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Entanglement in Spin Chains Vladimir Korepin Yang Institute for Theoretical Physics, State University of New York at Stony Brook

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Entanglement in Spin Chains Vladimir Korepin Yang Institute for Theoretical Physics, State University of New York at Stony Brook 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].
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