T n Σn 1 12 Π 3 c S n 2 Σ 0 Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

21 The end? Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22

22 Bonus slide: a heuristic argument σ n universal response of S n to a small deformation of an originally smooth surface. In the case of EE, this response is determined by correlators of the stress tensor and the modular Hamiltonian H. H involves an integral of the stress tensor over V the calculation involves TT C T. Natural to expect σ C T. In the case n > 1, the calculation involves in turn T τ n h n. Natural to expect σ n h n. σn θ π Pablo Bueno (IFT, Madrid) Universality of corner entanglement in CFTs 16/07/ / 22