Highly entangled quantum spin chains Fumihiko Sugino Center for - PowerPoint PPT Presentation

Highly entangled quantum spin chains Fumihiko Sugino Center for Theoretical Physics of the Universe, Institute for Basic Science 10th Mathematical Physics Meeting: School and Conference on Modern Mathematical Physics, Belgrade, Sept. 10, 2019 Mainly based on Bravyi et al, Phys. Rev. Lett. 118 (2012) 207202, arXiv: 1203.5801 R. Movassagh and P. Shor, Proc. Natl. Acad. Sci. 113 (2016) 13278, arXiv: 1408.1657 F.S. and V. Korepin, Int. J. Mod. Phys. B 32 (2018) no.28, 1850306, arXiv:1806.04049

Outline Introduction Motzkin spin model Colored Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion

Introduction 1 Quantum entanglement ◮ Most surprising feature of quantum mechanics, No analog in classical mechanics

Introduction 1 Quantum entanglement ◮ Most surprising feature of quantum mechanics, No analog in classical mechanics ◮ From pure state of the full system S : ρ = | ψ �� ψ | , reduced density matrix of a subsystem A: ρ A = Tr S − A ρ can become mixed states, and has nonzero entanglement entropy S A = − Tr A [ ρ A ln ρ A ] . This is purely a quantum property.

Introduction 2 Area law of entanglement entropy ◮ Ground states of quantum many-body systems with local interactions typically exhibit the area law behavior of the entanglement entropy: S A ∝ (area of A ) ◮ Gapped systems in 1D are proven to obey the area law. [Hastings 2007]

Introduction 2 Area law of entanglement entropy ◮ Ground states of quantum many-body systems with local interactions typically exhibit the area law behavior of the entanglement entropy: S A ∝ (area of A ) ◮ Gapped systems in 1D are proven to obey the area law. [Hastings 2007] (Area law violation) ⇒ Gapless ◮ For gapless case, (1 + 1)-dimensional CFT violates logarithmically: S A = c 3 ln (volume of A ). [Calabrese, Cardy 2009]

Introduction 2 Area law of entanglement entropy ◮ Ground states of quantum many-body systems with local interactions typically exhibit the area law behavior of the entanglement entropy: S A ∝ (area of A ) ◮ Gapped systems in 1D are proven to obey the area law. [Hastings 2007] (Area law violation) ⇒ Gapless ◮ For gapless case, (1 + 1)-dimensional CFT violates logarithmically: S A = c 3 ln (volume of A ). [Calabrese, Cardy 2009] ◮ Belief for gapless case in D -dim. (over two decades) : S A = O ( L D − 1 ln L ) ( L : length scale of A )

Introduction 2 Area law of entanglement entropy ◮ Ground states of quantum many-body systems with local interactions typically exhibit the area law behavior of the entanglement entropy: S A ∝ (area of A ) ◮ Gapped systems in 1D are proven to obey the area law. [Hastings 2007] (Area law violation) ⇒ Gapless ◮ For gapless case, (1 + 1)-dimensional CFT violates logarithmically: S A = c 3 ln (volume of A ). [Calabrese, Cardy 2009] ◮ Belief for gapless case in D -dim. (over two decades) : S A = O ( L D − 1 ln L ) ( L : length scale of A ) ◮ Recently, 1D solvable spin chain models which exhibit significant area-law violation have been discovered. ◮ Beyond logarithmic violation: S A ∝ � (volume of A ) [Movassagh, Shor 2014] , [Salberger, Korepin 2016] Counterexamples of the belief!

Introduction Motzkin spin model Colored Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion

Motzkin spin model 1 [Bravyi et al 2012] ◮ 1D spin chain at sites i ∈ { 1 , 2 , · · · , 2 n } ◮ Spin-1 state at each site can be regarded as up, down and flat steps; | u � ⇔ , | d � ⇔ , | 0 � ⇔

Motzkin spin model 1 [Bravyi et al 2012] ◮ 1D spin chain at sites i ∈ { 1 , 2 , · · · , 2 n } ◮ Spin-1 state at each site can be regarded as up, down and flat steps; | u � ⇔ , | d � ⇔ , | 0 � ⇔ ◮ Each spin configuration ⇔ length-2 n walk in ( x , y ) plane Example) y | u � 1 | 0 � 2 | d � 3 | u � 4 | u � 5 | d � 6 x

Motzkin spin model 2 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy , H bdy = | d � 1 � d | + | u � 2 n � u |

Motzkin spin model 2 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy , H bdy = | d � 1 � d | + | u � 2 n � u | ◮ Bulk part: H bulk = � 2 n − 1 j =1 Π j , j +1 , Π j , j +1 = | D � j , j +1 � D | + | U � j , j +1 � U | + | F � j , j +1 � F | (local interactions) with 1 √ | D � ≡ ( | 0 , d � − | d , 0 � ) , 2 1 √ | U � ≡ ( | 0 , u � − | u , 0 � ) , 2 1 √ | F � ≡ ( | 0 , 0 � − | u , d � ) . 2

Motzkin spin model 2 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy , H bdy = | d � 1 � d | + | u � 2 n � u | ◮ Bulk part: H bulk = � 2 n − 1 j =1 Π j , j +1 , Π j , j +1 = | D � j , j +1 � D | + | U � j , j +1 � U | + | F � j , j +1 � F | (local interactions) with ⇔ ∼ 1 √ | D � ≡ ( | 0 , d � − | d , 0 � ) , 2 1 √ | U � ≡ ( | 0 , u � − | u , 0 � ) , 2 1 √ | F � ≡ ( | 0 , 0 � − | u , d � ) . 2

Motzkin spin model 2 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy , H bdy = | d � 1 � d | + | u � 2 n � u | ◮ Bulk part: H bulk = � 2 n − 1 j =1 Π j , j +1 , Π j , j +1 = | D � j , j +1 � D | + | U � j , j +1 � U | + | F � j , j +1 � F | (local interactions) with ⇔ ∼ 1 √ | D � ≡ ( | 0 , d � − | d , 0 � ) , 2 ⇔ ∼ 1 √ | U � ≡ ( | 0 , u � − | u , 0 � ) , 2 1 √ | F � ≡ ( | 0 , 0 � − | u , d � ) . 2

Motzkin spin model 2 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy , H bdy = | d � 1 � d | + | u � 2 n � u | ◮ Bulk part: H bulk = � 2 n − 1 j =1 Π j , j +1 , Π j , j +1 = | D � j , j +1 � D | + | U � j , j +1 � U | + | F � j , j +1 � F | (local interactions) with ⇔ ∼ 1 √ | D � ≡ ( | 0 , d � − | d , 0 � ) , 2 ⇔ ∼ 1 √ | U � ≡ ( | 0 , u � − | u , 0 � ) , 2 ⇔ ∼ 1 √ | F � ≡ ( | 0 , 0 � − | u , d � ) . 2 “gauge equivalence”.

Motzkin spin model 3 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy ⇓

Motzkin spin model 3 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy ⇓ ◮ H Motzkin is the sum of projection operators. ⇒ Positive semi-definite spectrum ◮ We find the unique zero-energy ground state.

Motzkin spin model 3 [Bravyi et al 2012] Hamiltonian: H Motzkin = H bulk + H bdy ⇓ ◮ H Motzkin is the sum of projection operators. ⇒ Positive semi-definite spectrum ◮ We find the unique zero-energy ground state. ◮ Each projector in H Motzkin annihilates the zero-energy state. ⇒ Frustration free ◮ The ground state corresponds to randoms walks starting at (0 , 0) and ending at (2 n , 0) restricted to the region y ≥ 0 (Motzkin Walks (MWs)).

Motzkin spin model 4 [Bravyi et al 2012] Example) 2 n = 4 case, MWs: + + + + + + + + � Ground state: 1 | P 4 � = √ [ | 0000 � + | ud 00 � + | 0 ud 0 � + | 00 ud � 9 + | u 0 d 0 � + | 0 u 0 d � + | u 00 d � + | udud � + | uudd � ] .

Motzkin spin model 5 [Bravyi et al 2012] Note Forbidden paths for the ground state 1. Path entering y < 0 region ∼ Forbidden by H bdy 2. Path ending at nonzero height ∼ ∼ ∼ Forbidden by H bdy

Motzkin spin model 6 [Bravyi et al 2012] In terms of S = 1 spin matrices       1 1 1  ,  ,  , S z = 0 S ± ≡ √ ( S x ± iS y ) = 1  1   2 − 1 1 2 n − 1 H bulk = 1 � 1 j 1 j +1 − 1 4 S z j S z j +1 − 1 z j S z j +1 + 1 � 4 S 2 4 S z j S 2 z j +1 2 j =1 − 3 4 S 2 z j S 2 z j +1 + S + j ( S z S − ) j +1 + S − j ( S + S z ) j +1 − ( S − S z ) j S + j +1 � − ( S z S + ) j S − j +1 − ( S − S z ) j ( S + S z ) j +1 − ( S z S + ) j ( S z S − ) j +1 , H bdy = 1 1 + 1 S 2 S 2 � z − S z � � � z + S z 2 n 2 2 Quartic spin interactions

Motzkin spin model 7 [Bravyi et al 2012] Entanglement entropy of the subsystem A = { 1 , 2 , · · · , n } : ◮ Normalization factor of the ground state | P 2 n � is given by � 2 n � the number of MWs of length 2 n : M 2 n = � n . k =0 C k 2 k � 2 k � 1 C k = : Catalan number k +1 k

Motzkin spin model 7 [Bravyi et al 2012] Entanglement entropy of the subsystem A = { 1 , 2 , · · · , n } : ◮ Normalization factor of the ground state | P 2 n � is given by � 2 n � the number of MWs of length 2 n : M 2 n = � n . k =0 C k 2 k � 2 k � 1 C k = : Catalan number k +1 k ◮ Consider to trace out the density matrix ρ = | P 2 n �� P 2 n | w.r.t. the subsystem B = { n + 1 , · · · , 2 n } . Schmidt decomposition: � � � � � � p ( h ) � P (0 → h ) � P ( h → 0) | P 2 n � = ⊗ � � n , n n n h ≥ 0 � 2 � M ( h ) with p ( h ) n n , n ≡ . ↑ M 2 n Paths from (0 , 0) to ( n , h )

Motzkin spin model 8 [Bravyi et al 2012] ◮ M ( h ) is the number of paths in P (0 → h ) . n n For n → ∞ , Gaussian distribution √ ( h + 1) 2 n , n ∼ 3 6 ( h +1)2 p ( h ) e − 3 √ π × [1 + O (1 / n )] . 2 n n 3 / 2 ◮ Reduced density matrix � � � � p ( h ) � P (0 → h ) P (0 → h ) � ρ A = Tr B ρ = � � n , n n n � h ≥ 0 ◮ Entanglement entropy � p ( h ) n , n ln p ( h ) = − S A n , n h ≥ 0 1 2 ln n + 1 2 ln 2 π 3 + γ − 1 = ( γ : Euler constant) 2 up to terms vanishing as n → ∞ .