Mutual Information in Conformal Field Theories in Higher Dimensions

Transcription

1 Mutual Information in Conformal Field Theories in Higher Dimensions John Cardy University of Oxford Conference on Mathematical Statistical Physics Kyoto 2013 arxiv: ; J. Phys. : Math. Theor. 46 (2013)

2 Outline Quantum entanglement in general and its quantification Path integral approach rea law in higher dimensions Mutual information for a general CFT Results for a gaussian free field Universal logarithmic corrections

3 Quantum Entanglement (Bipartite, Pure State) quantum system in a pure state Ψ, density matrix ρ = Ψ Ψ H = H H B lice can make unitary transformations and measurements only in, Bob only in the complement B in general lice s measurements are entangled with those of Bob example: two spin- 1 2 degrees of freedom ψ = cos θ B + sin θ B

4 Measuring bipartite entanglement in pure states Schmidt decomposition: Ψ = j c j ψ j ψ j B with c j 0, j c2 j = 1. one quantifier of the amount of entanglement is the entropy S j c j 2 log c j 2 = S B if c 1 = 1, rest zero, S = 0 and Ψ is unentangled if all c j equal, S log min(dimh, dimh B ) maximal entanglement

5 equivalently, in terms of lice s reduced density matrix: ρ Tr B Ψ Ψ S = Tr ρ log ρ = S B the von Neumann entropy: similar information is contained in the Rényi entropies S (n) = (1 n) 1 log Tr ρ n S = lim n 1 S (n)

6 other measures of entanglement exist, but entropy has several nice properties: additivity, convexity,... it increases under Local Operations and Classical Communication (LOCC) it gives the amount of classical information required to specify ρ (important for numerical computations) it gives a basis-independent way of identifying and characterising quantum phase transitions in a relativistic theory the entanglement in the vacuum encodes all the data of the theory (spectrum, anomalous dimensions,...)

7 Entanglement entropy in a (lattice) QFT In this talk we consider the case when: the degrees of freedom are those of a local relativistic QFT in large region R in R d the whole system is in the vacuum state 0 is the set of degrees of freedom in some large (compact) subset of R, so we can decompose the Hilbert space as H = H H B in fact this makes sense only in a cut-off QFT (e.g. a lattice), and some of the results will in fact be cut-off dependent How does S depend on the size and geometry of and the universal data of the QFT?

8 Rényi entropies from the path integral (d = 1) 0 τ B _ wave functional Ψ({a}, {b}) is proportional to the conditioned path integral in imaginary time from τ = to τ = 0: Ψ({a}, {b}) = Z 1/2 1 a(0)=a,b(0)=b [da(τ)][db(τ)] e (1/ )S[{a(τ)},{b(τ)}] where S = 0 L( a(τ), b(τ) ) dτ similarly Ψ ({a}, {b}) is given by the path integral from τ = 0 to +

9 Example: n = 2 ρ (a 1, a 2 ) = db Ψ(a 1, b)ψ (a 2, b) Tr ρ 2 = da 1 da 2 db 1 db 2 Ψ(a 1, b 1 )Ψ (a 2, b 1 )Ψ(a 2, b 2 )Ψ (a 1, b 2 ) B Tr ρ 2 = Z (C (2) )/Z 2 1 where Z (C (2) ) is the euclidean path integral (partition function) on an 2-sheeted conifold C (2)

10 in general Tr ρ n = Z (C (n) )/Z n 1 where the half-spaces are connected as B to form C (n). conical singularity of opening angle 2πn at the boundary of and B on τ = 0 in 1+1 dimensions many results are known, e.g for a single interval of length l in a CFT (Holzhey et al., Calabrese-JC) S (n) (c/6)(1 + n 1 ) log(l/ɛ)

11 Higher dimensions d > 1 B the conifold C (n) is now locally {2d conifold} Rd 1, formed by sewing together n copies of {τ > 0} R d 1 to n copies of {τ < 0} R d 1 along τ = 0, so that copy j is sewn to j + 1 for r, and j to j for r B S (n) log(z (C(n) )/Z n ) Vol( ) ɛ (d 1) this is the area law in 3+1 dimensions [Srednicki 1992] coefficient is non-universal

12 Mutual Information of multiple regions B 2 1 the non-universal area terms cancel in I (n) ( 1, 2 ) = S (n) 1 + S (n) 2 S (n) 1 2 this mutual Rényi information is expected to be universal depending only on the geometry and the data of the CFT however this dependence is very difficult to compute, even in 1+1 dimensions (Calabrese-JC-Tonni)

13 Operator Expansion Method For any region X So S (n) X I (n) ( 1, 2 ) S (n) 1 +S (n) 2 S (n) ( (n) Z (C = (1 n) 1 X log ) ) Z n 1 2 = (n 1) 1 log Z (C(n) 1 2 )Z n Z (C (n) 1 )Z (C (n) 2 ) Write where Σ (n) Z (C (n) 1 2 ) Z n = Σ (n) 1 Σ (n) 2 (R d+1 ) n = Z (C(n) ) Z n n 1 C{k j } {k j } j=0 Φ kj (r (j) )

14 Z (C (n) 1 2 ) Z n = Z (C (n) 1 ) Z (C Z n = Z (C(n) 1 ) Z (C Z n (n) 2 ) Z n {k j },{k (n) 2 ) Z n {k j } C 1 j } n 1 {k j } C 2 {k j } j=0 C 1 {k j } C 2 {k j } r 2 j x k j Φ kj (r (j) 1 )Φ k j (r (j) 2 ) last equation flows from orthonormality of 2-point functions, valid in any CFT this gives an expansion of I (n) ( 1, 2 ) in increasing powers of 1/r, valid for large r first term comes from the identity operator with x kj = 0 j, but this cancels in I (n) ( 1, 2 ) leading terms come from taking either 1 or 2 of the x kj 0

15 The coefficients C {k j } B These may be computed by inserting a complete set of operators on a single conifold C (n) : Φ k (r (j ) ) j j (n) C = Φ k (r (j ) ) n 1 C j j {k j } Φ kj (r (j) ) {kj } j=0 Using orthonormality C {k j } = lim r (j) j x k j {r (j) } j j Φ kj (r (j) ) C (n) (R d+1 ) n

16 note that C {k j } R j x k j by dimensional analysis the 1- and 2-point functions on C (n) are still very hard to compute, and we have succeeded only for a free field theory

17 Free scalar field theory (gaussian free field) ction is proportional to ( φ) 2 d d+1 x, and we normalise so 2-point function in R d+1 is We need to compute φ(x)φ(x ) G 0 (x x ) = x x (d 1). lim x,x (xx ) d 1 φ j (x)φ j (x ) C (n) (j j ) lim x x 2(d 1) :φ 2 j (x): C (n) where φ j (x, 0 ) = φ j+1 (x, 0+) for x, and φ j (x, 0 ) = φ j (x, 0+) for x /. These can be though of as the potential at x on copy j due to a unit charge at x on copy j, and the self-energy of a unit charge at x.

18 The case n = 2 Define φ ± = 2 1/2 (φ 0 ± φ 1 ) φ + is continuous everywhere and so φ + (x)φ + (x ) = G 0 (x x ) φ changes sign across {τ = 0}; on the other hand, if the source x lies on τ = 0 then φ (x)φ (x ) must be symmetric under τ τ, so it vanishes on {τ = 0} x 1 x 2 φ (x)φ (x ) is the potential at x due to a unit charge at x in the presence of a conductor held at zero potential at {τ = 0}

19 s x, x φ (x)φ (x ) G 0 (x x ) C x (d 1) x (d 1) where C is the electrostatic capacitance of {τ = 0}. This gives I (2) ( 1, 2 ) C 1 C 2 2r 2(d 1) If is a sphere of radius R, the generalisation of a classic result of W. Thomson gives C = Γ(d/2)Γ(1/2) πγ((d + 1)/2) R d 1 but in general the result depends on the shape of.

20 Case when 1 and 2 are both spheres, general n If is the interior of a sphere S d, we can make a conformal mapping in R d+1 so that the boundary of becomes R 2 B B the conifold is now a 2d conical singularity R d 1 so we have cylindrical symmetry. We want the potential G (n) (ρ, θ, z) due to the unit charge at ((2R ) 1, 0, 0) for the moment suppose that n = 1/m, where m is a positive integer, so the cone has opening angle 2π/m

21 Method of images gives G (1/m) (ρ, θ, z) = Specialising to ρ = 1, z = 0, G (1/m) (1, θ, 0) = m 1 k=0 m 1 k=0 G 0 (ρ, θ + 2πk/m, z) 1 ( 2 2 cos(θ + 2πk/m) ) (d 1)/2 This is straightforward for d + 1 even, a little harder for d + 1 odd. E.g. for d = 3 G (1/m) (1, θ, 0) = m cos mθ This can now be continued back to n = 1/m > 1.

22 Self-energy ( :φ 2 0 (1): 1/n 2 ) C (n) = lim θ cos(θ/n) 1 = 1 n2 2 2 cos θ 12n 2 The leading term in the mutual entropy involves (this piece) 2 and n 1 j=1 G (n) (1, 2πj/n, 0) 2 = 1 n 1 n 4 j=1 Once again this can be done analytically. 1 ( 2 2 cos(2πj/n) ) 2

23 Final result in d = 3 I (n) n 4 ( ) 1 R1 R 2 2 ( 1, 2 ) 15n 3 (n 1) r 2 Taking the limit n 1 gives the mutual information I( 1, 2 ) 4 ( ) R1 R r 2 this can computed another way: for a gaussian state, the correlation functions determine the density matrix (Bombelli et al., Casini-Huerta) but the matrix computations must still be carried out numerically for finite r and extrapolated this was carried out by N. Shiba who found 0.26 compared with 4 = For d = 2 we find 15 I( 1, 2 ) 1 3 ( ) R1 R 2 r 2

24 Logarithmic corrections to area law B we can use the same methods to compute the stress tensor in the cylindrically symmetric geometry. e.g. in d = 3 T ρρ (1 1/n4 )a a-anomaly ρ 4 ɛ( / ɛ) log Z (C (n) ) = n T ρρ ρdρdθd 2 z ɛ 2 rea( )

25 but when we map back to the sphere B ( ) T ρρ a (1 1/n 4 1 ) ρ R 2 + ρ2 ɛ( / ɛ) log Z (C (n) ) ɛ 2 (4πR 2 ) + universal O(1) term S (n) [Casini/Huerta, Fursaev/Soludukhin,...] ɛ 2 rea( ) + #a (n 1/n 3 ) log(r /ɛ) similar result whenever d + 1 is even relation to a-theorem?

26 Summary mutual information in the ground state of relativistic field theory encodes data (scaling dimensions, OPE coefficients...) of general CFTs (= critical systems) in higher dimensions we have treated example of free field theory, difficult to go further quantitatively universal log corrections to area law in even d + 1 encode the a-anomaly