Entanglement in Spin Chains Vladimir Korepin Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook -10 Entanglement is a resource for quantum computation. How much quantum effects we can use to control one quantum sys- tem by another. OUTLINE 1. Entanglement of two quantum subsystems A and B. Sim- plest case when the whole system fA; Bg is in a pure state: a unique wave function jΨA;Bi. Entropy of a subsystem is a measure of entanglement. 2. The wave function will be a unique ground state of a dynam- ical model: interacting spins, Bose gas or strongly correlated electrons. Universal properties of entropy of a large subsystem. -9 Easy to explain when entanglement is absent: jΨA;Bi = jΨAi ⊗ jΨBi () no entanglement But if the wave function is a sum of several such terms d jΨA;Bi = X jΨAi ⊗ jΨBi entangled. j=1 j i A B Here d > 1 and jΨj i are linear independent; as well as jΨj i. Measure of entanglement? C.H. Bennett, H.J. Bernstein, S.Popescu, and B.Schumacher 1996 entropy of a subsystem: 0 A;B A;B 1 S = −trA (ρA ln ρA) ; ρA = trB @jΨ ihΨ jA -8 Contra-intuitive features: Two spins 1=2 forming spin 1. j "i ⊗ j "i no entanglement but middle component with Sz = 0 is maximally entangled: j "i ⊗ j #i + j "i ⊗ j #i maximum entanglement Entanglement depends on Sz. Ferromagnetic XXX: L X x x y y z z Hf = − fσ σ + σj σj+1 + σ σ g j=1 j j+1 j j+1 0 1 PL z σ are Pauli matrices. [Hf;Sz] = 0, Sz = @ j=1 σj A We can fix Sz to fix ground state uniquely: 0 1 PL z Maximum of Sz = @ j=1 σj A j "i ⊗ j "i ::: ⊗ j "i no entanglement. -7 0 1 − PL − We can lower Sz by applying the operator S = @ j=1 σj A: − M jΨfmi = (S ) (j "i ⊗ j "i ::: ⊗ j "i) entangled Non-trivial magnetization Sz=L if M ! 1 proportionally to L ! 1. We shall artificially represent the ground state as by partite system: Block of spins on an space interval [1; x] is the subsystem A , the rest of the ground state is the subsystem B. Entanglement of a block of spins on a space interval [1; x] with the rest of the ground state jΨfmi. Density matrix of the block ρ(x) the entropy of the block S(x). 1 S(x) ! ln(x); x ! 1; F 2 Double scaling limit 1 << x << L. Salerno and Popkov 2004 -6 Anti - ferromagnetic XXX: X x x y y z z Haf = fσ σ + σj σj+1 + σ σ g j j j+1 j j+1 The ground state is unique and has spin 0. Also a gapless case, but for low lying excitations energy is proportional to momentum ∼ p. So one can use conformal field theory for evaluation of asymptotic of entropy of a large block of spins, belonging to the ground state: 1 S(x) ! ln(x); x ! 1; AF 3 Holzhey, Larsen and Wilczek 1994. Finite size corrections 1 X j S(x) = ln(x)=3 + x0 + aj=x j=1 -5 Zamolodchikov, Fatteev, Takhatajan and Babujian: H = X fX − X2g spin s = 1 1 n n n ~ ~ x x y y z z Xn = SnSn+1 = SnSn+1 + SnSn+1 + SnSn+1 Solvable by Bethe Ansatz. Higher spin s Faddeev 1983: H = X F (X );F (X)is a polynomial of degree 2s s n n 2s 2s 1 2s X − yj F (X) = 2 X X Y ; y = l(l+1)=2−s(s+1) j = 0 l l=0 k=l+1 k j 6= l yl − yj The entropy of a block of x spins V.K. 2003: s S (x) = ln x; as x ! 1 s + 1 -4 Other models: Bose gas with delta interaction: 2 3 Z y y y H = dx 4@ x@ x + g 5 : Here is a canonical Bose field and g > 0 is a coupling constant. @2 NX X HN = − + 2g δ(xk − xl) j=1 @xj N≥k>l≥1 Bethe Ansaz: E. Lieb, W.Liniger in 1963. The entropy of gas on a space interval [0; x] also scales as 1 S(x) ! ln(x); x ! 1 3 V.K, 2004 -3 The Hubbard model H: X y y X H = − (cj,σcj+1,σ + cj+1,σcj,σ) + u nj;"nj;# j=1 j=1 σ=";# y Here cj,σ is a canonical Fermi operator on the lattice [creates y of an electron] and nj,σ = cj,σcj,σ is an operator on number of electrons in cite number j with spin σ, u > 0. Lieb and Wu in 1968. Below half filling [less then one electron per lattice cite] both charge and spin degrees of freedom can be described by Virasoro algebra with central charge equal to 1, Korepin, Frahm 1990 : cc = 1 and cs = 1. 2 S (x) = ln x 3 V.K. 2004. At half filed band S (x) = (1=3) ln x -2 R´enyi entropy also scales logarithmically 0 −1 1 1 α B1+α C Sα = 1−α ln T r(ρA) = @ 6 A ln x Bai Qi Jin , Korepin 2004 -1 The entropy of a block of spins for a model with a gap ap- proaches a constant S1 as the size of the block increases: S(x) ! S1 as x ! 1 The Hamiltonian of XY model : 1 X x x y y z H = − (1 + γ)σ σ + (1 − γ)σ σn+1 + hσ n=−∞ n n+1 n n Here γ is anisotropy parameter: 0 < γ < 1 and 0 < h is a magnetic field and σn are Pauli matrices. Solution, phase transitions: Lieb, Schultz, Mattis, Barouch and McCoy. Toeplitz determinants and integrable Fredholm operators were used for evaluation of correlation functions, 0 The model has three different cases: s 1. Case 1a: 2 1 − γ2 < h < 2. medium magnetic field s 2. Case 1b: 0 ≤ h < 2 1 − γ2 , small magnetic field 3. Case 2: h > 2 , strong magnetic field A.Its, B.-Q. Jin, V. Korepin 2004: Case 1b 2 0 2 1 0 2 1 0 3 6 B C B C 7 1 6 B k C B k C 4I(k)I(k )7 6 B C B C 7 6 B C B C 7 S1 = 6 ln B C + B1 − C 7 + ln 2 6 4 @16k0 A @ 2 A π 5 p with k0 = 1 − k2, and v u u u 2 2 u u1 − (h=2) − γ Z π=2 dα u u s k = u I(k) = 0 t 1 − (h=2)2 1 − k2 sin2 α I(k) is the complete elliptic integral of the first kind: 1 I. Peschel 2004: Case 1a: medium magnetic field 2 0 2 1 0 2 1 0 3 6 B C B C 7 1 6 B k C B k C 4I(k)I(k )7 6 B C B C 7 6 B C B C 7 S1 = 6 ln B C + B1 − C 7 + ln 2; 6 4 @16k0 A @ 2 A π 5 and in Case 2 : strong magnetic field 2 0 3 6 7 1 6 16 4I(k)I(k )7 6 2 02 7 6 7 S1 = 6 ln + (k − k ) 7 ; (1) 12 4 (k2k02) π 5 8 > s > > 2 2 > > (h=2) + γ − 1 = γ; Phase 1a <> k = > (2) > s > > 2 2 > :> γ = (h=2) + γ − 1; Phase 2 Consider range of variation: S1 reaches minimum as ordered states and it reaches maximum at phase transitions [disordered states]. 2 Local minimum S1 = ln 2 at boundary between cases 1a and s 1b: h = 2 1 − γ2 The ground state is doubly degenerated: Y n jG1i = [cos(θ)j "ni + (−1) sin(θ)j #ni] n2lattice Y n jG2i = [cos(θ)j "ni − (−1) sin(θ)j #ni] n2lattice Each state is factorized and has no entropy. cos2(2θ) = (1 − γ)=(1 + γ) Absolute minimum of is reached at h ! 1 : S1 ! 0 as the state becomes ferromagnetic " ::: ". These are most ordered states. These is a gap. 3 Phase transitions: as the gap closes S1 ! +1 1. Critical magnetic field: h ! 2 and γ 6= 0: 1 1 2 S1 ! − ln j2 − hj + ln 4γ + O(j2 − hj ln j2 − hj) 6 3 2. An approach to XX model: γ ! 0 and 0 < h < 2: 1 1 2 1 2 S1 ! − ln γ + ln(4 − h ) + ln 2 + O(γ ln γ) 3 6 3 This agrees with conformal approach of P. Calabrese, J. Cardy and Toeplitz determinant approach of B.-Q. Jin, V.Korepin. 4 Let us consider another gapped spin chain introduced by Affleck, Kennedy, Lieb, and Tasaki: AKLT model also known as VBS. It consists of a chain of N spin-1's in the bulk, and two spin-1/2 ~ on the boundary. We shall denote by Sk the vector of spin-1 operators and by ~sb spin-1/2 operators at boundaries. 0 1 N−1 B 1 C X B 2C B ~ ~ ~ ~ C H = @SkSk+1 + (SkSk+1) A + π0;1 + πN;N+1: k=1 3 1 ~ ~ 1 ~ ~ 2 1 Notice that 2SkSk+1 + 6(SkSk+1) + 3 is a projector on a state of spin 2. The terms π describe interaction of boundary spin 1/2 and next spin 1; π is a projector on a state with spin 3/2: 2 0 1 2 0 1 ~ ~ π0;1 = @1 + ~s0S1A ; π = @1 + ~s S A : 3 N;N+1 3 N+1 N 5 Ground state is unique and there is a gap: # # # # # # v vv vv vv vv vv vv v 1"!"!"!"!"!"! a dot is spin-2; circle means symmetrisation [makes spin 1].