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Highly entangled quantum spin chains and their extensions by semigroups

Fumihiko Sugino

Center for Theoretical Physics of the Universe, Institute for Basic Science

Workshop on “Matrix Models for Noncommutative Geometry and String Theory” Erwin Schr¨

• dinger Institute (ESI),

July 12, 2018

Highly entangled quantum spin chains and their extensions by semigroups

Fumihiko Sugino

Center for Theoretical Physics of the Universe, Institute for Basic Science

Workshop on “Matrix Models for Noncommutative Geometry and String Theory” Erwin Schr¨

• dinger Institute (ESI),

July 12, 2018

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 P. Padmanabhan, J. Stat. Mech. 1801 (2018) 013101, arXiv: 1710.10426

• P. Padmanabhan, F.S. and V. Korepin, arXiv: 1804.00978

F.S. and V. Korepin, arXiv:1806.04049

Outline

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

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 = |ψψ|, reduced

density matrix of a subsystem A: ρA = Tr S−A ρ can become mixed states, and has nonzero entanglement entropy SA = −Tr A [ρA ln ρA] . 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: SA ∝ (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: SA ∝ (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: SA = 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: SA ∝ (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: SA = c

3 ln (volume of A). [Calabrese, Cardy 2009] ◮ Belief for gapless case in D-dim. (over two decades) :

SA = O(LD−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: SA ∝ (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: SA = c

3 ln (volume of A). [Calabrese, Cardy 2009] ◮ Belief for gapless case in D-dim. (over two decades) :

SA = O(LD−1 ln L) (L: length scale of A)

◮ Recently, 1D solvable spin chain model which exhibit

extensive entanglement entropy have been discussed.

◮ Beyond logarithmic violation: SA ∝

• (volume of A)

[Movassagh, Shor 2014], [Salberger, Korepin 2016]

Counterexamples of the belief!

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

Motzkin spin model 1

[Bravyi et al 2012] ◮ 1D spin chain at sites i ∈ {1, 2, · · · , 2n} ◮ 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, · · · , 2n} ◮ Spin-1 state at each site can be regarded as up, down and flat

steps; |u ⇔ , |d ⇔ , |0 ⇔

◮ Each spin configuration ⇔ length-2n walk in (x, y) plane

Example) y x |u1 |02 |d3 |u4 |u5 |d6

Motzkin spin model 2

[Bravyi et al 2012]

Hamiltonian: HMotzkin = Hbulk + Hbdy

◮ Bulk part: Hbulk = 2n−1 j=1 Πj,j+1,

Πj,j+1 = |Dj,j+1D| + |Uj,j+1U| + |Fj,j+1F| (local interactions) with |D ≡ 1 √ 2 (|0, d − |d, 0) , |U ≡ 1 √ 2 (|0, u − |u, 0) , |F ≡ 1 √ 2 (|0, 0 − |u, d) .

Motzkin spin model 2

[Bravyi et al 2012]

Hamiltonian: HMotzkin = Hbulk + Hbdy

◮ Bulk part: Hbulk = 2n−1 j=1 Πj,j+1,

Πj,j+1 = |Dj,j+1D| + |Uj,j+1U| + |Fj,j+1F| (local interactions) with |D ≡ 1 √ 2 (|0, d − |d, 0) , |U ≡ 1 √ 2 (|0, u − |u, 0) , |F ≡ 1 √ 2 (|0, 0 − |u, d) . ⇔ ∼

Motzkin spin model 2

[Bravyi et al 2012]

Hamiltonian: HMotzkin = Hbulk + Hbdy

◮ Bulk part: Hbulk = 2n−1 j=1 Πj,j+1,

Πj,j+1 = |Dj,j+1D| + |Uj,j+1U| + |Fj,j+1F| (local interactions) with |D ≡ 1 √ 2 (|0, d − |d, 0) , |U ≡ 1 √ 2 (|0, u − |u, 0) , |F ≡ 1 √ 2 (|0, 0 − |u, d) . ⇔ ∼ ⇔ ∼

Motzkin spin model 2

[Bravyi et al 2012]

Hamiltonian: HMotzkin = Hbulk + Hbdy

◮ Bulk part: Hbulk = 2n−1 j=1 Πj,j+1,

Πj,j+1 = |Dj,j+1D| + |Uj,j+1U| + |Fj,j+1F| (local interactions) with |D ≡ 1 √ 2 (|0, d − |d, 0) , |U ≡ 1 √ 2 (|0, u − |u, 0) , |F ≡ 1 √ 2 (|0, 0 − |u, d) . ⇔ ∼ ⇔ ∼ ⇔ ∼ “gauge equivalence”.

Motzkin spin model 3

[Bravyi et al 2012]

Hamiltonian: HMotzkin = Hbulk + Hbdy

◮ Boundary part: Hbdy = |d1d| + |u2nu|

⇓

Motzkin spin model 3

[Bravyi et al 2012]

Hamiltonian: HMotzkin = Hbulk + Hbdy

◮ Boundary part: Hbdy = |d1d| + |u2nu|

⇓

◮ HMotzkin 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: HMotzkin = Hbulk + Hbdy

◮ Boundary part: Hbdy = |d1d| + |u2nu|

⇓

◮ HMotzkin is the sum of projection operators.

⇒ Positive semi-definite spectrum

◮ We find the unique zero-energy ground state.

◮ Each projector in HMotzkin annihilates the zero-energy state.

⇒ Frustration free

◮ The ground state corresponds to randoms walks starting at

(0, 0) and ending at (2n, 0) restricted to the region y ≥ 0 (Motzkin Walks (MWs)).

Motzkin spin model 4

[Bravyi et al 2012]

In terms of S = 1 spin matrices Sz =   1 −1   , S± ≡ 1 √ 2 (Sx±iSy) =   1 1   ,  1 1   , Hbulk = 1 2

2n−1

• j=1
• 1j1j+1 − 1

4Sz jSz j+1 − 1 4S2

z jSz j+1 + 1

4Sz jS2

z j+1

−3 4S2

z jS2 z j+1 + S+ j (SzS−)j+1 + S− j (S+Sz)j+1 − (S−Sz)j S+ j+1

− (SzS+)j S− j+1− (S−Sz)j (S+Sz)j+1 − (SzS+)j (SzS−)j+1

• ,

Hbdy = 1 2

• S2

z − Sz

• 1 + 1

2

• S2

z + Sz

• 2n

Quartic spin interactions

Motzkin spin model 5

[Bravyi et al 2012]

Example) 2n = 4 case, MWs: + + + + + + + +

• Ground state:

|P4 = 1 √ 9 [|0000 + |ud00 + |0ud0 + |00ud +|u0d0 + |0u0d + |u00d + |udud +|uudd] .

Motzkin spin model 6

[Bravyi et al 2012]

Note

Forbidden paths for the ground state

• 1. Path entering y < 0 region

∼ Forbidden by Hbdy

• 2. Path ending at nonzero height

∼ ∼ ∼ Forbidden by Hbdy

Motzkin spin model 7

[Bravyi et al 2012]

Entanglement entropy of the subsystem A = {1, 2, · · · , n}:

◮ Normalization factor of the ground state |P2n is given by

the number of MWs of length 2n: M2n = n

k=0 Ck

2n 2k

• .

Ck =

1 k+1

2k k

• : Catalan number

Motzkin spin model 7

[Bravyi et al 2012]

Entanglement entropy of the subsystem A = {1, 2, · · · , n}:

◮ Normalization factor of the ground state |P2n is given by

the number of MWs of length 2n: M2n = n

k=0 Ck

2n 2k

• .

Ck =

1 k+1

2k k

• : Catalan number

◮ Consider to trace out the density matrix ρ = |P2nP2n| w.r.t.

the subsystem B = {n + 1, · · · , 2n}. Schmidt decomposition: |P2n =

• h≥0
• p(h)

n,n

• P(0→h)

n

• ⊗
• P(h→0)

n

• with p(h)

n,n ≡

• M(h)

n

2 M2n

. ↑ Paths from (0, 0) to (n, h)

Motzkin spin model 8

[Bravyi et al 2012] ◮ M(h) n

is the number of paths in P(0→h)

n

. For n → ∞, Gaussian distribution p(h)

n,n ∼ 3

√ 6 √π (h + 1)2 n3/2 e− 3

2 (h+1)2 n

× [1 + O(1/n)] .

◮ Reduced density matrix

ρA = Tr Bρ =

• h≥0

p(h)

n,n

• P(0→h)

n

P(0→h)

n

• ◮ Entanglement entropy

SA = −

• h≥0

p(h)

n,n ln p(h) n,n

= 1 2 ln n + 1 2 ln 2π 3 + γ − 1 2 (γ: Euler constant) up to terms vanishing as n → ∞.

Motzkin spin model 9

[Bravyi et al 2012]

Notes

◮ The system is critical (gapless).

SA 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).

SA is similar to the (1 + 1)-dimensional CFT with c = 3/2.

◮ But, gap scales as O(1/nz) 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).

SA is similar to the (1 + 1)-dimensional CFT with c = 3/2.

◮ But, gap scales as O(1/nz) 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

• uk

⇔ k ,

• dk

⇔ 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

• uk

⇔ k ,

• dk

⇔ k , |0 ⇔ Color d.o.f. decorated to Motzkin Walks

◮ Hamiltonian HcMotzkin = Hbulk + Hbdy

◮ Bulk part consisting of local interactions:

Hbulk =

2n−1

• j=1
• Πj,j+1 + Πcross

j,j+1

• ,

Πj,j+1 =

s

• k=1
• Dk

j,j+1

• Dk

+

• Uk

j,j+1

• Uk

+

• F k

j,j+1

• F k
• with

Colored Motzkin spin model 2

[Movassagh, Shor 2014]

• Dk

≡ 1 √ 2

• 0, dk

−

• dk, 0
• ,
• Uk

≡ 1 √ 2

• 0, uk

−

• uk, 0
• ,
• F k

≡ 1 √ 2

• |0, 0 −
• uk, dk

, and Πcross

j,j+1 =

• k=k′
• uk, dk′

j,j+1

• uk, dk′
• .

⇒ Colors should be matched in up and down pairs.

◮ Boundary part

Hbdy =

s

• k=1
• dk

1

• dk
• +
• uk

2n

• uk
• .

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) 2n = 4 case,

+ k k + k k + k k + k k + k k + k k + k k k′ k′ + k k′ k′ k |P4 = 1 √ 1 + 6s + 2s2

• |0000 +

s

• k=1
• ukdk00
• + · · · +
• uk00dk

+

s

• k,k′=1
• ukdkuk′dk′

+

• ukuk′dk′dk

.

Colored Motzkin spin model 4

[Movassagh, Shor 2014]

Entanglement entropy

◮ Paths from (0, 0) to (n, h), P(0→h) n

, have h unmatched up steps. Let ˜ P(0→h)

n

({κm}) be paths with the colors of unmatched up steps frozen. (unmatched up from height (m − 1) to m) → uκm

◮ Similarly,

P(h→0)

n

→ ˜ P(h→0)

n

({κm}), (unmatched down from height m to (m − 1)) → dκm.

◮ The numbers satisfy M(h) n

= sh ˜ M(h)

n .

Colored Motzkin spin model 5

[Movassagh, Shor 2014]

Example

2n = 8 case, h = 2 x y 1 2 3 4 5 6 7 8 1 2 3 k k k′ k′ uκ1 dκ1 uκ2 dκ2 A B

Colored Motzkin spin model 6

[Movassagh, Shor 2014] ◮ Schmidt decomposition

|P2n =

• h≥0

s

• κ1=1

· · ·

s

• κh=1
• p(h)

n,n

×

• ˜

P(0→h)

n

({κm})

• ⊗
• ˜

P(h→0)

n

({κm})

• with

p(h)

n,n =

• ˜

M(h)

n

2 M2n .

◮ Reduced density matrix

ρA =

• h≥0

s

• κ1=1

· · ·

s

• κh=1

p(h)

n,n

×

• ˜

P(0→h)

n

({κm})

• ˜

P(0→h)

n

({κm})

• .

Colored Motzkin spin model 7

[Movassagh, Shor 2014] ◮ For n → ∞,

p(h)

n,n ∼

√ 2 s−h √π (σn)3/2 (h + 1)2 e− (h+1)2

2σn

× [1 + O(1/n)] with σ ≡

√s 2√s+1.

Note: Effectively h O(√n).

◮ Entanglement entropy

SA = −

• h≥0

sh p(h)

n,n ln p(h) n,n

Colored Motzkin spin model 7

[Movassagh, Shor 2014] ◮ For n → ∞,

p(h)

n,n ∼

√ 2 s−h √π (σn)3/2 (h + 1)2 e− (h+1)2

2σn

× [1 + O(1/n)] with σ ≡

√s 2√s+1.

Note: Effectively h O(√n).

◮ Entanglement entropy

SA = −

• h≥0

sh p(h)

n,n ln p(h) n,n

= (2 ln s)

• 2σn

π + 1 2 ln n + 1 2 ln(2πσ) + γ − 1 2− ln s up to terms vanishing as n → ∞. Grows as √n.

Colored Motzkin spin model 8

[Movassagh, Shor 2014]

Comments

◮

Matching color ⇒ s−h factor in p(h)

n,n

⇒ crucial to O(√n) behavior in SA

Colored Motzkin spin model 8

[Movassagh, Shor 2014]

Comments

◮

Matching color ⇒ s−h factor in p(h)

n,n

⇒ crucial to O(√n) behavior in SA

◮ Typical configurations:

k k′ k′ k h = O(√n) + (equivalence moves).

Colored Motzkin spin model 8

[Movassagh, Shor 2014]

Comments

◮

Matching color ⇒ s−h factor in p(h)

n,n

⇒ crucial to O(√n) behavior in SA

◮ Typical configurations:

k k′ k′ k h = O(√n) + (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]

Sz, 1Sz, 2nconnected → −0.034... × s3 − s 6 = 0 (n → ∞) ⇒ 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]

Sz, 1Sz, 2nconnected → −0.034... × s3 − s 6 = 0 (n → ∞) ⇒ Violation of cluster decomposition property for s > 1 (Strong correlation due to color matching)

◮ Deformation of models to achieve the volume law behavior

(SA ∝ 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 Sk Sk

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 Sk Sk

p (p = 1, · · · , k) ◮ xa,b ∈ Sk 1 maps a to b. (a, b ∈ {1, · · · , k})

Product rule: xa,b ∗ xc,d = δb,c xa,d x1,2 ∗ x2,1 = x1,1, x2,1 ∗ x1,2 = x2,2 տ ր (partial identities) (x1,2)−1 = x2,1 (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 Sk Sk

p (p = 1, · · · , k) ◮ xa,b ∈ Sk 1 maps a to b. (a, b ∈ {1, · · · , k})

Product rule: xa,b ∗ xc,d = δb,c xa,d x1,2 ∗ x2,1 = x1,1, x2,1 ∗ x1,2 = x2,2 տ ր (partial identities) (x1,2)−1 = x2,1 (unique inverse)

◮ xa1,a2; b1,b2 ∈ Sk 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 Sk Sk

p (p = 1, · · · , k) ◮ xa,b ∈ Sk 1 maps a to b. (a, b ∈ {1, · · · , k})

Product rule: xa,b ∗ xc,d = δb,c xa,d x1,2 ∗ x2,1 = x1,1, x2,1 ∗ x1,2 = x2,2 տ ր (partial identities) (x1,2)−1 = x2,1 (unique inverse)

◮ xa1,a2; b1,b2 ∈ Sk 2 etc, ...

Sk

k ≡ Sk

SIS Motzkin model 1

[Sugino, Padmanabhan 2017] ◮ Change the spin d.o.f. as |xa,b with a, b ∈ {1, 2, · · · , k}. ◮ a < b case: ‘up’ ⇔ a

b a > b case: ‘down’ ⇔ a b a = b case: ‘flat’ ⇔ a b

SIS Motzkin model 1

[Sugino, Padmanabhan 2017] ◮ Change the spin d.o.f. as |xa,b with a, b ∈ {1, 2, · · · , k}. ◮ a < b case: ‘up’ ⇔ a

b a > b case: ‘down’ ⇔ a b a = b case: ‘flat’ ⇔ a b

◮ We regard the configuration of adjacent sites

|(xa,b)j |(xc,d)j+1 as a connected path for b = c. c.f.) Analogous to the product rule of Symmetric Inverse Semigroup (Sk

1 ):

xa,b ∗ xc,d = δb,c xa,d a, b: semigroup indices

◮ Inner product: xa,b|xc,d = δa,cδb,d ◮ Let us consider the k = 3 case.

SIS Motzkin model 2

[Sugino, Padmanabhan 2017] ◮ Maximum height is lower than the original Motzkin case.

x y 1 2 3 4 5 1 2 3 1 2 3 1 2 3

SIS Motzkin model 3

[Sugino, Padmanabhan 2017]

Hamiltonian HS31Motzkin = Hbulk + Hbulk,disc + Hbdy

◮ Hbulk: local interactions corresponding to the following moves:

(Down) a a b ∼ a b b (a > b) (Up) a a b ∼ a b b (a < b) (Flat) a a a ∼ a b a (a < b) (Wedge) 3 1 3 ∼ 3 2 3

SIS Motzkin model 4

[Sugino, Padmanabhan 2017] ◮ Hbulk,disc lifts disconnected paths to excited states.

Π|ψ: projector to |ψ Hbulk,disc =

2n−1

• j=1

3

• a,b,c,d=1;b=c

Π|(xa,b)j,(xc,d)j+1

SIS Motzkin model 4

[Sugino, Padmanabhan 2017] ◮ Hbulk,disc lifts disconnected paths to excited states.

Π|ψ: projector to |ψ Hbulk,disc =

2n−1

• j=1

3

• a,b,c,d=1;b=c

Π|(xa,b)j,(xc,d)j+1

◮

Hbdy =

• a>b

Π|(xa,b)1 +

• a<b

Π|(xa,b)2n +Π|(x1,3)1,(x3,2)2,(x2,1)3 + Π|(x1,2)2n−2,(x2,3)2n−1,(x3,1)2n 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

(0, 0), ending at (2n, 0) and not entering y < 0. S3

1 MWs

SIS Motzkin model 5

[Sugino, Padmanabhan 2017] ◮ Ground states correspond to connected paths starting at

(0, 0), ending at (2n, 0) and not entering y < 0. S3

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: x3,3x3,3 · · · x3,3.

SIS Motzkin model 5

[Sugino, Padmanabhan 2017] ◮ Ground states correspond to connected paths starting at

(0, 0), ending at (2n, 0) and not entering y < 0. S3

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: x3,3x3,3 · · · x3,3.

◮ The number of paths can be obtained by recursion relations.

For length-n paths from the semigroup index a to b (Pn,a→b), Pn,1→1 = x1,1Pn−1,1→1 + x1,2

n−2

• i=1

Pi,2→2 x2,1Pn−2−i,1→1 +x1,3

n−2

• i=1

Pi,3→3 x3,1Pn−2−i,1→1 +x1,3

n−2

• i=1

Pi,3→3 x3,2Pn−2−i,2→1, etc.

SIS Motzkin model 6

[Sugino, Padmanabhan 2017]

Result

◮ The entanglement entropies SA,1→1, SA,1→2, SA,2→1 and

SA,2→2 take the same form as in the case of the Motzkin model. Logarithmic violation of the area law

◮ The form of p(h)

n

∼ (h+1)2

n3/2 e−(const.) (h+1)2

n

is universal.

◮ SA,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 2n = 6 case, y x 1 2 2 1 2 3 2 1

SIS Motzkin model 7

Localization

[Padmanabhan, F.S., Korepin 2018] ◮ There are excited states corresponding to disconnected paths.

Example) One such path in 2n = 6 case, y x 1 2 2 1 2 3 2 1 Corresponding excited state: |P3, 1→1⊗

• P(1→0)

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 2n = 6 case, y x 1 2 2 1 2 3 2 1 Corresponding excited state: |P3, 1→1⊗

• P(1→0)

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 S3

2 ◮ 18 elements xab,cd with ab ∈ {12, 23, 31} and

cd ∈ {12, 23, 31, 21, 32, 13} satisfying xab,cd ∗ xef ,gh = δc,eδd,f xab,gh + δc,f δd,e xab,hg.

◮ can be regarded as 2 sets of S3 1.

⇒ color d.o.f.

Colored SIS Motzkin model 1

[Sugino, Padmanabhan 2017]

The SIS S3

2 ◮ 18 elements xab,cd with ab ∈ {12, 23, 31} and

cd ∈ {12, 23, 31, 21, 32, 13} satisfying xab,cd ∗ xef ,gh = δc,eδd,f xab,gh + δc,f δd,e xab,hg.

◮ can be regarded as 2 sets of S3 1.

⇒ color d.o.f.

◮ Spin variables: xs a,b (s = 1, 2) (a, b = 1, 2, 3) ◮ The new moves (C moves) introduced to the Hamiltonian.

a a 1 ∼ a a 2

Colored SIS Motzkin model 2

[Sugino, Padmanabhan 2017]

Hamiltonian: HcS31Motzkin = Hbulk + Hbulk,disc + Hbdy

◮ In Hbulk, (Down), (Up) and (Flat) are essentially the same as

before. (Down) a a b s s ∼ a b b s s (a > b) (Up) a a b s s ∼ a b b s s (a < b) (Flat) a a a s s ∼ a b a s s (a < b)

Colored SIS Motzkin model 3

[Sugino, Padmanabhan 2017] ◮ Wedge move:

(Wedge) 3 1 3 s s′ ∼ 3 2 3 s s′

◮

(Cross)j,j+1 =

• b>a,c
• Π|(x1

a,b)j,(x2 b,c)j+1 + Π|(x2 a,b)j,(x1 b,c)j+1

forbids unmatched up and down steps in ground states. ⇓ Hbulk = µ

2n

• j=1

Cj +

2n−1

• j=1

[(Down)j,j+1 + (Up)j,j+1 +(Flat)j,j+1 + (Wedge)j,j+1 + (Cross)j,j+1]

Colored SIS Motzkin model 4

[Sugino, Padmanabhan 2017] ◮

Hbulk,disc =

2n−1

• j=1

3

• a,b,c,d=1;b=c

2

• s,t=1

Π|(xs

a,b)j,(xt c,d)j+1

◮

Hbdy =

• a>b

2

• s=1

Π|(xs

a,b)1 +

• a<b

2

• s=1

Π|(xs

a,b)2n

+

2

• s,t=1

Π|(xs

1,3)1,(xs 3,2)2,(xt 2,1)3

+

2

• s,t=1

Π|(xs

1,2)2n−2,(xt 2,3)2n−1,(xt 3,1)2n

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,

SA = (2 ln 2)

• 2σn

π + 1 2 ln n + 1 2 ln(2πσ) + γ − 1 2 + ln 3 21/3 with σ ≡

√ 2−1 9 √ 2 .

◮ For µ = 0, colors 1 and 2 decouple.

SA ∝ 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: SA, α = 1 1 − α ln Tr A ρα

A

with α > 0 and α = 1.

R´ enyi entropy

[R´ enyi, 1970] ◮ R´

enyi entropy has further importance than the von Neumann entanglement entropy: SA, α = 1 1 − α ln Tr A ρα

A

with α > 0 and α = 1.

◮ Generalization of the von Neumann entanglement entropy:

limα→1 SA, α = SA

R´ enyi entropy

[R´ enyi, 1970] ◮ R´

enyi entropy has further importance than the von Neumann entanglement entropy: SA, α = 1 1 − α ln Tr A ρα

A

with α > 0 and α = 1.

◮ Generalization of the von Neumann entanglement entropy:

limα→1 SA, α = SA

◮ Reconstructs the whole spectrum of the entanglement

Hamiltonian Hent, A ≡ − ln ρA.

R´ enyi entropy

[R´ enyi, 1970] ◮ R´

enyi entropy has further importance than the von Neumann entanglement entropy: SA, α = 1 1 − α ln Tr A ρα

A

with α > 0 and α = 1.

◮ Generalization of the von Neumann entanglement entropy:

limα→1 SA, α = SA

◮ Reconstructs the whole spectrum of the entanglement

Hamiltonian Hent, A ≡ − ln ρA.

◮ For SA, α (0 < α < 1), the gapped systems in 1D is proven to

• bey the area law.

[Huang, 2015]

R´ enyi entropy

[R´ enyi, 1970] ◮ R´

enyi entropy has further importance than the von Neumann entanglement entropy: SA, α = 1 1 − α ln Tr A ρα

A

with α > 0 and α = 1.

◮ Generalization of the von Neumann entanglement entropy:

limα→1 SA, α = SA

◮ Reconstructs the whole spectrum of the entanglement

Hamiltonian Hent, A ≡ − ln ρA.

◮ For SA, α (0 < α < 1), the gapped systems in 1D is proven to

• bey 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

SA, α = 1 1 − α ln

n

• h=0

sh p(h)

n,n

α .

R´ eyni entropy of Motzkin model 1

[F.S., Korepin, 2018] ◮ What we compute is the asymptotic behavior of

SA, α = 1 1 − α ln

n

• h=0

sh p(h)

n,n

α .

◮ For colorless case (s = 1), we obtain

SA,α = 1 2 ln n + 1 1 − α ln Γ

• α + 1

2

• −

1 2(1 − α)

• (1 + 2α) ln α + α ln π

24 + ln 6

• 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

SA, α = 1 1 − α ln

n

• h=0

sh p(h)

n,n

α .

◮ For colorless case (s = 1), we obtain

SA,α = 1 2 ln n + 1 1 − α ln Γ

• α + 1

2

• −

1 2(1 − α)

• (1 + 2α) ln α + α ln π

24 + ln 6

• up to terms vanishing as n → ∞.

◮ Logarithmic growth ◮ Reduces to SA 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)

◮ The summand sh

p(h)

n,n

α has a factor s(1−α)h.

R´ eyni entropy of Motzkin model 2

[F.S., Korepin, 2018]

Colored case (s > 1)

◮ The summand sh

p(h)

n,n

α has a factor s(1−α)h. 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)

◮ The summand sh

p(h)

n,n

α has a factor s(1−α)h. 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

SA,α = n 2α 1 − α ln

• σ
• s

1−α 2α + s− 1−α 2α + s−1/2

+ 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)

◮ The summand sh

p(h)

n,n

α has a factor s(1−α)h. 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

SA,α = n 2α 1 − α ln

• σ
• s

1−α 2α + s− 1−α 2α + s−1/2

+ 1 + α 2(1 − α) ln n + C(s, α) with C(s, α) being n-independent terms.

◮ The saddle point value is h∗ = n

s

1 2α −s1− 1 2α

s

1 2α +s1− 1 2α +1 + O(n0).

R´ eyni entropy of Motzkin model 2

[F.S., Korepin, 2018]

Colored case (s > 1)

◮ The summand sh

p(h)

n,n

α has a factor s(1−α)h. 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

SA,α = n 2α 1 − α ln

• σ
• s

1−α 2α + s− 1−α 2α + s−1/2

+ 1 + α 2(1 − α) ln n + C(s, α) with C(s, α) being n-independent terms.

◮ The saddle point value is h∗ = n

s

1 2α −s1− 1 2α

s

1 2α +s1− 1 2α +1 + O(n0).

◮ Linear growth in n.

R´ eyni entropy of Motzkin model 2

[F.S., Korepin, 2018]

Colored case (s > 1)

◮ The summand sh

p(h)

n,n

α has a factor s(1−α)h. 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

SA,α = n 2α 1 − α ln

• σ
• s

1−α 2α + s− 1−α 2α + s−1/2

+ 1 + α 2(1 − α) ln n + C(s, α) with C(s, α) being n-independent terms.

◮ The saddle point value is h∗ = n

s

1 2α −s1− 1 2α

s

1 2α +s1− 1 2α +1 + O(n0).

◮ 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

◮ For α > 1, the factor s(1−α)h in the summand sh

p(h)

n,n

α exponentially decays.

R´ eyni entropy of Motzkin model 3

[F.S., Korepin, 2018]

R´ enyi entropy for α > 1

◮ For α > 1, the factor s(1−α)h in the summand sh

p(h)

n,n

α exponentially decays. ⇒ h O

• 1

(α−1) ln s

• = O(n0) dominantly contributes to the

sum.

R´ eyni entropy of Motzkin model 3

[F.S., Korepin, 2018]

R´ enyi entropy for α > 1

◮ For α > 1, the factor s(1−α)h in the summand sh

p(h)

n,n

α exponentially decays. ⇒ h O

• 1

(α−1) ln s

• = O(n0) dominantly contributes to the

sum.

◮ The result:

SA, α = 3α 2(α − 1) ln n + O(n0).

R´ eyni entropy of Motzkin model 3

[F.S., Korepin, 2018]

R´ enyi entropy for α > 1

◮ For α > 1, the factor s(1−α)h in the summand sh

p(h)

n,n

α exponentially decays. ⇒ h O

• 1

(α−1) ln s

• = O(n0) dominantly contributes to the

sum.

◮ The result:

SA, α = 3α 2(α − 1) ln n + O(n0).

◮ Logarithmic growth

R´ eyni entropy of Motzkin model 3

[F.S., Korepin, 2018]

R´ enyi entropy for α > 1

◮ For α > 1, the factor s(1−α)h in the summand sh

p(h)

n,n

α exponentially decays. ⇒ h O

• 1

(α−1) ln s

• = O(n0) dominantly contributes to the

sum.

◮ The result:

SA, α = 3α 2(α − 1) ln n + O(n0).

◮ 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

◮ SA 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

◮ SA 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

◮ SA 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 ρα

A = Tr A e−αHent, A

α: “inverse temperature”

R´ eyni entropy of Motzkin model 4

[F.S., Korepin, 2018]

Phase transition

◮ SA 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 ρα

A = Tr A e−αHent, 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(n0))

R´ eyni entropy of Motzkin model 4

[F.S., Korepin, 2018]

Phase transition

◮ SA 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 ρα

A = Tr A e−αHent, 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(n0))

◮ The transition point α = 1 itself forms the third phase.

1/α 1 SA, α: O(ln n) O(√n) O(n) h: O(n0) O(√n) O(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

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 (S3

1), log. violation v.s. area law O(1) for SA

In colored case (S3

2), √n v.s. ln n for SA.

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 (S3

1), log. violation v.s. area law O(1) for SA

In colored case (S3

2), √n v.s. ln n for SA.

◮ Semigroup extension of the Fredkin model [Padmanabhan, F.S., Korepin 2018]

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 (S3

1), log. violation v.s. area law O(1) for SA

In colored case (S3

2), √n v.s. ln n for SA.

◮ Semigroup extension of the Fredkin model [Padmanabhan, F.S., Korepin 2018] ◮ As a feature of the extended models,

Anderson-like localization occurs in excited states corresponding to disconnected paths.

◮ “2nd quantized paths”.

Summary and discussion 2

Summary

◮ We have analytically computed the R´

enyi entropy of half-chain in the Motzkin model.

◮ Phase transition at α = 1

(New phase transition!) No analog for other spin chains investigated so far (XX, XY, AKLT,...).

Summary and discussion 2

Summary

◮ We have analytically computed the R´

enyi entropy of half-chain in the Motzkin model.

◮ Phase transition at α = 1

(New phase transition!) No analog for other spin chains investigated so far (XX, XY, AKLT,...).

◮ For 0 < α < 1 (“high temperature”), SA, α = O(n). ◮ For α > 1 (“low temperature”), SA, α = O(ln n).

Summary and discussion 2

Summary

◮ We have analytically computed the R´

enyi entropy of half-chain in the Motzkin model.

◮ Phase transition at α = 1

(New phase transition!) No analog for other spin chains investigated so far (XX, XY, AKLT,...).

◮ For 0 < α < 1 (“high temperature”), SA, α = O(n). ◮ For α > 1 (“low temperature”), SA, α = O(ln n).

◮ We also have a similar result for the Fredkin spin chain. [F.S., Korepin, 2018]

Summary and discussion 2

Summary

◮ We have analytically computed the R´

enyi entropy of half-chain in the Motzkin model.

◮ Phase transition at α = 1

(New phase transition!) No analog for other spin chains investigated so far (XX, XY, AKLT,...).

◮ For 0 < α < 1 (“high temperature”), SA, α = O(n). ◮ For α > 1 (“low temperature”), SA, α = O(ln n).

◮ We also have a similar result for the Fredkin spin chain. [F.S., Korepin, 2018] ◮ R´

enyi entropy of chain of general length (in progress) Our conjecture: the same phase transition occurs for chain of general length

Summary and discussion 2

Summary

◮ We have analytically computed the R´

enyi entropy of half-chain in the Motzkin model.

◮ Phase transition at α = 1

(New phase transition!) No analog for other spin chains investigated so far (XX, XY, AKLT,...).

◮ For 0 < α < 1 (“high temperature”), SA, α = O(n). ◮ For α > 1 (“low temperature”), SA, α = O(ln n).

◮ We also have a similar result for the Fredkin spin chain. [F.S., Korepin, 2018] ◮ R´

enyi entropy of chain of general length (in progress) Our conjecture: the same phase transition occurs for chain of general length

◮ Similar computation for semigroup extensions

(in progress)

[F.S., Padmanabhan, 2018], [Padmanabhan, F.S., Korepin, 2018]

Summary and discussion 3

Future directions

◮ Continuum limit? (In particular, for colored case) [Chen, Fradkin, Witczak-Krempa 2017]

Summary and discussion 3

Future directions

◮ Continuum limit? (In particular, for colored case) [Chen, Fradkin, Witczak-Krempa 2017] ◮ Holography? Application to quantum gravity or black holes? [Alexander, Klich 2018] ◮ Higher-dimensional models (d = 2, 3, ...)?

Summary and discussion 3

Future directions

◮ Continuum limit? (In particular, for colored case) [Chen, Fradkin, Witczak-Krempa 2017] ◮ Holography? Application to quantum gravity or black holes? [Alexander, Klich 2018] ◮ Higher-dimensional models (d = 2, 3, ...)?

Thank you very much for your attention!

• App. SIS Motzkin model

[Sugino, Padmanabhan 2017] ◮ By adding the balancing term to the Hamiltonian

λ2

2n−1

• j=1
• Π|(x1,3)j,(x3,2)j+1 + Π|(x2,3)j,(x3,1)j+1

with λ1 put to the term 3 1 3 ∼ 3 2 3 , quantum phase transition takes place in the 4 sectors except (3, 3): λ1 λ2 SA = O(1) (area law) SA ∝ ln n λ1, λ2 > 0 is not frustration free (here, we do not consider).