Entanglement and Quantum Spin Chains

ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems

ENTANGLEMENT AND QUANTUM SPIN CHAINS

V. Korepin

YITP of Stony Brook University

New exactly solvable spin chain came out of quantum information

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Outline

1 ABitofHistoryofQuantumEntanglement

2 Quantum Information and Dynamical Systems

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems The dawn

Aug 14, 1935 Shroedinger submitted his paper to Proceedings of Cambridge Philosophical Society. 1964 John Bell: entanglement distinguishes quantum mechanics from classical at ﬁxed value of Plank constant. 1981 Richard Feynman: quantum computation. 1996 Bennett, Bernstein, Popescu and Schumacher introduced entropy of subsystem as a measure of entanglement discovery of entanglement entropy

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Application to physics: area Law

1986 L. Bombelli, R. Koul, J. Lee, R. Sorkin; 1993 M. Srednicki 1994 Holzhey, Larsen and Wilczek: ent entr of block of spins of c size x for 1D gapless models scales as S = 3 log x . 2003 Logarithmic scaling of Renyi entropy. B.-Q.Jin, Korepin, Journal Statistical Physics; arXiv.org April 15 of 2003. ln tr(ρα) −1 S (α)= 1+α ln x R 1−α → 6 c v πTx 2004 For ﬁnite temperature S = 3 log πT sinh v Korepin, PRL, vol 92, ei 096402, 05 March 2004,! ! "" arXiv:cond-mat/0311056

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Environment and Error Correcting Codes

Q-bits should be entangled with other qubits in quantum computer, but not entangled with the environment. Error correcting codes discovered by Peter Shor 1995. Gottesman developed Stabilizer Codes 1997.

Eˆn Ψ = Ψ | ⟩ | ⟩

Set of Eˆn are error operators: bit ﬂip, phase ﬂip... Linear combination of all Ψ is a error-free subspace. Sometimes Eˆn | ⟩ can be expressed in terms of local spins and Eˆn 1. For | | ≤ H = Eˆn error-free subspace is the ground state. The − n most famous! example is KITAEV MODEL. Recently Peter Shor designed another solvable model of statistical mechanics starting from error correcting codes.

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems New spin chain

New spin chain come out of error correcting codes. Ramis Movassagh and Peter Shor: arXiv:1408.1657 The Hamiltonian is translation invariant. It describes interaction of nearest neighbors in 1 dimension:

H = Hj,j+1 "j

Can be consturcted for any integer spins.. The ground state is unique and know explicitly. It is described in terms of enumerative combinatorics [Motzkin numers] . In the inﬁnite volume L the model is gapless. The energy gap decays as L−∆ and ∆ 2. ≥

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Combinatoric and Spin Chains

Spin 2. In r.h.s I wrote the value of Sz l1 = 2 , l2 = 1 , r 1 = 1 , r 2 = 2 .Statesontwo |lattice⟩ | sites:⟩ | ⟩ | ⟩ | ⟩ |− ⟩ | ⟩ |− ⟩ 1 Rk = 0, r k r k , 0 (1) | ⟩ √2 #| ⟩−| ⟩$ 1 Lk = 0, lk lk , 0 (2) | ⟩ √2 #| ⟩−| ⟩$ 1 ϕk = 0, 0 lk , r k (3) | ⟩ √2 #| ⟩−| ⟩$ k = 1, 2 (4)

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems The Hamiltonian

2n−1 cr H = ⎛ Πj,j+1 + Πj,j+1⎞ + Πboundary "j=1 ⎝ ⎠ 2 k k k k k k Π , = R R + L L + ϕ ϕ j j+1 | ⟩j ⟨ |j+1 | ⟩j ⟨ |j+1 | ⟩j ⟨ |j+1 "k=1 Πcr = lk r i lk r i j,j+1 | ⟩j ⟨ |j+1 "k≠ i

2 Π = r k r k + lk lk boundary | ⟩1⟨ |1 | ⟩2n⟨ |2n "k=1

V. Ko r e p in F-H Formula 2

[ ] ( 0 [ ( 0 ) ] )

Figure 2: A Motzkin walk with s =2colors of length 2n = 10.

Local projectors as interactions have the advantage of being robust against certain perturbations. This is important from a practical point of view and experimental realizations. The local Hamiltonian, with projectors as interactions, that has the Motzkin state as its unique zero energy ground state is

2n 1 2n 1 cross H = ⇧j,j+1 + ⇧j,j+1, (1) Xj=1 Xj=1 where ⇧j,j+1 implements the local changes discussed above and is deﬁned by

s ⇧ Rk Rk + Lk Lk + 'k 'k j,j+1 ⌘ ij,j+1h ij,j+1h ij,j+1h Xk=1 h i with Rk = 1 0rk rk0 , Lk = 1 0`k `k0 and 'k = 1 00 `krk . The i p2 | i| i i p2 | i i i p2 i i s k k k k projectors ⇧boundary⇥ k=1 ⇤r 1 r + ` 2⇥n ` select⇤ out the Motzkin state⇥ by excluding⇤ all ⌘ | i h | | i h | cross k i k i walks that start and end at non-zero heights. Lastly, ⇧j,j+1 = k=i ` r j,j+1 ` r ensures that P ⇥ ⇤ 6 | i h | balancing is well ordered (i.e., prohibits 00 `kri when k = i); these projectors are required only $ 6 P when s>1 and do not appear in the s =1model (PRL 109, 207202 (2012)).

Lastly for the Hamiltonian to : 1. Be local (nearest neighbors), which H deﬁned above is, 2. Translationally invariant, which H is, and 3. Have a unique ground state, which H has not, that violates the area law by log (n) for s =1 and pn for s =2, 3,... , we add an external ﬁeld, where the model is described by the new Hamiltonian

H˜ H + ✏ F ⌘ 2n s F [ r i r + ` i ` ] ⌘ ( | i h | | i h | ) Xi=1 Xk=1 where, H is as before and 0 < ✏ 1. H˜ has all the features above and has a unique ground ⌧ state. The gap = ⇥ (n c), where provably c is a constant c 2, which cannot have CFT as its continuum limit. Notes on the spin-operator representation of the model

1, 2, Ramis Movassagh ⇤

1Department of Mathematics, Northeastern University, Boston MA 02115 2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139 (Dated: May 19, 2015)

We will write the spin interactions in the spin-representation for the model that gives a square root violation of the area law (joint work with Peter W. Shor arXiv:1408.1657 [quant-ph]).

A. Review of the local Hamiltonian [arXiv:1408.1657 [quant-ph]]

Let us consider an integer spin s chain of length 2n. It is convenient to label the d =2s +1 spin states by 0, `1, `2, , `s,r1,r2, ,rs where ` means a left parenthesis (or a step up) and ··· ··· r a right parenthesis (or a step down) as shown in Fig. 1. We distinguish each type of steps (or parenthesis) by associating a color from the s colors shown as superscripts on ` and r. A Motzkin walk on 2n steps is any walk from (x, y)=(0, 0) to (x, y)=(2n, 0) with steps (1, 0), (1, 1) and (1, 1) that never passes below the x-axis, i.e., y 0. An example of such a walk is shown in Fig. 2. The height in the middle is 0 m n which results from m steps up with the balancing   steps down on the second half of the chain. In our model the unique ground state is the s colored Motzkin state which is deﬁned to be the uniform superposition of all s colorings of Motzkin walks on 2n steps. The nonzero heights in the middle is the source of the mutual information between the two halves and the large entanglement entropy of the half-chain (Fig. 2). Consider the following local operations to any Motzkin walk: interchanging zero with a non-ﬂat step (i.e., 0rk rk0 or 0`k `k0) or interchanging a consecutive pair of zeros with a peak of a $ $ given color (i.e., 00 `krk). Any s colored Motzkin walk can be obtained from another one by a $ sequence of these local changes. To construct a local Hamiltonian with projectors as interactions that has the uniform superposition of the Motzkin walks as its zero energy ground state, each of the local terms of the Hamiltonian has to annihilate states that are symmetric under these interchanges.

s = 2 ℓ1 ( ℓ2 [ 0 0 d = 5 r1 ) r2 ]

Figure 1: Labeling the states for s =2. 3

B. Hamiltonian in Spin-Operator language

Recall that in the spin operator representation, taking ~ =1:

S2 s, m = s (s + 1) s, m | i | i Sz s, m = m s, m | i | i S± s, m = s (s + 1) m (m 1) s, m 1 | i ± | ± i p where S = sx isy. ± ± For a general spin s model, let k =1, 2,...s.Wehave

1 Lk Lk = ( 0,k k, 0 )( 0,k k, 0 ) | ih | 2 { | i| i h | h | } 1 = ( 0 0 k k k 0 0 k 0 k k 0 + k k 0 0 ) , 2 { | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | } 1 Rk Rk = ( 0, k k, 0 )( 0, k k, 0 ) | ih | 2 { | i| i h | h | } 1 = ( 0 0 k k k 0 0 k 0 k k 0 + k k 0 0 ) 2 { | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | } 1 'k 'k = ( 0, 0 k, k )( 0, 0 k, k ) | ih | 2 { | i| i h | h | } 1 = ( 0 0 0 0 k 0 k 0 0 k 0 k + k k k k ) . 2 { | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | } Moreover the ’cross terms’, such as k, j k, j where k = j, can simply be wrtten as k k j j . | ih | 6 | ih | ⌦ | ih | Let us ﬁrst work out s =1, 2 before giving the general form applicable to any s. Label the states by m where m = 1, 0, +1 and we drop the dependence on s and denote m, s simply by m . | i | i | i We identify ` = +1 , r = 1 and 0 = 0 , whereby | i | i | i | i | i | i sz 0 =0 | i sz +1 =++1 | i | i sz 1 = 1 | i | i Our local terms on any pair of nearest neighbor spins become

1 L L = ( 0, 1 1, 0 )( 0, 1 1, 0 ) | ih | 2 { | i| i h | h | } 1 = ( 0 0 1 1 1 0 0 1 0 1 1 0 + 1 1 0 0 ) , 2 { | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | } 1 R R = ( 0, 1 1, 0 )( 0, 1 1, 0 ) | ih | 2 { | i| i h | h | } 1 = ( 0 0 1 1 1 0 0 1 0 1 1 0 + 1 1 0 0 ) 2 { | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | } 1 ' ' = ( 00 1, 1 )( 00 1, 1 ) | ih | 2 { | i| i h | h | } 1 = ( 0 0 0 0 1 0 1 0 0 1 0 1 + 1 1 1 1 ) . 2 { | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | | ih | ⌦ | ih | } 4

The two qudit projectors will be span of vectors labeled by m, n . Let us compute | i 0 0 =1(sz)2 1 1 = 1 (1 sz) sz | ih | | ih | 2 1 1 = 1 (1 + sz) sz 0 1 = 1 S+ (1 sz) sz | ih | 2 | ih | 2p2 0 1 = 1 S (1 + sz) sz 1 0 = 1 sz (1 sz) S | ih | 2p2 | ih | 2p2 1 0 = 1 sz (1 + sz) S+ | ih | 2p2 These expressions can be plugged into the right hand side of the equations that deﬁne L L , | ih | R R and ' '| above. | ih | | ih Now suppose s =2and label the states by m with m = 2, 1, 0, +1, +2. We identify | i `2 = +2 , `1 = +1 , 0 = 0 , r1 = 1 and r2 = 2 , whereby | i | i | i | i | i | i | i | i | i | i sz 0 =0 | i sz +1 =++1 sz +2 =+2 +2 | i | i | i | i sz 1 = 1 sz 2 = 2 2 | i | i | i | i The two qudit projectors will be the span of vectors labeled by m, n , which can be written as | i tensor products of the following projectors. For example when s =2,weﬁndthat

0 0 = 1 1 (sz)2 4 (sz)2 s2(s 1)2 | ih | 1 1 = 1 h4 (sz)2 ih(1 + sz) sz i | ih | s(s+1) 1 z 2 z z 1 1 = h4 (s ) i (1 s ) s | ih | s(s+1) +2 +2 = 1 h (sz)2 i1 (2 + sz) sz | ih | (s+2)(s+1)s 1 z 2 z z 2 2 = h(s ) 1i (2 s ) s | ih | (s+2)(s+1)s h i and the ’oﬀ-diagonal’ terms are

1 z z 2 z 0 1 = S (1 + s ) 4 (s ) s | ih | [s(s+1)]3/2 1 0 = 1 sz (1 + sz) h4 (sz)2 iS+ | ih | [s(s+1)]3/2 1 z z z 2 1 0 = s (1 s ) h4 (s ) i S | ih | [s(s+1)]3/2 1 + z z 2 z 0 1 = S (1 s )h 4 (s )i s | ih | [s(s+1)]3/2 1 2 z z 2 z 0 2 = (S) (s +h 2) (s ) i 1 s | ih | s(s+2)[s(s+1)]3/2 2 0 = 1 sz (sz + 2) (szh)2 1 (S+i )2 | ih | s(s+2)[s(s+1)]3/2 0 2 = 1 (S+)2 (sz h 2) (sz)2i 1 sz | ih | s(s+2)[s(s+1)]3/2 1 z z z 2 2 2 0 = s (s 2) (s h) 1 (Si ) | ih | s(s+2)[s(s+1)]3/2 h i

Suppose now that the spin, s, is an (positive) integer. This general form of the diagonal terms is, below N and N 0 are just normalizations: 5

s 1 2 z 2 0 0 = s 2 k (s ) | ih | k=1 k kY=1 h i Q

m S sgn(m) | | (sz + m) sz k2 (sz)2 k= m 0 m =0 = { | | } N h i | ih 6 | 0 Q c m | | 2 2 2 N 0 =2m s (s + 1) k (k 1) k m . 8 9 { }

⇤ Electronic address: [email protected] ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Magnetic Field

2n−1 cr H = ⎛ Πj,j+1 + Πj,j+1⎞ + F "j=1 ⎝ ⎠ 2n 2 F = h r k r k + lk lk | ⟩j ⟨ |j | ⟩j ⟨ |j "j=1 "k=1 # $ 0 < h << 1 Same ground state.

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Entropy of a block

Shor and Movassagh maped the model to random walk. Entropy of half of the spin chain for spin 2.

1 1 1 S = coeff√n + log n + γ + (log 2 + log π + log σ) 2 − 2 2

2σ coeff = 2log(2) ) π

√2 σ = 2√2 + 1

V. Ko r e p in F-H Formula Measures of Entanglement Two blocks in VBS Afﬂeck, Lieb, Kennedy, Tasaki

Spin chain consists of N spin-1’s in the bulk and two spin-1/2 ~ on the boundary: Sk is spin-1 and ~sb is spin-1/2.

N 1 1 2 = S~ S~ + (S~ S~ )2 + + ⇡ + ⇡ . H k k+1 3 k k+1 3 0,1 N,N+1 kX=1 ✓ ◆ 1 S~ S~ + 1 (S~ S~ )2 + 1 is a projector on a state of spin 2. b 2 k k+1 6 k k+1 3 c The ⇡ is a projector on a state with spin 3/2:

2 2 ⇡ = 1 + ~s S~ , ⇡ = 1 + ~s S~ . 0,1 3 0 1 N,N+1 3 N+1 N ⇣ ⌘ ⇣ ⌘

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Ground state is VBS

Ground state is unique and there is a gap:

◆⇣ ◆⇣ ◆⇣ ◆⇣ ◆⇣ ◆⇣ ssssssssssssss a dot is✓⌘ spin-1/✓⌘2; circle means✓⌘ symmetrisation✓⌘ ✓⌘ [makes✓⌘ spin 1]. A line is a anti-symmetrisation . Each projector | "#i | #"i annihilates the ground state (no frustration). The ground state is called VBS. Correlation function at distance x is: 3 1 x < S~ S~ >= = p(x) 4 x 1 3 ✓ ◆

V. E. Korepin Negativity in VBS ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Spectrum of density matrix of AKLT

Fan, Korepin and Roychowdhury 2004 PRL: the density matrix of ﬁnite block of spins on the ﬁnite latticehas only 4non-zeroeigenvalues. One

1 + 3p(x) p = 1 4 and three

1 p(x) p = − 3 4

x 1 = p(x) − 3 * +

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Entropy of a Subsystem

Fan, Korepin and Roychowdhury 2004 PRL calculated exactly the entropy of the block of the size x on a ﬁnite lattice:

3(1−p(x)) 1−p(x) S(x)= 4 log 4 − # $ − 1+3p(x) 1+3p(x) 4 log 4 − # $ It does not depend on the size of the lattice. S(x) quickly changes from ln3 to ln4 as x goes from 1 to . ∞

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Partial transposion

We shall consider a block of x sequential spins in the ground state of AKLT as subsystem A and the rest of the ground state as subsystem B. The density matrix of the ground state ρAB is a projector to the unique wave function of the ground state. Let us make transposition only in the Hilbert space of the block to obtain: TA ρAB

TA What is the spectrum? Let us denote eigenvalues of ρAB by λ.

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Spectrum

TA Ten non-negative eigenvalues of ρAB

1 1 x p = 1 , 6-fold degeneracy, 3 4 , − ,−3- - 1 1 x p = 1 + 3 , no deg. 1 4 , ,−3- -

p4 = √p1p3, 3-fold deg.

Six negative eigenvalues

p = p , 3-fold deg. 2 − 3 p5 = p , 3-fold deg. − 4

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Negativity

N = 3p3 + 3p4 Changes from 1 at x = 1to3/2atx = . Sometimes people consider ln N. ∞

V. Ko r e p in F-H Formula Measures of Entanglement Two blocks in VBS Negativity

3 1 x N = 3p3 + 3p4 = 1 + 2 3  ✓ ◆ Changes from 1 at x = 1 to 3/2 at x = . 1 Negativity qualitatevly agrees with entropy for one block in the ground state of AKLT. It conﬁrms that the block is entangled with environment.

V. E. Korepin Negativity in VBS Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Heisenberg XY model

H = (1 + γ)σxσx +(1 γ)σyσy + hσz. − j j+1 − j j+1 j !j i σj are Pauli operators.

Critical ﬁeld hc =2. XX model obtained at isotropic line γ =0.

V. E. Korepin Entanglement in spin chains 24/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Phase diagram of XY model

Γ 1 2

0 0 1 2 3 4 h Gapped everywhere except critical lines (red). On Barouch–McCoy circle γ2 + h2/4=1ground state is a product state.Nophasetransitionbetween1aand1b. Entropy on Barouch–McCoy circle is S = log 2.

V. E. Korepin Entanglement in spin chains 25/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Multicritical point of XY model

For long blocks, entropy S(k) is a function of elliptic parameter k(γ,h):

√(h/2)2+γ2−1 γ : region 1a ⎧ ⎪ 1−(h/2)2−γ2 k(γ,h)=⎪ 2 : region 1b ⎪ 1−(h/2) ⎨⎪ % γ : region 2. ⎪ √(h/2)2+γ2−1 ⎪ ⎪ ⎩⎪

V. E. Korepin Entanglement in spin chains 27/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Multicritical point of XY model

Plot of k(γ,h) : | | 2

Γ 0

!2 !4 !2 0 2 4 h Constant entropy ellipses/hyperbolae on XY phase diagram. Entropy essentially singular at isotropic point (γ =0) and critical ﬁeld (h = 2). ± arXiv:quant-ph/0609098 Franchini, Its, Jin, Korepin Journal of Phys. A: Math. Theor. 40 (2007) 8467-8478

V. E. Korepin Entanglement in spin chains 28/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Entanglement spectrum of XY model

Density matrix of block has full rank 2x. Eigenvalues form geometric sequence λ e−nπτ (large block n ∝ x ). →∞ Parameter τ is ratio of elliptic integrals I(√1 k2)/I(k), 1 − I(k)= dx 0 √(1−x2)(1−k2x2) Degeneracy! g of λ grows subexponentially g eπ√n/3. n n n ∼

V. E. Korepin Entanglement in spin chains 29/67 ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Application to physics: area Law

Area law. 1986 L. Bombelli, R. Koul, J. Lee, and R. Sorkin 1993 M. Srednicki 1994 Holzhey, Larsen and Wilczek: ent entr of block of spins of c size x for 1D gapless models scales as S = 3 log x . 2003 Logarithmic scaling of Renyi entropy. B.-Q.Jin, Korepin, Journal Statistical Physics; arXiv.org April 15 of 2003. c v πTx 2004 For ﬁnite temperature S = 3 log πT sinh v Korepin, PRL, vol 92, ei 096402, 05 March 2004,* * ++ arXiv:cond-mat/0311056

V. Ko r e p in F-H Formula Appendix For Further Reading For Further Reading I

Fisher M E & Hartwig R E Toeplitz Determinants, Some Applications, Theorems and Conjectures. Adv. Chem. Phys. 15 333-353 1968. Bötter A & Silbermann B Analysis of Toeplitz Operators. Springer-Verlag, Berlin 1990. Korepin V E, Izergin G, Bogoliubov N M Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz. Cambridge University Press, Cambridge 1993

V. Ko r e p in F-H Formula Measures of Entanglement Two blocks in VBS

MEASURES OF ENTANGLEMENT IN SPIN CHAINS

Sougato Bose, Raul Santos and Vladimir Korepin

5th Asia-Paciﬁc Workshop on Quantum Information Science

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Outline

1 Measures of Entanglement

2 Two blocks in VBS

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Entanglement

Entanglement is pure quantum relation of two sub-systems. What are good measures? Entropy of a subsystem is a good measure for pure state. Classical case: if the entropy of a system is zero, then there is no entropy in any subsystem.

34 Not true in quantum case: } = h/2⇡ 10 Js. ⇡ Entropy of a subsystem can be positive, while the entropy of the whole system is zero.

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Notations

Subsystems will be denoted by A and B. Pure case means that entropy is zero: SA&B = 0. In classical case subsystems also have no entropy SA = SB = 0, but

SA = SB 0 in quantum case Entropy of a subsystem distinguishes quantum form classical.

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Mixed case

In thermodynamics SA&B > 0

Mutual information is

IAB = SA + SB SA&B is used as a measure of entanglement, but it contains classical correlations. It can be positive for classical systems: In probability theory we can ﬁnd two correlated random variables

A and B with IAB > 0. For example it is used to measure capacity of noisy classical communication channels.

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Negativity

Introduced by A.Peres in 1996 , developed by S. Bose

⇢AB actes in a tensor product of two Hilbert spaces and HA HB Partial transposition only in one subspace A:

TA ⇢AB

can have negative eigenvalues. The magnitude of sum of negative eigenvalues is negativity N. It is a measure of entanglement. Very few results for dynamical systems.

V. E. Korepin Negativity in VBS Efﬁcient representation of Ground states Matrix Product and Tensor Network States AKLT model

N 1 AKLT Hamiltonian HBulk = i=1 P(Si + Si+1), with S = 1. Here P(S + S ) a projector onto spin 2 states, given by i i+1 P 1 P(S + S )= 3S S +(S S )2 + 2 , k k+1 6 k · k+1 k · k+1 ⇣ ⌘

1 2 ...... N Figure: Graphic representation of the 1D VBS ground state.

In order to construct the ground state VBS of H we can associate | i Bulk two spin 1/2 variables at each lattice site and create the spin 1 state symmetrizing them. To prevent the formation of spin 2, we antisymmetrize states between different neighbor lattice sites.

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 13 / 21 Measures of Entanglement Two blocks in VBS Density matrix of a block

gs = A B | i | [ i Block of x sequential spins in the ground state of AKLT is subsystem A and the rest of the ground state as subsystem B. density matrix of the ground state state

⇢ = gs gs | ih | Density matrix of the block

⇢A = Tr B (⇢)

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Eigenvalues of the density matrix

Fan, Korepin and Roychowdhury 2004 PRL: the density matrix of ﬁnite block of spins on the ﬁnite lattice has only 4 non-zero eigenvalues. One

1 1 x p = 1 + 3 1 4 3  ✓ ◆ and three

1 1 x p = 1 3 4 3  ✓ ◆

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Entropy of a Subsystem

The entropy of the block of the size x on a ﬁnite lattice:

S(x)= 3[1 p(x)] 1 p(x) [1+3p(x)] 1+3p(x) = log log 4 4 4 4 ⇣ ⌘ ⇣ ⌘ Here p(x)=( 1/3)x . Block entropy does not depend on the size of the lattice. S(x) quickly changes from ln 3 to ln 4 as x goes from 1 to . Fan, Korepin and Roychowdhury 2004 . 1 Let us compare this with negativity [for one block also].

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Partial transposion

Block of x sequential spins in the ground state of AKLT is subsystem A and the rest of the ground state is subsystem B. The density matrix of the ground state

⇢ = gs gs AB | ih | is a projector to the unique wave function of the ground state. Let us make transposition only in the Hilbert space of the block : TA ⇢AB

TA What is the spectrum? Eigenvalues of ⇢AB are p( ) ·

V. E. Korepin Negativity in VBS ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Two blocks

Now let us consider entanglement of two blocks in the VBS state. They are in a mixed state [no unique wave function describes their state]. We can measure the entanglement either by negativity or by mutual information. Both are zero. The two block are not entangled, unless they touch. In this case partially transposed density matrix has 5 negative eigenvalues.

V. Ko r e p in F-H Formula Measures of Entanglement Two blocks in VBS Spectrum

TA Ten non-negative eigenvalues of ⇢AB

x p = 1 1 1 , 6-fold degeneracy, 3 4 3 ⇣ ⌘x p = 1 1 + 3 1 , unique 1 4 3 p4 = p⇣p1p3, ⌘3-fold deg.

Six negative eigenvalues

p = p , 3-fold deg. 2 3 p = p , 3-fold deg. 5 4

V. E. Korepin Negativity in VBS Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Area law for XY model

1 Γ = 4

S log!2"

0.5 Γ = 1 BM circle

Γ = 1/3 0 0 1 2 3 4 h Entropy curves for fixed anisotropy γ (long blocks x ). →∞ Entropy diverges at critical field hc =2. Below critical field, entropy has a minimum log 2 on Barouch–McCoy circle.

V. E. Korepin Entanglement in spin chains 26/67 ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Two blocks

V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Two blocks

V. Ko r e p in F-H Formula Appendix Referencies Bibliography II

V.Vedral, New J.Phys.6,10 (2004). Eisert J, Cramer M & Plenio M B Area laws for the entanglement entropy - a review. Pre-print arXiv:0808.3773 Jin B -Q & Korepin V E Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture. Jour. Stat. Phys. 116 79-95 2004

V. E. Korepin Negativity in VBS Measures of entanglement 1D Gapless models Gapped models Outline

1 Measures of entanglement Pure states Mixed states Area law

2 1D Gapless models Hubbard model Heisenberg XX model High spin XXX model

3 Gapped models Heisenberg XY model Isotropic AKLT model Anisotropic AKLT model 2D AKLT model

V. E. Korepin Entanglement in spin chains 2/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Entanglement

Entanglement is pure quantum relation of two subsystems. Discovred at the dawn of quantum mechanics in 1935.

What are good measures that distinguish quantum correlations from classical ones?

Entropy of a subsystem is a good measure for a pure state.

V. E. Korepin Entanglement in spin chains 3/67 Measures of entanglement 1D Gapless models Gapped models

Measures of entanglement in spin chains

Vladimir E Korepin

C.N. Yang Institute for Theoretical Physics Stony Brook University

G¨oteberg, Sweden, 2012

V. E. Korepin Entanglement in spin chains 1/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Entropy

Classical case If the entropy of a whole state is zero, then there is no entropy in any subsystem.

Quantum case If the entropy of a whole state is zero, a subsystem can have positive entropy.

V. E. Korepin Entanglement in spin chains 4/67 Appendix Referencies Bibliography III

Its A R, Jin B -Q & Korepin V E Entropy of XY Spin Chain and Block Toeplitz Determinants. Fields Institute Communications, Universality and Renormalization [editors Bender I & Kreimer D] 50 151, 2007

V. E. Korepin Negativity in VBS Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Notation

Subsystems denoted by A and B. Consider state vector ψ = A B | ⟩ | ∪ ⟩ Density matrix of whole state: ρ = ψ ψ AB | ⟩⟨ | Density matrix of subsystem A: ρA =TrBρAB von Neumann entropy of subsystem A

S = Tr(ρ log ρ ) A − A A R´enyi entropy of subsystem A 1 S (α)= log Trρα , 0 < α < 1 A 1 α A −

V. E. Korepin Entanglement in spin chains 5/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Pure states

Given a pure state with zero entropy:

SAB =0.

In classical case SA = SB =0,but in quantum case S = S 0 due to quantum ﬂuctuations. A B ≥ Entropy of a subsystem can distinguish quantum system from classical one.

V. E. Korepin Entanglement in spin chains 6/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Entropy as an entanglement measure

von Neumann entropy satisﬁes the properties of an entanglement measure for pure bipartite systems: Vanishes for separable states (ρ = ρ ρ , no quantum correlation). AB A ⊗ B Does not increase under local operations and classical communication (LOCC). Additive for copies of the system.

V. E. Korepin Entanglement in spin chains 7/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Entanglement spectrum

Li & Haldane (2008) proposed to measure entanglement by full spectrum of ρA. Write

ρ e−βentHent . A ≡

V. E. Korepin Entanglement in spin chains 8/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Mixed states

S > 0= thermodynamics. AB ⇒ Classical and quantum correlations diﬃcult to distinguish.

V. E. Korepin Entanglement in spin chains 9/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Mutual information

Mutual information

I = S + S S AB A B − AB used in the literature as a measure of entanglement. Does not distinguish classical from quantum correlations. Can be positive for classical systems. In probability theory we can ﬁnd two correlated random variables A and B with IAB > 0.Example:capacityofnoisy channels.

V. E. Korepin Entanglement in spin chains 10/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Negativity

Introduced by A. Peres in 1996. Consider whole density matrix

ρ = α i k j l AB ijkl| ⟩A⟨ |A ⊗ | ⟩B ⟨ |B acting in a tensor product! of two Hilbert spaces A and B. H H Make partial transposition only in subspace A: H ρTA = α k i j l AB ijkl| ⟩A⟨ |A ⊗ | ⟩B ⟨ |B ! The result can have negative eigenvalues.

V. E. Korepin Entanglement in spin chains 11/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Negativity

Negativity is N = (negative eigenvalues of ρTA ). N − AB is a measure of entanglement! for mixed states. N [Bose 2000; Vidal, Werner 2001]

Comparison to concurrence of 2 qubits: Verstraete et al. JPA 2001. Few analytical results for dynamical systems.

V. E. Korepin Entanglement in spin chains 12/67 Measures of entanglement Pure states 1D Gapless models Mixed states Gapped models Area law Area law for pure states

A Γ B Srednicki 1993.

At zero temperature entropy of a subsystem SA for gapfull model is proportional to area of boundary Γ (large systems). Unlike the second law of thermodynamic.

V. E. Korepin Entanglement in spin chains 13/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Outline

1 Measures of entanglement Pure states Mixed states Area law

2 1D Gapless models Hubbard model Heisenberg XX model High spin XXX model

3 Gapped models Heisenberg XY model Isotropic AKLT model Anisotropic AKLT model 2D AKLT model

V. E. Korepin Entanglement in spin chains 14/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Conformal approach for 1D gapless model

(periodic box of length L) Finite-size corrections to ground state energy πv E = Lϵ + c + (L−2). 0 6L O Central charge c (universal). Spectrum of low-lying excitations:

ε(k)=v k k + ( k k 2). | − F| O | − F| dε Fermi velocity ν = (depends on model). dk k=kF ! ! !

V. E. Korepin Entanglement in spin chains 15/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Universal entropy scaling in 1D gapless models

Entropy for blocks of length x on inﬁnite chain is →∞ c S log x vN ∼ 3 Holzhey, Larsen, Wilczek 1994; Vidal, Latorre, Rico, Kitaev 2003; Korepin 2004; Calabrese, Cardy 2004 Renyi entropy c S (α) (1 + α−1) log x, . R ∼ 6 B.-Q.Jin, V.E.Korepin 2003; Calabrese, Cardy 2004

V. E. Korepin Entanglement in spin chains 16/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Hubbard model

† † H = (c c +c c )+u n ↑n ↓ h (n ↑ n ↓). − j,σ j+1,σ j+1,σ j,σ j, j, − j, − j, !j,σ !j !j † Fermion operators cj,σ. Gapless below critical ﬁeld hc.

Spin-charge separation. Entropy for long blocks x is →∞ c c 2 S(x) spin log x + charge log x log x. ∼ 3 3 → 3

V. E. Korepin Entanglement in spin chains 17/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Heisenberg XX model

H = σxσx + σyσy + hσz. − j j+1 j j+1 j !j i σj are Pauli operators. Critical ﬁeld hc =2.

Ferromagnetic above critical ﬁeld h h . | | ≥ c Gapless below critical ﬁeld h

V. E. Korepin Entanglement in spin chains 18/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Critical XX model

Above critical ﬁeld, entropy vanishes (ferromagnet).

Below critical ﬁeld, entropy of long blocks x is →∞ 1 S(x) log x(4 h2)1/2 +0.4950179 . ∼ 3 − ··· [Jin & Korepin JSP 2004, Peschel! JSM 2004]"

Central charge c =1. Magnetic length (4 h2)−1/2 diverges at critical ﬁeld h =2. − c

V. E. Korepin Entanglement in spin chains 19/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Critical spin-s XXX model

H = F S S . j · j+1 j ! " # S are spin-s operators. X = S S j j · j+1 Function F (X) is a polynomial of order 2s:

2s 2s 2s 1 X y F (X)=2 − j , k yl yj !l=0 k!=l+1 j=0$,j≠ l − y = l(l +1)/2 s(s +1).SolvablebyBethe ansatz. l − For s =1/2, antiferromagnetic XXX model obtained. For s =1,Takhtajan-Babujianmodelobtained.

V. E. Korepin Entanglement in spin chains 20/67 Measures of entanglement Hubbard model 1D Gapless models Heisenberg XX model Gapped models High spin XXX model Entropy of spin-s XXX model

Central charge of model depends on spin s: 3s c = . s +1

von Neumann entropy for large blocks x scales as →∞ s S(x) log x. ∼ s +1

V. E. Korepin Entanglement in spin chains 21/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Outline

1 Measures of entanglement Pure states Mixed states Area law

2 1D Gapless models Hubbard model Heisenberg XX model High spin XXX model

3 Gapped models Heisenberg XY model Isotropic AKLT model Anisotropic AKLT model 2D AKLT model

V. E. Korepin Entanglement in spin chains 22/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Gapped spin chains (1D)

Correlations decay exponentially.

Area law for entropy of long blocks x implies →∞ S(x)=constant,x . →∞ The constant represents boundary contribution to entropy.

V. E. Korepin Entanglement in spin chains 23/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Entanglement Hamiltonian of XY model

† Hent = ω ncncn n=1 ! † Here cn is creation operator for n-th species of fermions.

E = ω positive integers ! The number of diﬀerent partitions is known from number theory. It was ﬁrst calculated by Srinivasa Ramanujan in 1918. We denoted it by gn

V. E. Korepin Entanglement in spin chains 30/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model AKLT model

Aﬄeck, Kennedy, Lieb, Tasaki 1987 Ground state is valence-bond-solid (VBS) state. Haldane gap separates ground state from excited states. Unique ground state of a ﬁnite graph: Kirillov-Korepin 1989

V. E. Korepin Entanglement in spin chains 31/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model AKLT Hamiltonian

Spin chain of L spin-1’s in the bulk and two boundary spin-1/2. Sk is spin-1 and sb is spin-1/2.

L−1 1 2 H = S S + (S S )2 + + π + π . k k+1 3 k k+1 3 0,1 L,L+1 !k=1 " # 1 1 2 1 2 SkSk+1 + 6 (SkSk+1) + 3 is a projector on a state of spin-2.

The π is a projector on a state with spin-3/2: 2 2 π = (1 + s S ) , π = (1 + s S ) . 0,1 3 0 1 L,L+1 3 L+1 L

V. E. Korepin Entanglement in spin chains 32/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Ground state is VBS

Ground state is unique and there is a gap:

✓✏ ✓✏ ✓✏ ✓✏ ✓✏ ✓✏ ssssssssssssss ✒✑ ✒✑ ✒✑ ✒✑ ✒✑ ✒✑ 1 Adotisspin-2 ;circlemeanssymmetrisation(makesspin1). Alinemeansanti-symmetrisation( ). |↑↓⟩ − |↓↑⟩ Each projector annihilates the ground state (no frustration). Exact spin correlation function at any distance is:

3 1 x S1Sx = = p(x). 4⟨ ⟩ −3 ! "

V. E. Korepin Entanglement in spin chains 33/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Eigenvectors for open boundary conditions

If all spins in the chain are equal to 1, then the ground state is four times degenerated. Four bulk eigenstates labelled by the total spin at the boundary 2 2 z z z stot =(s1 + sL) ,stot = s1 + sL with sa,spin-1/2 at site a

V. E. Korepin Entanglement in spin chains 34/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Density matrix

0 1 ... m ... m+x-1 ... L L+1 ! ! ! B A=x sites B

Abasisin is A ,withµ =1,...,4, similarly for B HA | µ⟩ Density matrix ρˆ = VBS VBS = ρ A B A B | ⟩⟨ | αβγδ αβγδ| α⟩| β⟩⟨ γ |⟨ δ| !

V. E. Korepin Entanglement in spin chains 35/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Reduced density matrix

Wavefunction: VBS = A B | ⟩ | ∪ ⟩ Block of x sequential spins in the ground state of AKLT is subsystem A and the rest of the ground state as subsystem B. Density matrix of whole ground state

ρ = VBS VBS AB | ⟩⟨ | Density matrix of the block

ρA =TrBρAB

V. E. Korepin Entanglement in spin chains 36/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Eigenvalues of the density matrix

Fan, Korepin, and Roychowdhury PRL 2004: Density matrix of ﬁnite block of spins on the ﬁnite lattice has only 4 non-zero eigenvalues.One

1+3p(x) λ = 1 4 and thrice degenerate

1 p(x) λ = − 3 4 x 1 = p(x) − 3 ! "

V. E. Korepin Entanglement in spin chains 37/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Entropy of a subsystem

Exact entropy of the block of the size x on a ﬁnite lattice:

3 1 p(x) 1 p(x) S(x)= − log − − 4 4 ! " # $ 1+3p(x) 1+3p(x) log . − 4 4 # $ It does not depend on the lattice size L.

For x =1the entropy is log 3. R´enyi, von Neumann entropy rapidly approach log 4 as x . →∞

V. E. Korepin Entanglement in spin chains 38/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Mixed states in VBS (partial transposition)

Recall the density matrix of the ground state ρAB as a projector onto VBS . Let us make transposition only in the Hilbert space | ⟩ A of the block to obtain: H TA ρAB

TA What is the spectrum? Let us denote eigenvalues of ρAB by λ.

V. E. Korepin Entanglement in spin chains 39/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Spectrum

TA Ten non-negative eigenvalues of ρAB 1 1 x λ = 1 , 6-fold degeneracy, 3 4 − −3 ! ! " " 1 1 x λ = 1+3 , no deg. 1 4 −3 ! ! " " λ4 = λ1λ3, 3-fold deg. # Six negative eigenvalues

λ = λ , 3-fold deg. 2 − 3 λ = λ , 3-fold deg. 5 − 4

V. E. Korepin Entanglement in spin chains 40/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Negativity

log! N " 0.5

0.4

0.3

0.2

0.1

x 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Negativity =3λ +3λ N 3 4 Negativity changes from 0 at x =1to log(3/2) as x . N →∞ [R. Santos, V. Korepin, S. Bose PRA 2011]

V. E. Korepin Entanglement in spin chains 41/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Two blocks in isotropic VBS state

...... { { { { { E1 A E2 B E 3

Mixed state results after tracing away E = E E E . 1 ∪ 2 ∪ 3 Negativity of two blocks is zero unless they touch.

V. E. Korepin Entanglement in spin chains 42/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Bounded entangled states

The state is still entangled in the sense that it can not be written as a product state. Vanishing negativity on an entangled state means: Bounded entangled state These states can not be distilled by local action to create an useful entanglement for quantum communication tasks such as teleportation. It can be used to generate a secret quantum key, or to enhance the ﬁdelity of conclusive teleportation using another state. [K. Horodecki, M. Horodecki, P. Horodecki, J. Oppenheim, PRL2005, P. Horodecki, M. Horodecki, R. Horodecki, PRL 1999]. It has been found in other physical systems. [D. Patane, R. Fazio, L. Amico, NJP 2007]

V. E. Korepin Entanglement in spin chains 43/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Two blocks touch

Mutual negativity of two block vanish unless they touch. If they do touch then: 1 3 = ( 1/3)2xA +( 1/3)2xB x ,x 1 N 2 − 4{ − − } A B ≫

Here xA and xB are the respective block sizes. Mutual information on the contrary decays exponentially with distance d between the blocks: for inﬁnite lengths xA and xB

1 2d I 3 → 3 ! " [R. A. Santos, V. Korepin, JPA 45 (2012)]

V. E. Korepin Entanglement in spin chains 44/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Homogeneous spin-s AKLT

Generalization of AKLT model for general integer spin s on each site in the bulk and spin s/2 at the boundary with Hamiltonian

L−1 2s H = CJ πJ (j, j +1)+H(0, 1) + H(L, L +1) !j=1 J!=s+1 where CJ > 0 and the projector πJ (j, j +1)projects the bond spin Jj,j+1 = Sj + Sj+1 onto the subspace with total spin J (J = s +1,...,2s). Boundary terms project the bond spin at the boundaries J and J onto the subspace of spin J 3s/2. 0,1 L,L+1 ≤

V. E. Korepin Entanglement in spin chains 45/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model The projector

Sj +Sj+1 (S + S )2 l(l +1) π (j, j +1)= j j+1 − J J(J +1) l(l +1) l=|Sj −Sj+1| − !l̸=J

Projects two quantum spins Sj and Sj+1 one the total spin J.

V. E. Korepin Entanglement in spin chains 46/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Reduced density matrix in the large block limit

Density matrix ρ = VBS VBS , tracing out spins outside a block | ⟩⟨ | of consecutive x spins, we obtain the reduced density matrix of the block ρx. 2 The rank of ρx is (s +1) . In the large block limit, x , →∞ 1 ρ = I 2 . x→∞ (s +1)2 (s+1) The entropy is 2log(s +1) [Ying Xu, Hosho Katsura, Takaaki Hirano, Vladimir E. Korepin, 2008]

V. E. Korepin Entanglement in spin chains 47/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model SU(N) AKLT model

Construction of SU(N) VBS state.

White dots denote fundamental representations of SU(N) ! (like spins 1/2 in the SU(2) spin-1 case). Black dots denote adjoint representations of SU(N) !. Alineconnectingtwodotsrepresentsthesingletstate 0, 0 . | ⟩ Large circles represent a projection onto the adjoint representation.

V. E. Korepin Entanglement in spin chains 48/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model SU(N) AKLT model

Hamiltonian is a projector. Von Neumman entropy :

2 2 1+(N − 1)pN (x) 2 S(x)=logN − log(1 + (N − 1)pN (x)) N 2 2 1 − pN (x) − (N − 1) log(1 − pN (x)). N 2

−1 x pN (x)=[ ] N 2 − 1 Katsura, Hirano, Korepin JPA 41 2008

V. E. Korepin Entanglement in spin chains 49/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model q-deformed AKLT model (AKLTq)

AKLT is deﬁned in terms of representation theory of Lie algebras. One can consider AKLT based on a quantum group SU(2)q [Drinfel’d 1985, Jimbo 1985] Batchelor , Nepomechie, Mezinchescu and Rittenberg JPA 1990, Kl¨umper, Schadschneider, Zittartz JPA 1991.

V. E. Korepin Entanglement in spin chains 50/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model

SUq(2) quantum group

Quantum algebra of generators

z z q2S q−2S [S+, S−]= − , [Sz, S±]= S±. q q−1 ± − Co-multiplication of two spin-1:

Sz Sz Sz Sz Sz S± = S± q 2 + q− 1 S±,qtot = q 1 q 2 . tot 1 ⊗ ⊗ 2 SU(2) obtained at q =1.

V. E. Korepin Entanglement in spin chains 51/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model q-deformed AKLT hamiltonian

Interaction of nearest neighbors, still a projector

1 −1 z z = b c S⃗j S⃗j+1 + S⃗j S⃗j+1 + (1 c)(q + q 2)S S +1 H j · · 2 − − j j 1 ! " −1 z # z 2 1 −1 2 z z 2 + 4 (1 + c)(q q )(Sj+1 Sj ) + 4 c (1 c)(q + q 2) (Sj Sj+1) − −− − − − + 1 c (1 + c)(q q 1)(q + q 1 2)SzSz (Sz Sz) 4 − $ − j j+1 j+1 − j 1 1 2 z z + 4 (c 3) c 1+ 2 (1 + c) Sj Sj+1 − − − +2 c 1 (1+c)2 (Sz )2+(Sz)2 +(c 1)+ 1 c (q2 q 2)(Sz Sz) , − 8 j+1 #%j − 2 & − j+1− j with% c =1+q2&%+ q−2 and b =[c&$(c 1)]−1 ' −

V. E. Korepin Entanglement in spin chains 52/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Ground state of q-deformed AKLT

Unique ground state is VBSq state (periodic boundary conditions). VBSq state separated from excited states by Haldane gap. Wavefunction of L spins as matrix product:

VBS =tr[g ·g ·...·g ] | q⟩ 1 2 L with −1 −1 q 0 j q + q + j gj = | ⟩ − | ⟩ . q + q−1 q 0 ! |−⟩j " − | ⟩j # 0 , are spin-1"Sz states at site j. | ⟩j |±⟩j

V. E. Korepin Entanglement in spin chains 53/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Eigenvalues of the density matrix

Density matrix of ﬁnite block of spins on inﬁnite lattice has 4 non-zero eigenvalues. Nondegenerate

2 −2 2 q + q +2pq(x) 1 1 [pq(x)] λ± = − . 2(q + q−1)2 ± !4 − (q + q−1)2 and twice degenerate

1 p (x) λ = − q , 0

V. E. Korepin Entanglement in spin chains 54/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Eigenvalues of the density matrix

Double scaling limit (inﬁnite block) x : →∞ Eigenvalues:

q∓2 λ± → (q + q−1)2 and 1 λ (degeneracy 2). 0 → (q + q−1)2

V. E. Korepin Entanglement in spin chains 55/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Entanglement entropy (double scaling limit)

log!4" von Neumann entropy

0 0.0 0.2 0.4 0.6 0.8 1.0 q

(left) Eigenvalues of reduced density matrix.

(right) von Neumann entropy of VBSq state. Entropy vanishes as q 0 (classical Ising limit). → V. E. Korepin Entanglement in spin chains 56/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Entanglement spectrum

−βHent −βHent ρA = e /Tr e . Santos, Paraan, Korepin, Kl¨umper, EPL 2012: For VBSq in double scaling limit

βH = log q σx σx + σy σy . ent | | 1 ⊗ 2 1 ⊗ 2 ! " Entanglement boundary Hamiltonian is Heisenberg XX model. Eﬀective temperature 1/ log q . | | No deformation inﬁnite temperature. → Full anisotropy zero temperature. →

V. E. Korepin Entanglement in spin chains 57/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Spin-sq-deformed AKLT

A natural generalization of the q-deformed AKLT model is to consider higher spin on each site. The matrix product state representation of the VBSq state is

VBS (s) = tr (g g ...g ) , (1) | q ⟩ 1 · 2 · L with

s/2 s/2 s b −b (gi)ab = ( 1) q s, m i, a −bm q − | ⟩ m ! " # where s/2 s/2 s ( s a, b s ) is the Clebsch-Gordan a −bm q − 2 ≤ ≤ 2 coeﬃcient" relating# the tensor product of two spin-s/2 states to the direct sum of spin-s states.

V. E. Korepin Entanglement in spin chains 58/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Entanglement entropy

After tracing out the spin degrees of freedom outside a block of length x, the density matrix in the double scaling limit (x, L ) →∞ is ( s/2 a, b, c, d s/2) − ≤ ≤

q−2(a+b) qs+1 q−(s+1) (ρ ) = δ δ with [s +1]= − . ∞ ab;cd [s +1]2 ac bd q q−1 −

−1 s+1 −(s+1) q + q q + q 2 SvN =2log [s +1] + (s +1) log q . q q−1 − qs+1 q−(s+1) # − $ ! " − [Santos, Paraan, Korepin, Kl¨umper, JPA 2012]

V. E. Korepin Entanglement in spin chains 59/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Entanglement properties in mixed systems

Theorem

Negativity between two blocks in the q-deformed case AKLTq vanish if the blocks do not touch.

Theorem 2

Mutual information AKLTq decay exponentially with distance between blocks.

V. E. Korepin Entanglement in spin chains 60/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model 2D AKLT model

Entanglement spectra in 2D VBS states. [Lou, Tanaka, Katsura, Kawashima, Kirillov, Korepin, JPA 2010, PRB 2011]

Square lattice AFM Heisenberg XXX chain (nonzero → temperature).

Hexagonal lattice FM Heisenberg XXX chain (nonzero → temperature).

V. E. Korepin Entanglement in spin chains 61/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model VBS states on ladders

AB ......

Example: Ladders with n steps.

Symmetric partitioning into subsystem A and B.

V. E. Korepin Entanglement in spin chains 62/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Reduced density matrix of A

A B Hamiltonian: Heﬀ = J ⃗σ 1 ⃗σ 2, Square (n ): − · ... →∞ J 1 1+2√19 = log 0.217. n T 2 15 ≈− A Hexagonal (n ): →∞ ... J 1 49 + 2√1627 = log 0.078. T 2 111 ≈

V. E. Korepin Entanglement in spin chains 63/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model VBS state on square lattice ...

Example: Square lattice with periodic boundaries.

Ly ...... y Symmetric partitioning into subsystem A and B. ...

V. E. Korepin Entanglement in spin chains 64/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Reduced density matrix of A ...

ρ =exp( βH ) A − ent

Ly ...... Eﬀective Hamiltonian y H = ⃗σ ⃗σ , ent i · i+1 !i ...

V. E. Korepin Entanglement in spin chains 65/67 Appendix For Further Reading

For Further Reading I

1 Fan H, V Korepin, V Roychowdhury PRL 93 227203 (2004). 2 Jin B-Q & V Korepin JSP 116 79 (2004). 3 Franchini F, A Its, V Korepin JPA 41 025302 (2008). 4 Katsura H, N Kawashima, A Kirillov, V Korepin, S Tanaka JPA 43 255303 (2010). 5 Santos R, V Korepin, S Bose PRA 84 062307 (2011). 6 Santos R, V Korepin JPA 45 125307 (2012). 7 Santos R, F Paraan, V Korepin, A Kl¨umper JPA 45 175303 (2012), EPL 98 (2012).

V. E. Korepin Entanglement in spin chains 67/67 Heisenberg XY model Measures of entanglement Isotropic AKLT model 1D Gapless models Anisotropic AKLT model Gapped models 2D AKLT model Summary

Entropy is a good entanglement measure. Entropy of 1D gapless models have universal conformal scaling (logarithmic in block size). Entropy of gapped models satisfy area law (constant entropy). Entanglement spectrum provides picture of interacting boundary spins.

V. E. Korepin Entanglement in spin chains 66/67 Measures of Entanglement Two blocks in VBS Two blocks

Let us consider two blocks on a distance in VBS state. They are in a mixed state: no unique wave function describes them. Let us make transposition in the Hilbert space of one block only. Calculations show that negativity vanish, iff two blocks does not touch: N = 0 In case of touching blocks of large sizes L , L , the A B !1 negativity is

1 3 1 2LA 1 2LB NL ,L = + A B!1 2 4 3 3 "✓ ◆ ✓ ◆ #

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Mutual information of two blocks

Consider two blocks at a distance x. Mutual information is: 3 1 I = [1 p(x)] ln[1 p(x)] + [1 + 3p(x)] ln[1 + 3p(x)] 4 4 here 1 x p(x)= 3 ✓ ◆ It does not vanish unlike negativity. These properties of AKLT is unique and different from other models like Heisenberg spin chains.

V. E. Korepin Negativity in VBS Measures of Entanglement Two blocks in VBS Open problems

~ # Negativity for VBS with higher spins?

⇣ Negativity of VBS in two space dimensions?

V. E. Korepin Negativity in VBS Appendix Referencies Bibliography I

A.Afﬂeck, T.Kennedy, E.H.Lieb and H.Tasaki, Commun. Math. Phys. 115, 477 (1988) Valence Bond Solid in Quasicrystals A. Kirillov and V. Korepin ALGEBRA and ANALYSIS , vol 1, issue 2, page 47, 1989 Entanglement in a Valence-Bond-Solid State H. Fan, V. Korepin, V. Roychowdhury, Physical Review Letters, vol 93, issue 22, 227203, 2004 Density Matrix of a Block of Spins in AKLT Model Y. Xu, H. Katsura, T. Hirano, V Korepin, Journal-ref: Jour. Stat. Phys. vol 133, no. 2, 347-377 (2008)

V. E. Korepin Negativity in VBS Appendix Referencies Bibliography II

V. E. Korepin Negativity in VBS Appendix Referencies Bibliography III

Its A R, Jin B -Q & Korepin V E Entropy of XY Spin Chain and Block Toeplitz Determinants. Fields Institute Communications, Universality and Renormalization [editors Bender I & Kreimer D] 50 151, 2007

V. E. Korepin Negativity in VBS Efﬁcient representation of Ground states Matrix Product and Tensor Network States AKLT model

N 1 AKLT Hamiltonian HBulk = i=1 P(Si + Si+1), with S = 1. Here P(S + S ) a projector onto spin 2 states, given by i i+1 P 1 P(S + S )= 3S S +(S S )2 + 2 , k k+1 6 k · k+1 k · k+1 ⇣ ⌘

1 2 ...... N Figure: Graphic representation of the 1D VBS ground state.

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 13 / 21 Efﬁcient representation of Ground states Matrix Product and Tensor Network States 1D AKLT model, spin 1 VBS VBS ⇢ = | VBSihVBS | , This is a one dimensional projector on the ground state h | i of the Hamiltonian HBulk . B BA

0 1 ... m ... m+L-1 ... N N+1 { L sites

⇢(A)=tr (⇢)= Aˆ Aˆ 7 B µ| µih µ| with = 1 (1 z(L)), = 1 (1 + 3z(L)) and z(L)=( 1 )L 0,1,3 4 2 4 3

S(⇢(A)) = 3 ln ln 2 ln 2 3z2 L 1 0 0 2 2 ⇡ 7H. Fan, V. Korepin, V. Roychowdhury, Phys. Rev. Lett., 93, 22, 227203, (2004) Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 15 / 21 Efﬁcient representation of Ground states Matrix Product and Tensor Network States 1D AKLT model, spin 1 VBS VBS ⇢ = | VBSihVBS | , This is a one dimensional projector on the ground state h | i of the Hamiltonian HBulk . B BA

0 1 ... m ... m+L-1 ... N N+1 { L sites

⇢(A)=tr (⇢)= Aˆ Aˆ 7 B µ| µih µ| with = 1 (1 z(L)), = 1 (1 + 3z(L)) and z(L)=( 1 )L 0,1,3 4 2 4 3

Vladimir Korepin, Raul A. Santos

Department

July 21, 2015

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 1 / 21 Outline

1 Entanglement Measures Entanglement Entropy Area Law in Gapped systems Gapless systems

2 Efﬁcient representation of Ground states Matrix Product and Tensor Network States

3 Entanglement Spectrum and Boundary theories Entanglement Spectrum Topological systems Entanglement Measures What is Entanglement?

Let’s consider a system made up of two components A and B. The states of system A and B . This is called a bipartite state. 2 H1 2 H2 Quantum Mechanics dictates that the state of the system will be

A B = cij i A j B 1 2. (1) | i [ | i ⌦ | i 2 H ⌦ H Xi,j for some cij C. 2 A B If cij = ↵i j , A B =( i ↵i i ) j j j is a separable state. If | i [ | i A ⌦ | i B c = ↵AB the state is inseparable ⇣it is called⌘ entangled state. ij 6 i j P ! P Entangled states refuse a classical description!.

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 4 / 21 Entanglement Measures What is Entanglement?

Let’s consider a system made up of two components A and B. The states of system A and B . This is called a bipartite state. 2 H1 2 H2 Quantum Mechanics dictates that the state of the system will be

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 4 / 21 Entanglement Measures Entanglement is a pure quantum effect

Let’s take = 1 ( ), | i p2 | "iA| #iB | #iA| "iB The density matrix of the system A B is ⇢ = . The von [ AB | ih | Neumann entropy of the system is

S (A B)= tr⇢ ln ⇢ = 0. vN [ AB AB The entropy of the subsystems does not vanish!

S (A)= tr⇢ ln ⇢ = tr⇢ ln ⇢ = ln 2 = S (B), vN A A B B vN with ⇢A = trB⇢AB and ⇢B = trA⇢AB.

Classically this is not possible.

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 5 / 21 Entanglement Measures Entanglement is a pure quantum effect

Let’s take = 1 ( ), | i p2 | "iA| #iB | #iA| "iB The density matrix of the system A B is ⇢ = . The von [ AB | ih | Neumann entropy of the system is

S (A B)= tr⇢ ln ⇢ = 0. vN [ AB AB The entropy of the subsystems does not vanish!

S (A)= tr⇢ ln ⇢ = tr⇢ ln ⇢ = ln 2 = S (B), vN A A B B vN with ⇢A = trB⇢AB and ⇢B = trA⇢AB.

Classically this is not possible.

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 5 / 21 Entanglement Measures How do we quantify how entangled are two systems?

An entanglement measure E has to have the following conditions 1 E has to be invariant under local Unitary tranformations U E has to be continuous. E has to be additive when several copies of the system are present E( )=E( )+E( ) | ABi⌦| ABi | ABi | ABi A measure E compatible with the above conditions is the von Neumann Entropy for bipartite pure state.

S(⇢ )= tr(⇢ ln ⇢ ) A/B A/B A/B

1L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008). Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 6 / 21 Entanglement Measures How do we quantify how entangled are two systems?

S(⇢ )= tr(⇢ ln ⇢ ) A/B A/B A/B

Schmidt decomposition of bipartite pure states = p A B , | i i i | i i| i i P

A S(⇢)= Tr (⇢ ln ⇢) ⇢ = Tr ( ) A B | ih | B S(⇢A)=S(⇢B)

Figure: A B is a pure state [ Scaling of entropy in a system (correspondingly the entanglement) was conjectured 2 to scale with the area (not the volume) of the system.

2M. Srednicki, , Phys. Rev. Lett., 71, 666-669 (1993) Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 7 / 21 Entanglement Measures Area Law in Gapped systems Gapped systems

Given a local Hamiltonian H, Lieb-Robinson proved3 a bound for the transmition of signals,

a(d(X,Y ) v t ) [A(t), B] ce | | , || ||  itH itH 4 where A(t)=e Ae . Using this bound, Hastings , proved that Local gapped systems follow an area law for entanglement entropy in 1D.

S = A + ... a C a In higher dimensions, sensible gapped systems seem to follow and area law of entanglement entropy.

3Lieb, E. H., and D. W. Robinson, 1972, Commun. Math. Phys.28, 251. 4Hastings, M. B., 2007, J. Stat. Mech.: Theory Exp. P08024. Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 8 / 21 Entanglement Measures Area Law in Gapped systems Gapped systems

Given a local Hamiltonian H, Lieb-Robinson proved3 a bound for the transmition of signals,

a(d(X,Y ) v t ) [A(t), B] ce | | , || ||  itH itH 4 where A(t)=e Ae . Using this bound, Hastings , proved that Local gapped systems follow an area law for entanglement entropy in 1D.

S = A + ... a C a In higher dimensions, sensible gapped systems seem to follow and area law of entanglement entropy.

Entanglement entropy in generic gapless systems (with sensible interactions), does NOT follows an area law. At the bottom of the dispersion curve, a good approximation is conformal ﬁeld theory (CFT). From CFT in 1+1 dimensions, it has been found that 5

a) b) @ c ` Tr⇢↵ = S(`)= ln +O(1) @↵ 3 a ↵=1 ✓ ◆ No general expression for D > 1 spatial dimensions

5Holzhey, C., F. Larsen, and F. Wilczek, 1994, Nucl. Phys. B 424, 443; Calabrese, P., and J. Cardy, 2007, J. Stat. Mech. , P10004. Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 9 / 21 Entanglement Measures Gapless systems Higher dimensions and Holographic Principle

From the holographic principle and its AdS/CFT realization, it has been proposed 6 that the EE in a region A of a d dimensional state can be computed as the minimal surface with A as its boundary of a AdS space in d + 1 dimensions. From the MERA perspective

6M. Nozaki, S. Ryu, T. Takayanagi arXiv:1208.3469 Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 10 / 21 Outline

1 Entanglement Measures Entanglement Entropy Area Law in Gapped systems Gapless systems

2 Efﬁcient representation of Ground states Matrix Product and Tensor Network States

3 Entanglement Spectrum and Boundary theories Entanglement Spectrum Topological systems Efﬁcient representation of Ground states Matrix Product and Tensor Network States Representation of Ground states

From the area Law of entanglement entropy ’Small’ subset of ! generic quantum states Useful as an input for variational ansatz ground states. Systems satisfy this law by construction Matrix Product States i | i

↵ A[i] GS = i Tr(A[i1]A[i2] ...A[in]) i1i2 ...in , | i { } | i Projected Entangled Pair StatesP (PEPS) a d = i i P = A a ↵ • • • • | i i | i| i ↵,,a ↵| ih | P P

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 12 / 21 Outline

1 Entanglement Measures Entanglement Entropy Area Law in Gapped systems Gapless systems

2 Efﬁcient representation of Ground states Matrix Product and Tensor Network States

3 Entanglement Spectrum and Boundary theories Entanglement Spectrum Topological systems

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 2 / 21 Efﬁcient representation of Ground states Matrix Product and Tensor Network States Representation of Ground states

↵ A[i] GS = i Tr(A[i1]A[i2] ...A[in]) i1i2 ...in , | i { } | i Projected Entangled Pair StatesP (PEPS) a d = i i P = A a ↵ • • • • | i i | i| i ↵,,a ↵| ih | P P

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 12 / 21 Efﬁcient representation of Ground states Matrix Product and Tensor Network States VBS state as a Matrix product state

M = z , M = p2+, VBS = Tr(M M ..M ) s ..s with 0 1 | i s1 s2 sn | 1 ni M = p2 s ..s 1 X1 n 1, 0 p2 1, 1 with M = | i| i p2 1, 1 1, 0  | i| i

= tr(A A )tr(A A ) n , n O m , O m , h |Oi Oj | i n1 n2 ··· m1 m2 ··· h i j | i i j j i n m X{ i } {Xj } a) b) Ai Aj

Oi Oj

Ai Aj

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 14 / 21 Figure: (color online) (a) Transfer matrix (see text). (b) Computation of correlation functions with MPS state. The matrix T is obtained by inserting Oi the operator between matrices A. Oi

N VBS = (a†b† a† b†) 0 a = a . | i i i+1 i+1 i | i i+N i Yi=0 Outline

1 Entanglement Measures Entanglement Entropy Area Law in Gapped systems Gapless systems

2 Efﬁcient representation of Ground states Matrix Product and Tensor Network States

3 Entanglement Spectrum and Boundary theories Entanglement Spectrum Topological systems Entanglement Spectrum and Boundary theories Entanglement Spectrum Entanglement Spectrum

In topological systems ⇠i spectrum resembles the spectrum of a CFT in the edge of the partition

8H. Li and F. D. M. Haldane, PRL 101, 010504 (2008) Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 17 / 21 Entanglement Spectrum and Boundary theories Entanglement Spectrum Entanglement Spectrum

In topological systems ⇠i spectrum resembles the spectrum of a CFT in the edge of the partition

8H. Li and F. D. M. Haldane, PRL 101, 010504 (2008) Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 17 / 21 Entanglement Spectrum and Boundary theories Topological systems Topological systems

Topological systems are characterized by energy gap, cannot be deformed adiabatically to trivial gapped insulators, not characterized by symmetry breaking mechanism but instead, a topological property like presence of (symmetry protected) edge states or ground state degeneracy depending on the topology of the surface where the system resides.

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 18 / 21 Entanglement Spectrum and Boundary theories Topological systems CFT boundary excitations9

H = H + H + H H()=H + H + H A B int ! A B int H + H + H H E H + H + H A B int ! | int | ⌧ bulk ! L R int If H is a relevant operator, then H( 1) H( = 1) int ⌧ ! Then Partial density matrix ⇢A can be 4⌧0HL written as a thermal mixture of a ⇢A = a Pae Pa boundary CFT 9X. Qi, H. Katsura, A. Ludwig PRL 108, 196402 (2012), P N. Schuch, D. Poilblanc, J. I. Cirac, D. Perez-Garcia, arXiv:1210.5601 Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 19 / 21 Entanglement Spectrum and Boundary theories Topological systems CFT boundary excitations9

Finite size corrections to ⇢A eigenvalues in general spin S AKLT chain 10 (`/⇠ = ` ln[(S + 2)/S] 1) 1 3 S ` p 1 (2J(J + 1) S(S + 2)) . JM ⇡ (S + 1)2 S(S + 2) S + 2 ( ✓ ◆ )

Parametrizing ⇢ = e Heff , and deﬁning J S + S as the sum of A ⌘ 1 ` two spin- S/2 operators on the block boundaries, the effective hamiltonian takes the form

H = (S, `)( 1)` S S , eff 1 · ` 12 `/⇠ with (S, `)= S(S+2) e .

10 Raul A. Santos, F. N. C. Paraan, V. E. Korepin, A. Klumper,¨ Phys. A: Math. Theor. 45 175303 (2012) Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 20 / 21 Entanglement Spectrum and Boundary theories Topological systems General spin S AKLT chain

Finite size corrections to ⇢A eigenvalues in general spin S AKLT chain 10 (`/⇠ = ` ln[(S + 2)/S] 1) 1 3 S ` p 1 (2J(J + 1) S(S + 2)) . JM ⇡ (S + 1)2 S(S + 2) S + 2 ( ✓ ◆ )

Parametrizing ⇢ = e Heff , and deﬁning J S + S as the sum of A ⌘ 1 ` two spin- S/2 operators on the block boundaries, the effective hamiltonian takes the form

H = (S, `)( 1)` S S , eff 1 · ` 12 `/⇠ with (S, `)= S(S+2) e .

Ground states of gapped systems have an entanglement entropy which scales with the Area of the partition Indication of a correspondence between bulk physics and boundary description. Topological theories in the bulk CFT on the boundary. ! Similar trend in spin systems.

Vladimir Korepin, Raul A. Santos () Review of Entanglement July 21, 2015 21 / 21 Appendix For Further Reading For Further Reading II

Eßler F H L, Frahm H, Izergin A G & Korepin V E 1995 Integro-Difference Equation for a correlation function of the 1 spin- 2 Heisenberg XXZ chain Nucl. Phys. B 446 448-460 Frahm H, Its A R & Korepin V E 1994 1 Differential equation for a correlation function of the spin- 2 Heisenberg chain Nucl. Phys. B 428 FS 694 Korepin V E, Izergin A G, Eßler F H L, Uglov D B 1994 Correlation Function of the Spin-1/2 XXX Antiferromagnet Phys. Lett. A 190 182-184 Eisert J, Cramer M & Plenio M B Area laws for the entanglement entropy - a review. Pre-print arXiv:0808.3773

V. Ko r e p in F-H Formula Appendix For Further Reading For Further Reading III

Jin B -Q & Korepin V E Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture. Jour. Stat. Phys. 116 79-95 2004 Its A R, Jin B -Q & Korepin V E Entropy of XY Spin Chain and Block Toeplitz Determinants. Fields Institute Communications, Universality and Renormalization [editors Bender I & Kreimer D] 50 151, 2007

V. Ko r e p in F-H Formula