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 → ∞ .
Motzkin spin model 9 [Bravyi et al 2012] Notes ◮ The system is critical (gapless). S A is similar to the (1 + 1)-dimensional CFT with c = 3 / 2.
Motzkin spin model 9 [Bravyi et al 2012] Notes ◮ The system is critical (gapless). S A is similar to the (1 + 1)-dimensional CFT with c = 3 / 2. ◮ But, gap scales as O (1 / n z ) with z ≥ 2. The system cannot be described by relativistic CFT. Lifshitz type ? Different z depending on excited states (Multiple dynamics)? [Chen, Fradkin, Witczak-Krempa 2017]
Motzkin spin model 9 [Bravyi et al 2012] Notes ◮ The system is critical (gapless). S A is similar to the (1 + 1)-dimensional CFT with c = 3 / 2. ◮ But, gap scales as O (1 / n z ) with z ≥ 2. The system cannot be described by relativistic CFT. Lifshitz type ? Different z depending on excited states (Multiple dynamics)? [Chen, Fradkin, Witczak-Krempa 2017] ◮ Excitations have not been much investigated.
Introduction Motzkin spin model Colored Motzkin model SIS Motzkin model Colored SIS Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion
Colored Motzkin spin model 1 [Movassagh, Shor 2014] ◮ Introducing color d.o.f. k = 1 , 2 , · · · , s to up and down spins as k k , � u k � � � � d k � ⇔ , ⇔ | 0 � ⇔ � � Color d.o.f. decorated to Motzkin Walks
Colored Motzkin spin model 1 [Movassagh, Shor 2014] ◮ Introducing color d.o.f. k = 1 , 2 , · · · , s to up and down spins as k k , � � u k � � � d k � ⇔ , ⇔ | 0 � ⇔ � � Color d.o.f. decorated to Motzkin Walks ◮ Hamiltonian H cMotzkin = H bulk + H bdy ◮ Bulk part consisting of local interactions: 2 n − 1 � Π j , j +1 + Π cross � � H bulk = , j , j +1 j =1 s �� � � � + � + � D k � � D k � � U k � � � U k � � � F k � � F k � Π j , j +1 = � j , j +1 j , j +1 j , j +1 k =1 with
Colored Motzkin spin model 2 [Movassagh, Shor 2014] 1 � �� � � D k � � 0 , d k � �� � d k , 0 √ ≡ − , � � � 2 1 � � U k � �� � 0 , u k � � �� � u k , 0 ≡ √ − , � � � 2 1 � � � F k � � � u k , d k �� ≡ √ | 0 , 0 � − , � � 2 and � u k , d k ′ � � � u k , d k ′ � Π cross � j , j +1 = � . � � j , j +1 k � = k ′ ⇒ Colors should be matched in up and down pairs. ◮ Boundary part s �� d k � � u k � � d k � � � u k � � � � H bdy = � + . � � � � � 1 2 n k =1
Colored Motzkin spin model 3 [Movassagh, Shor 2014] ◮ Still unique ground state with zero energy
Colored Motzkin spin model 3 [Movassagh, Shor 2014] ◮ Still unique ground state with zero energy ◮ Example) 2 n = 4 case, + k k k k k k + + k k ′ k ′ + k k k k + k k + k + k ′ k ′ + k k � s 1 �� � � u k 00 d k �� � � � u k d k 00 | P 4 � = √ | 0000 � + + · · · + � � 1 + 6 s + 2 s 2 k =1 s � u k u k ′ d k ′ d k ��� �� � � u k d k u k ′ d k ′ � � + + . � � k , k ′ =1
Colored Motzkin spin model 4 [Movassagh, Shor 2014] Entanglement entropy ◮ Paths from (0 , 0) to ( n , h ), P (0 → h ) , have h unmatched up n steps. P (0 → h ) Let ˜ ( { κ m } ) be paths with the colors of unmatched up n steps frozen. (unmatched up from height ( m − 1) to m ) → u κ m ◮ Similarly, P ( h → 0) P ( h → 0) → ˜ ( { κ m } ) , n n (unmatched down from height m to ( m − 1)) → d κ m . ◮ The numbers satisfy M ( h ) = s h ˜ M ( h ) n . n
Colored Motzkin spin model 5 [Movassagh, Shor 2014] Example 2 n = 8 case, h = 2 y A B 3 k ′ k ′ 2 u κ 2 d κ 2 k k 1 u κ 1 d κ 1 x 0 1 2 3 4 5 6 7 8
Colored Motzkin spin model 6 [Movassagh, Shor 2014] ◮ Schmidt decomposition s s � p ( h ) � � � | P 2 n � = · · · n , n h ≥ 0 κ 1 =1 κ h =1 � � � � P (0 → h ) P ( h → 0) � ˜ � ˜ × ( { κ m } ) ⊗ ( { κ m } ) � � n n with � 2 � M ( h ) ˜ n p ( h ) n , n = . M 2 n ◮ Reduced density matrix s s p ( h ) � � � · · · ρ A = n , n h ≥ 0 κ 1 =1 κ h =1 � � P (0 → h ) �� P (0 → h ) � ˜ ˜ × ( { κ m } ) ( { κ m } ) � . � � n n
Colored Motzkin spin model 7 [Movassagh, Shor 2014] ◮ For n → ∞ , √ 2 s − h √ π ( σ n ) 3 / 2 ( h + 1) 2 e − ( h +1)2 p ( h ) n , n ∼ × [1 + O (1 / n )] 2 σ n √ s Effectively h � O ( √ n ). with σ ≡ 2 √ s +1 . Note: ◮ Entanglement entropy s h p ( h ) n , n ln p ( h ) � S A = − n , n h ≥ 0
Colored Motzkin spin model 7 [Movassagh, Shor 2014] ◮ For n → ∞ , √ 2 s − h √ π ( σ n ) 3 / 2 ( h + 1) 2 e − ( h +1)2 p ( h ) n , n ∼ × [1 + O (1 / n )] 2 σ n √ s Effectively h � O ( √ n ). with σ ≡ 2 √ s +1 . Note: ◮ Entanglement entropy s h p ( h ) n , n ln p ( h ) � S A = − n , n h ≥ 0 � 2 σ n + 1 2 ln n + 1 2 ln(2 πσ ) + γ − 1 2 − ln s = (2 ln s ) π Grows as √ n . up to terms vanishing as n → ∞ .
Colored Motzkin spin model 8 [Movassagh, Shor 2014] Comments s − h factor in p ( h ) Matching color ⇒ n , n crucial to O ( √ n ) behavior in S A ◮ ⇒
Colored Motzkin spin model 8 [Movassagh, Shor 2014] Comments s − h factor in p ( h ) Matching color ⇒ n , n crucial to O ( √ n ) behavior in S A ◮ ⇒ ◮ Typical configurations: k ′ k ′ h = O ( √ n ) k k + (equivalence moves) .
Colored Motzkin spin model 8 [Movassagh, Shor 2014] Comments s − h factor in p ( h ) Matching color ⇒ n , n crucial to O ( √ n ) behavior in S A ◮ ⇒ ◮ Typical configurations: k ′ k ′ h = O ( √ n ) k k + (equivalence moves) . ◮ For spin 1 / 2 chain (only up and down), the model in which similar behavior exhibits in colored as well as uncolored cases has been constructed. (Fredkin model) [Salberger, Korepin 2016]
Colored Motzkin spin model 9 [Movassagh, Shor 2014] ◮ Correlation functions [Dell’Anna et al, 2016] � S z , 1 S z , 2 n � connected → − 0 . 034 ... × s 3 − s � = 0 ( n → ∞ ) 6 ⇒ Violation of cluster decomposition property for s > 1 (Strong correlation due to color matching)
Colored Motzkin spin model 9 [Movassagh, Shor 2014] ◮ Correlation functions [Dell’Anna et al, 2016] � S z , 1 S z , 2 n � connected → − 0 . 034 ... × s 3 − s � = 0 ( n → ∞ ) 6 ⇒ Violation of cluster decomposition property for s > 1 (Strong correlation due to color matching) ◮ Deformation of models to achieve the volume law behavior ( S A ∝ n ) Weighted Motzkin/Dyck walks [Zhang et al, Salberger et al 2016]
Introduction Motzkin spin model Colored Motzkin model SIS Motzkin model Colored SIS Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion
Symmetric Inverse Semigroups (SISs) ◮ Inverse Semigroup ( ⊂ Semigroup): An unique inverse exists for every element. But, no unique identity (partial identities).
Symmetric Inverse Semigroups (SISs) ◮ Inverse Semigroup ( ⊂ Semigroup): An unique inverse exists for every element. But, no unique identity (partial identities). ◮ SIS ( ⊂ Semigroup): Semigroup version of the symmetric group S k S k p ( p = 1 , · · · , k )
Symmetric Inverse Semigroups (SISs) ◮ Inverse Semigroup ( ⊂ Semigroup): An unique inverse exists for every element. But, no unique identity (partial identities). ◮ SIS ( ⊂ Semigroup): Semigroup version of the symmetric group S k S k p ( p = 1 , · · · , k ) ◮ x a , b ∈ S k 1 maps a to b . ( a , b ∈ { 1 , · · · , k } ) Product rule: x a , b ∗ x c , d = δ b , c x a , d x 1 , 2 ∗ x 2 , 1 = x 1 , 1 , x 2 , 1 ∗ x 1 , 2 = x 2 , 2 տ ր (partial identities) ( x 1 , 2 ) − 1 = x 2 , 1 (unique inverse)
Symmetric Inverse Semigroups (SISs) ◮ Inverse Semigroup ( ⊂ Semigroup): An unique inverse exists for every element. But, no unique identity (partial identities). ◮ SIS ( ⊂ Semigroup): Semigroup version of the symmetric group S k S k p ( p = 1 , · · · , k ) ◮ x a , b ∈ S k 1 maps a to b . ( a , b ∈ { 1 , · · · , k } ) Product rule: x a , b ∗ x c , d = δ b , c x a , d x 1 , 2 ∗ x 2 , 1 = x 1 , 1 , x 2 , 1 ∗ x 1 , 2 = x 2 , 2 տ ր (partial identities) ( x 1 , 2 ) − 1 = x 2 , 1 (unique inverse) ◮ x a 1 , a 2 ; b 1 , b 2 ∈ S k 2 etc, ...
Symmetric Inverse Semigroups (SISs) ◮ Inverse Semigroup ( ⊂ Semigroup): An unique inverse exists for every element. But, no unique identity (partial identities). ◮ SIS ( ⊂ Semigroup): Semigroup version of the symmetric group S k S k p ( p = 1 , · · · , k ) ◮ x a , b ∈ S k 1 maps a to b . ( a , b ∈ { 1 , · · · , k } ) Product rule: x a , b ∗ x c , d = δ b , c x a , d x 1 , 2 ∗ x 2 , 1 = x 1 , 1 , x 2 , 1 ∗ x 1 , 2 = x 2 , 2 տ ր (partial identities) ( x 1 , 2 ) − 1 = x 2 , 1 (unique inverse) ◮ x a 1 , a 2 ; b 1 , b 2 ∈ S k S k 2 etc, ... k ≡ S k
SIS Motzkin model 1 [Sugino, Padmanabhan 2017] ◮ Change the spin d.o.f. as | x a , b � with a , b ∈ { 1 , 2 , · · · , k } . b ◮ a < b case: ‘up’ ⇔ a a b a > b case: ‘down’ ⇔ a = b case: ‘flat’ ⇔ a b
SIS Motzkin model 1 [Sugino, Padmanabhan 2017] ◮ Change the spin d.o.f. as | x a , b � with a , b ∈ { 1 , 2 , · · · , k } . b ◮ a < b case: ‘up’ ⇔ a a b a > b case: ‘down’ ⇔ a = b case: ‘flat’ ⇔ a b ◮ We regard the configuration of adjacent sites | ( x a , b ) j � | ( x c , d ) j +1 � as a connected path for b = c . c.f.) Analogous to the product rule of Symmetric Inverse Semigroup ( S k 1 ): x a , b ∗ x c , d = δ b , c x a , d a , b : semigroup indices ◮ Inner product: � x a , b | x c , d � = δ a , c δ b , d ◮ Let us consider the k = 3 case.
SIS Motzkin model 2 [Sugino, Padmanabhan 2017] ◮ Maximum height is lower than the original Motzkin case. y 3 3 2 3 2 2 1 1 1 x 0 1 2 3 4 5
SIS Motzkin model 3 [Sugino, Padmanabhan 2017] Hamiltonian H S 31 Motzkin = H bulk + H bulk , disc + H bdy ◮ H bulk : local interactions corresponding to the following moves: a a a b b b ( Down ) ∼ ( a > b ) b b b a a a ( Up ) ∼ ( a < b ) b a a a a a ∼ ( Flat ) ( a < b ) 3 3 3 3 1 2 ( Wedge ) ∼
SIS Motzkin model 4 [Sugino, Padmanabhan 2017] ◮ H bulk , disc lifts disconnected paths to excited states. Π | ψ � : projector to | ψ � 2 n − 1 3 Π | ( x a , b ) j , ( x c , d ) j +1 � � � H bulk , disc = j =1 a , b , c , d =1; b � = c
SIS Motzkin model 4 [Sugino, Padmanabhan 2017] ◮ H bulk , disc lifts disconnected paths to excited states. Π | ψ � : projector to | ψ � 2 n − 1 3 Π | ( x a , b ) j , ( x c , d ) j +1 � � � H bulk , disc = j =1 a , b , c , d =1; b � = c ◮ Π | ( x a , b ) 1 � + Π | ( x a , b ) 2 n � � � H bdy = a > b a < b +Π | ( x 1 , 3 ) 1 , ( x 3 , 2 ) 2 , ( x 2 , 1 ) 3 � + Π | ( x 1 , 2 ) 2 n − 2 , ( x 2 , 3 ) 2 n − 1 , ( x 3 , 1 ) 2 n � The last 2 terms have no analog to the original Motzkin model.
SIS Motzkin model 5 [Sugino, Padmanabhan 2017] ◮ Ground states correspond to connected paths starting at S 3 (0 , 0), ending at (2 n , 0) and not entering y < 0. 1 MWs
SIS Motzkin model 5 [Sugino, Padmanabhan 2017] ◮ Ground states correspond to connected paths starting at S 3 (0 , 0), ending at (2 n , 0) and not entering y < 0. 1 MWs ◮ The ground states have 5 fold degeneracy according to the initial and finial semigroup indices: (1 , 1), (1 , 2), (2 , 1), (2 , 2) and (3 , 3) sectors The (3 , 3) sector is trivial, consisting of only one path: x 3 , 3 x 3 , 3 · · · x 3 , 3 .
SIS Motzkin model 5 [Sugino, Padmanabhan 2017] ◮ Ground states correspond to connected paths starting at S 3 (0 , 0), ending at (2 n , 0) and not entering y < 0. 1 MWs ◮ The ground states have 5 fold degeneracy according to the initial and finial semigroup indices: (1 , 1), (1 , 2), (2 , 1), (2 , 2) and (3 , 3) sectors The (3 , 3) sector is trivial, consisting of only one path: x 3 , 3 x 3 , 3 · · · x 3 , 3 . ◮ The number of paths can be obtained by recursion relations. For length- n paths from the semigroup index a to b ( P n , a → b ), n − 2 � P n , 1 → 1 = x 1 , 1 P n − 1 , 1 → 1 + x 1 , 2 P i , 2 → 2 x 2 , 1 P n − 2 − i , 1 → 1 i =1 n − 2 � + x 1 , 3 P i , 3 → 3 x 3 , 1 P n − 2 − i , 1 → 1 i =1 n − 2 � + x 1 , 3 P i , 3 → 3 x 3 , 2 P n − 2 − i , 2 → 1 , etc . i =1
SIS Motzkin model 6 [Sugino, Padmanabhan 2017] Result ◮ The entanglement entropies S A , 1 → 1 , S A , 1 → 2 , S A , 2 → 1 and S A , 2 → 2 take the same form as in the case of the Motzkin model. Logarithmic violation of the area law n 3 / 2 e − ( const . ) ( h +1)2 ∼ ( h +1) 2 ◮ The form of p ( h ) is universal. n n ◮ S A , 3 → 3 = 0.
SIS Motzkin model 7 Localization [Padmanabhan, F.S., Korepin 2018] ◮ There are excited states corresponding to disconnected paths. Example) One such path in 2 n = 6 case, y 3 2 2 2 2 1 1 1 x
SIS Motzkin model 7 Localization [Padmanabhan, F.S., Korepin 2018] ◮ There are excited states corresponding to disconnected paths. Example) One such path in 2 n = 6 case, y 3 2 2 2 2 1 1 1 x � � P (1 → 0) � Corresponding excited state: | P 3 , 1 → 1 �⊗ � 3 , 2 → 1 Each connected component has no entanglement with other components. “2nd quantization” of paths
SIS Motzkin model 7 Localization [Padmanabhan, F.S., Korepin 2018] ◮ There are excited states corresponding to disconnected paths. Example) One such path in 2 n = 6 case, y 3 2 2 2 2 1 1 1 x � � P (1 → 0) � Corresponding excited state: | P 3 , 1 → 1 �⊗ � 3 , 2 → 1 Each connected component has no entanglement with other components. “2nd quantization” of paths ⇒ 2pt connected correlation functions of local operators belonging to separate connected components vanish. ⇒ Localization!
Introduction Motzkin spin model Colored Motzkin model SIS Motzkin model Colored SIS Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion
Colored SIS Motzkin model 1 [Sugino, Padmanabhan 2017] The SIS S 3 2 ◮ 18 elements x ab , cd with ab ∈ { 12 , 23 , 31 } and cd ∈ { 12 , 23 , 31 , 21 , 32 , 13 } satisfying x ab , cd ∗ x ef , gh = δ c , e δ d , f x ab , gh + δ c , f δ d , e x ab , hg . ◮ can be regarded as 2 sets of S 3 1 . ⇒ color d.o.f.
Colored SIS Motzkin model 1 [Sugino, Padmanabhan 2017] The SIS S 3 2 ◮ 18 elements x ab , cd with ab ∈ { 12 , 23 , 31 } and cd ∈ { 12 , 23 , 31 , 21 , 32 , 13 } satisfying x ab , cd ∗ x ef , gh = δ c , e δ d , f x ab , gh + δ c , f δ d , e x ab , hg . ◮ can be regarded as 2 sets of S 3 1 . ⇒ color d.o.f. ◮ Spin variables: x s a , b ( s = 1 , 2) ( a , b = 1 , 2 , 3) ◮ The new moves ( C moves) introduced to the Hamiltonian. 1 2 a a ∼ a a
Colored SIS Motzkin model 2 [Sugino, Padmanabhan 2017] Hamiltonian: H cS 31 Motzkin = H bulk + H bulk , disc + H bdy ◮ In H bulk , (Down), (Up) and (Flat) are essentially the same as before. s s a a s a s b b b ( Down ) ∼ ( a > b ) s s s b b b s a a a ( Up ) ∼ ( a < b ) s s b s s a a a a a ( Flat ) ∼ ( a < b )
Colored SIS Motzkin model 3 [Sugino, Padmanabhan 2017] ◮ Wedge move: s ′ s ′ s s 3 3 3 3 1 2 ( Wedge ) ∼ ◮ � Π | ( x 1 a , b ) j , ( x 2 b , c ) j +1 � + Π | ( x 2 a , b ) j , ( x 1 b , c ) j +1 � � � ( Cross ) j , j +1 = b > a , c forbids unmatched up and down steps in ground states. ⇓ 2 n 2 n − 1 � � = µ C j + [( Down ) j , j +1 + ( Up ) j , j +1 H bulk j =1 j =1 +( Flat ) j , j +1 + ( Wedge ) j , j +1 + ( Cross ) j , j +1 ]
Colored SIS Motzkin model 4 [Sugino, Padmanabhan 2017] ◮ 2 n − 1 3 2 Π | ( x s c , d ) j +1 � a , b ) j , ( x t � � � H bulk , disc = s , t =1 j =1 a , b , c , d =1; b � = c ◮ 2 2 Π | ( x s a , b ) 1 � + Π | ( x s a , b ) 2 n � � � � � H bdy = s =1 s =1 a > b a < b 2 Π | ( x s 1 , 3 ) 1 , ( x s 3 , 2 ) 2 , ( x t 2 , 1 ) 3 � � + s , t =1 2 Π | ( x s 3 , 1 ) 2 n � 1 , 2 ) 2 n − 2 , ( x t 2 , 3 ) 2 n − 1 , ( x t � + s , t =1
Colored SIS Motzkin model 5 [Sugino, Padmanabhan 2017] ◮ 5 ground states of (1 , 1), (1 , 2), (2 , 1), (2 , 2), (3 , 3) sectors ◮ Quantum phase transition between µ > 0 and µ = 0 in the 4 sectors except (3 , 3). ◮ For µ > 0, � 2 σ n + 1 2 ln n + 1 2 ln(2 πσ ) + γ − 1 3 S A = (2 ln 2) 2 + ln π 2 1 / 3 √ 2 − 1 with σ ≡ 2 . √ 9 ◮ For µ = 0, colors 1 and 2 decouple. S A ∝ ln n .
Introduction Motzkin spin model Colored Motzkin model SIS Motzkin model Colored SIS Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion
R´ enyi entropy [R´ enyi, 1970] ◮ R´ enyi entropy has further importance than the von Neumann entanglement entropy: 1 1 − α ln Tr A ρ α S A , α = with α > 0 and α � = 1 . A
R´ enyi entropy [R´ enyi, 1970] ◮ R´ enyi entropy has further importance than the von Neumann entanglement entropy: 1 1 − α ln Tr A ρ α S A , α = with α > 0 and α � = 1 . A ◮ Generalization of the von Neumann entanglement entropy: lim α → 1 S A , α = S A
R´ enyi entropy [R´ enyi, 1970] ◮ R´ enyi entropy has further importance than the von Neumann entanglement entropy: 1 1 − α ln Tr A ρ α S A , α = with α > 0 and α � = 1 . A ◮ Generalization of the von Neumann entanglement entropy: lim α → 1 S A , α = S A ◮ Reconstructs the whole spectrum of the entanglement Hamiltonian H ent , A ≡ − ln ρ A .
R´ enyi entropy [R´ enyi, 1970] ◮ R´ enyi entropy has further importance than the von Neumann entanglement entropy: 1 1 − α ln Tr A ρ α S A , α = with α > 0 and α � = 1 . A ◮ Generalization of the von Neumann entanglement entropy: lim α → 1 S A , α = S A ◮ Reconstructs the whole spectrum of the entanglement Hamiltonian H ent , A ≡ − ln ρ A . ◮ For S A , α (0 < α < 1), the gapped systems in 1D is proven to obey the area law. [Huang, 2015]
R´ enyi entropy [R´ enyi, 1970] ◮ R´ enyi entropy has further importance than the von Neumann entanglement entropy: 1 1 − α ln Tr A ρ α S A , α = with α > 0 and α � = 1 . A ◮ Generalization of the von Neumann entanglement entropy: lim α → 1 S A , α = S A ◮ Reconstructs the whole spectrum of the entanglement Hamiltonian H ent , A ≡ − ln ρ A . ◮ For S A , α (0 < α < 1), the gapped systems in 1D is proven to obey the area law. [Huang, 2015] Here, I give a review of Motzkin spin chain and analytically compute its R´ enyi entropy of half-chain. New phase transition found at α = 1!
Introduction Motzkin spin model Colored Motzkin model SIS Motzkin model Colored SIS Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion
R´ eyni entropy of Motzkin model 1 [F.S., Korepin, 2018] ◮ What we compute is the asymptotic behavior of n 1 � α s h � � p ( h ) S A , α = 1 − α ln . n , n h =0
R´ eyni entropy of Motzkin model 1 [F.S., Korepin, 2018] ◮ What we compute is the asymptotic behavior of n 1 � α s h � � p ( h ) S A , α = 1 − α ln . n , n h =0 ◮ For colorless case ( s = 1), we obtain 1 1 � α + 1 � S A ,α = 2 ln n + 1 − α ln Γ 2 1 (1 + 2 α ) ln α + α ln π � � − 24 + ln 6 2(1 − α ) up to terms vanishing as n → ∞ .
R´ eyni entropy of Motzkin model 1 [F.S., Korepin, 2018] ◮ What we compute is the asymptotic behavior of n 1 � α s h � � p ( h ) S A , α = 1 − α ln . n , n h =0 ◮ For colorless case ( s = 1), we obtain 1 1 � α + 1 � S A ,α = 2 ln n + 1 − α ln Γ 2 1 (1 + 2 α ) ln α + α ln π � � − 24 + ln 6 2(1 − α ) up to terms vanishing as n → ∞ . ◮ Logarithmic growth ◮ Reduces to S A in the α → 1 limit. ◮ Consistent with half-chain case in the result in [Movassagh, 2017]
R´ eyni entropy of Motzkin model 2 [F.S., Korepin, 2018] Colored case ( s > 1) � α ◮ The summand s h � p ( h ) has a factor s (1 − α ) h . n , n
R´ eyni entropy of Motzkin model 2 [F.S., Korepin, 2018] Colored case ( s > 1) � α ◮ The summand s h � p ( h ) has a factor s (1 − α ) h . n , n For 0 < α < 1, exponentially growing (colored case ( s > 1)). ⇒ Saddle point value of the sum: h ∗ = O ( n )
R´ eyni entropy of Motzkin model 2 [F.S., Korepin, 2018] Colored case ( s > 1) � α ◮ The summand s h � p ( h ) has a factor s (1 − α ) h . n , n For 0 < α < 1, exponentially growing (colored case ( s > 1)). ⇒ Saddle point value of the sum: h ∗ = O ( n ) ◮ Saddle point analysis for the sum leads to S A ,α = n 2 α � � 2 α + s − 1 / 2 �� 1 − α 2 α + s − 1 − α 1 − α ln σ s 1 + α + 2(1 − α ) ln n + C ( s , α ) with C ( s , α ) being n -independent terms.
R´ eyni entropy of Motzkin model 2 [F.S., Korepin, 2018] Colored case ( s > 1) � α ◮ The summand s h � p ( h ) has a factor s (1 − α ) h . n , n For 0 < α < 1, exponentially growing (colored case ( s > 1)). ⇒ Saddle point value of the sum: h ∗ = O ( n ) ◮ Saddle point analysis for the sum leads to S A ,α = n 2 α � � 2 α + s − 1 / 2 �� 1 − α 2 α + s − 1 − α 1 − α ln σ s 1 + α + 2(1 − α ) ln n + C ( s , α ) with C ( s , α ) being n -independent terms. 2 α − s 1 − 1 1 ◮ The saddle point value is h ∗ = n s 2 α +1 + O ( n 0 ). 2 α 2 α + s 1 − 1 1 s
R´ eyni entropy of Motzkin model 2 [F.S., Korepin, 2018] Colored case ( s > 1) � α ◮ The summand s h � p ( h ) has a factor s (1 − α ) h . n , n For 0 < α < 1, exponentially growing (colored case ( s > 1)). ⇒ Saddle point value of the sum: h ∗ = O ( n ) ◮ Saddle point analysis for the sum leads to S A ,α = n 2 α � � 2 α + s − 1 / 2 �� 1 − α 2 α + s − 1 − α 1 − α ln σ s 1 + α + 2(1 − α ) ln n + C ( s , α ) with C ( s , α ) being n -independent terms. 2 α − s 1 − 1 1 ◮ The saddle point value is h ∗ = n s 2 α +1 + O ( n 0 ). 2 α 2 α + s 1 − 1 1 s ◮ Linear growth in n .
R´ eyni entropy of Motzkin model 2 [F.S., Korepin, 2018] Colored case ( s > 1) � α ◮ The summand s h � p ( h ) has a factor s (1 − α ) h . n , n For 0 < α < 1, exponentially growing (colored case ( s > 1)). ⇒ Saddle point value of the sum: h ∗ = O ( n ) ◮ Saddle point analysis for the sum leads to S A ,α = n 2 α � � 2 α + s − 1 / 2 �� 1 − α 2 α + s − 1 − α 1 − α ln σ s 1 + α + 2(1 − α ) ln n + C ( s , α ) with C ( s , α ) being n -independent terms. 2 α − s 1 − 1 1 ◮ The saddle point value is h ∗ = n s 2 α +1 + O ( n 0 ). 2 α 2 α + s 1 − 1 1 s ◮ Linear growth in n . ◮ Note: α → 1 or s → 1 limit does not commute with the n → ∞ limit.
R´ eyni entropy of Motzkin model 3 [F.S., Korepin, 2018] R´ enyi entropy for α > 1 � α ◮ For α > 1, the factor s (1 − α ) h in the summand s h � p ( h ) n , n exponentially decays.
R´ eyni entropy of Motzkin model 3 [F.S., Korepin, 2018] R´ enyi entropy for α > 1 � α ◮ For α > 1, the factor s (1 − α ) h in the summand s h � p ( h ) n , n exponentially decays. � � 1 = O ( n 0 ) dominantly contributes to the ⇒ h � O ( α − 1) ln s sum.
R´ eyni entropy of Motzkin model 3 [F.S., Korepin, 2018] R´ enyi entropy for α > 1 � α ◮ For α > 1, the factor s (1 − α ) h in the summand s h � p ( h ) n , n exponentially decays. � � 1 = O ( n 0 ) dominantly contributes to the ⇒ h � O ( α − 1) ln s sum. ◮ The result: 3 α 2( α − 1) ln n + O ( n 0 ) . S A , α =
R´ eyni entropy of Motzkin model 3 [F.S., Korepin, 2018] R´ enyi entropy for α > 1 � α ◮ For α > 1, the factor s (1 − α ) h in the summand s h � p ( h ) n , n exponentially decays. � � 1 = O ( n 0 ) dominantly contributes to the ⇒ h � O ( α − 1) ln s sum. ◮ The result: 3 α 2( α − 1) ln n + O ( n 0 ) . S A , α = ◮ Logarithmic growth
R´ eyni entropy of Motzkin model 3 [F.S., Korepin, 2018] R´ enyi entropy for α > 1 � α ◮ For α > 1, the factor s (1 − α ) h in the summand s h � p ( h ) n , n exponentially decays. � � 1 = O ( n 0 ) dominantly contributes to the ⇒ h � O ( α − 1) ln s sum. ◮ The result: 3 α 2( α − 1) ln n + O ( n 0 ) . S A , α = ◮ Logarithmic growth ◮ α → 1 or s → 1 limit does not commute with the n → ∞ limit.
R´ eyni entropy of Motzkin model 4 [F.S., Korepin, 2018] Phase transition ◮ S A α grows as O ( n ) for 0 < α < 1 while as O (ln n ) for α > 1.
R´ eyni entropy of Motzkin model 4 [F.S., Korepin, 2018] Phase transition ◮ S A α grows as O ( n ) for 0 < α < 1 while as O (ln n ) for α > 1. ⇒ Non-analytic behavior at α = 1 (Phase transition)
R´ eyni entropy of Motzkin model 4 [F.S., Korepin, 2018] Phase transition ◮ S A α grows as O ( n ) for 0 < α < 1 while as O (ln n ) for α > 1. ⇒ Non-analytic behavior at α = 1 (Phase transition) ◮ In terms of the entanglement Hamiltonian, Tr A ρ α A = Tr A e − α H ent , A α : “inverse temperature”
R´ eyni entropy of Motzkin model 4 [F.S., Korepin, 2018] Phase transition ◮ S A α grows as O ( n ) for 0 < α < 1 while as O (ln n ) for α > 1. ⇒ Non-analytic behavior at α = 1 (Phase transition) ◮ In terms of the entanglement Hamiltonian, Tr A ρ α A = Tr A e − α H ent , A α : “inverse temperature” ◮ 0 < α < 1: “high temperature” (Height of dominant paths h = O ( n )) ◮ α > 1: “low temperature” (Height of dominant paths h = O ( n 0 ))
R´ eyni entropy of Motzkin model 4 [F.S., Korepin, 2018] Phase transition ◮ S A α grows as O ( n ) for 0 < α < 1 while as O (ln n ) for α > 1. ⇒ Non-analytic behavior at α = 1 (Phase transition) ◮ In terms of the entanglement Hamiltonian, Tr A ρ α A = Tr A e − α H ent , A α : “inverse temperature” ◮ 0 < α < 1: “high temperature” (Height of dominant paths h = O ( n )) ◮ α > 1: “low temperature” (Height of dominant paths h = O ( n 0 )) ◮ The transition point α = 1 itself forms the third phase. O ( √ n ) S A , α : O (ln n ) O ( n ) 1 /α 0 1 O ( √ n ) O ( n 0 ) O ( n ) h :
Introduction Motzkin spin model Colored Motzkin model SIS Motzkin model Colored SIS Motzkin model R´ enyi entropy R´ enyi entropy of Motzkin model Summary and discussion
Summary and discussion 1 Summary ◮ We have reviewed the (colored) Motzkin spin models which yield large entanglement entropy proportional to the square root of the volume.
Summary and discussion 1 Summary ◮ We have reviewed the (colored) Motzkin spin models which yield large entanglement entropy proportional to the square root of the volume. ◮ We have extended the models by introducing additional d.o.f. based on Symmetric Inverse Semigroups. ◮ Quantum phase transitions In uncolored case ( S 3 1 ), log. violation v.s. area law O (1) for S A 2 ), √ n v.s. ln n for S A . In colored case ( S 3
Summary and discussion 1 Summary ◮ We have reviewed the (colored) Motzkin spin models which yield large entanglement entropy proportional to the square root of the volume. ◮ We have extended the models by introducing additional d.o.f. based on Symmetric Inverse Semigroups. ◮ Quantum phase transitions In uncolored case ( S 3 1 ), log. violation v.s. area law O (1) for S A 2 ), √ n v.s. ln n for S A . In colored case ( S 3 ◮ Semigroup extension of the Fredkin model [Padmanabhan, F.S., Korepin 2018]
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