Entanglement in Quantum Field Theory

Transcription

1 Entanglement in Quantum Field Theory John Cardy University of Oxford Landau Institute, June 2008 in collaboration with P. Calabrese; O. Castro-Alvaredo and B. Doyon

2 Outline entanglement entropy as a measure of quantum entanglement path integral approach to entanglement entropy in QFT universal results in 1+1 dimensions: conformal field theory form factor approach in massive QFT higher dimensions

3 A. Quantum Entanglement (Bipartite) quantum system in a pure state Ψ, density matrix ρ = Ψ Ψ H = H A H B A B A can make observations only in A, B in the complement B in general A s measurements are entangled with those of B

4 Measuring entanglement Schmidt decomposition: Ψ = j c j ψ j A ψ j B with c j 0, j c2 j = 1. one measure of the amount of entanglement is the entropy S A 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 A, dimh B ) maximal entanglement

5 equivalently, in terms of A s reduced density matrix: ρ A Tr B Ψ Ψ entropy: S A = Tr A ρ A log ρ A

6 other measures of entanglement exist, but entropy has several nice properties: additivity, convexity,... in quantum information theory, it gives the efficiency of conversion of partially entangled maximally entangled states by local operations (Bennet et al) it gives the amount of classical information required to specify ρ A (important for density-matrix RG) it gives a basis-independent way of identifying and characterising quantum phase transitions

7 other measures of entanglement exist, but entropy has several nice properties: additivity, convexity,... in quantum information theory, it gives the efficiency of conversion of partially entangled maximally entangled states by local operations (Bennet et al) it gives the amount of classical information required to specify ρ A (important for density-matrix RG) it gives a basis-independent way of identifying and characterising quantum phase transitions

8 In this talk we consider the case when: the degrees of freedom of the quantum system are extended over some large region R in R d the hamiltonian H contains only short-range interactions, e.g: a lattice quantum spin system a UV cut-off quantum field theory A is the set of degrees of freedom in some large (compact) subset of R the whole system is in a pure state, usually the ground state 0 of H (although it will be useful to consider this as the limit β of a thermal mixed state with ρ e βh )

9 since first papers (Osborne and Nielsen 2002; Osterloh et al 2002; Vidal et al 2003), > 150 papers, mostly on exact or numerical calculations in quantum spin models [Review: (Amico et al 2007)] in this talk I will focus on the universal properties, which can be extracted from the QFT description of critical behaviour How does S A depend on the size and geometry of A and the universality class of the critical behaviour?

10 since first papers (Osborne and Nielsen 2002; Osterloh et al 2002; Vidal et al 2003), > 150 papers, mostly on exact or numerical calculations in quantum spin models [Review: (Amico et al 2007)] in this talk I will focus on the universal properties, which can be extracted from the QFT description of critical behaviour How does S A depend on the size and geometry of A and the universality class of the critical behaviour?

11 Entanglement entropy from the path integral β τ φ ρ( φ,φ ) = 1 Z 0 φ x Z =

12 Reduced density matrix ρ A = 1 Z A sew together the edges along τ = 0, β, leaving slit(s) open along A now use replica trick S A = Tr ρ A log ρ A = n Tr ρ n A n=1 for integer n 1, Trρ n A is given by n copies of the path integral for S A, sewn together cyclically along the slits: an n-sheeted Riemann surface

13 Conformal field theory approach suppose the QFT is a 1+1-dimensional CFT (i.e. with a dispersion relation ω k ) uniformising transformation for the n-sheeted surface R n ( ) w 1/n x1 z = w x 2 stress tensor T (w) Rn = c 12 {z, w} = (c/12)(1 1/n2 )(x 2 x 1 ) 2 (w x 1 ) 2 (w x 2 ) 2 log Z n x 2 x 1 = n 2π so that Z n x 2 x 1 (c/6)(n 1/n). (c/6)(n 1/n) T xx (x 0, t) dt = x 2 x 1

14 Taking the derivative with respect to n at n = 1 we get the final result (Holzhey et al 1994) S A (c/3) log x 2 x 1 in d = 1, S grows only logarithmically, even at a critical point: responsible for success of DMRG logarithmic behaviour not restricted to z = 1 critical points but also eg in random quantum spin chains (Refael and Moore)

15 many more universal results, eg finite-temperature cross-over between entanglement and thermodynamic entropy (Korepin 2004, Calabrese + JC 2004): S A (c/3) log ( (β/π) sinh(π x 2 x 1 /β) )

16 Finite correlation length in d = 1 B A l B entanglement occurs only over a distance ξ of the contact points between A and B S l 2 (c/6) log ξ with universal corrections O(e 2l/ξ ) (JC,Castro-Alvaredo,Doyon 2007)

17 Form factor approach instead of QFT on n-sheeted surface, consider n copies of the theory in R 2 with fields (Ψ 1,..., Ψ n ) and twist operators T n at the location of the branch points, so that Ψ i (y)t n (x) = T n (x)ψ i+1 (y) (mod n) (x > y) Ψ i (y)t n (x) = T n (x)ψ i (y) (x < y) the S-matrix is S ii (θ) = S(θ) S ij (θ) = 1 (i j) define form factors F i1,i 2...(θ 1, θ 2,...) 0 T n i 1, θ 1 ; i 2, θ 2 ;... in

18 Form factor equations (2 particles for simplicity) F ij (θ) = S ij (θ)f ji ( θ) F ij (θ + 2πi) = F j,i+1 ( θ) Res θ=iπ F i+1,i (θ) = T n kinematic pole all the F ij are simply obtained from F 11 : F ij (θ) = F 11 (2π(j i)i θ) (j > i) F 11 (θ) = F 11 (2πni θ) = S(θ)F 11 ( θ) these can be solved in the standard way by first finding a function F11 min (θ) which has no poles in the region 0 < Im θ < nπ and then multiplying by factors which insert the correct kinematic poles at θ = ±(2nN 1)πi

19 Two-point function in the two-particle approximation T (l)t (0) = T 2 + The sum has the form n Fij(θ 1 θ 2 ) 2 e ml(cosh θ 1+cosh θ 2 ) dθ 1 dθ 2 + i,j=1 n 1 F 11 (θ) 2 + F 11 (2πji θ) 2 j=1 which we have to analytically continue to n 1 this can be done by replacing the sum over j by a contour integral the dominant term comes from the kinematic singularities at j = 1 2 ± θ 2πi and j = n 1 2 θ 2πi, which pinch the contour as n 1 and produce a term (n 1)δ(θ) the coefficient of this is universal and independent of the details of S(θ)

20 Universal result S A (x) c 3 log ξ + U 1 8 K 0(2mx) + O(e 3mx ) U is also universal and calculable but dependent on the theory (U Ising = ) heuristic arguments (Doyon) show this result should also hold for non-integrable models

21 Higher dimensions d > 1 in general the leading term in S A is nonuniversal a 1 d Area( A): the area law expect a universal term ξ 1 d area hidden behind this however these area terms cancel in S A B + S A B S A S B A B for z = 1 quantum critical points this quantity is expected to be universal and given by c geometrical factor where c depends on the universality class