Entanglement Behavior of 2D Quantum Models Shu Tanaka (YITP, Kyoto - - PowerPoint PPT Presentation

▶
entanglement behavior of 2d quantum models
SMART_READER_LITE
LIVE PREVIEW

Entanglement Behavior of 2D Quantum Models Shu Tanaka (YITP, Kyoto - - PowerPoint PPT Presentation

Oct 22, 2022 156 likes •510 views

Entanglement Behavior of 2D Quantum Models Shu Tanaka (YITP, Kyoto University) Collaborators : Hosho Katsura (Univ. of Tokyo, Japan) Anatol N. Kirillov (RIMS, Kyoto Univ., Japan) Vladimir E. Korepin (YITP, Stony Brook, USA) Naoki Kawashima

slide-1
SLIDE 1

Entanglement Behavior of 2D Quantum Models

Shu Tanaka (YITP, Kyoto University) Collaborators:

Hosho Katsura (Univ. of Tokyo, Japan) Anatol N. Kirillov (RIMS, Kyoto Univ., Japan) Vladimir E. Korepin (YITP, Stony Brook, USA) Naoki Kawashima (ISSP, Univ. of Tokyo, Japan) Lou Jie (Fudan Univ., China) Ryo Tamura (NIMS, Japan)

VBS on symmetric graphs, J. Phys. A, 43, 255303 (2010) “VBS/CFT correspondence”, Phys. Rev. B, 84, 245128 (2011) Quantum hard-square model, Phys. Rev. A, 86, 032326 (2012) Nested entanglement entropy, Interdisciplinary Information Sciences, 19, 101 (2013)

論文での使用姓 ミドルネーム 見込み 受入希望研究室 審査結果通知先 所属部署 職 名

slide-2
SLIDE 2

Digest

VBS state on 2D lattice Quantum lattice gas on ladder

VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder

Volume exclusion effect

Entanglement properties of 2D quantum systems Physical properties of 1D quantum systems

Total system

Entanglement Hamiltonian

Square lattice

1D AF Heisenberg

Hexagonal lattice 1D F Heisenberg Total system

Entanglement Hamiltonian

Square ladder

2D Ising

Triangle ladder

2D 3-state Potts

slide-3
SLIDE 3

Introduction

  • Entanglement
  • Motivation
  • Preliminaries
slide-4
SLIDE 4

Introduction

Total system

Subsystem

A

Subsystem

B

Schmidt decomposition

|Ψ =

  • α

λα|φ[A]

α |φ[B] α

{|φ[A]

α }, {|φ[B] α }

φ[A]

α

∈ HA, φ[B]

α

∈ HB

: Orthonormal basis ρA = TrB|ΨΨ| =

  • α

λ2

α|φ[A] α φ[A] α |

Reduced density matrix

Normalized GS

von Neumann entanglement entropy

EE is a measure to quantify entanglement.

|Ψ

Divide

S = Tr ρA ln ρA = −

  • α

λ2

α ln λ2 α

slide-5
SLIDE 5

Introduction

Entanglement properties in 1D quantum systems!!

Entanglement properties in 2D quantum systems??

10 20 30 40 NUMBER OF SITES − L − 1 1.5 2 2.5 ENTROPY − S − HXXZ =

  • i

(σx

i σx i+1 + σy i σy i+1 + ∆σz i σz i+1 − λσz i )

  • G. Vidal et al. PRL 90, 227902 (2003)

XY(a = ∞, γ = 0) XY(a = 1, γ = 1) X X Z ( ∆ = 1 , λ = ) X X Z ( ∆ = 2 . 5 , λ = ) XY(a = 1.1, γ = 1)

XXZ model under magnetic field XY model under magnetic field

1D gapped systems: EE converges to some value. 1D critical systems: EE diverges logarithmically with L. coefficient is related to the central charge.

B A

L

slide-6
SLIDE 6

Preliminaries: reflection symmetric case

Subsystem

A

Subsystem

B Reflection symmetry

Pre-Schmidt decomposition

|Ψ =

  • α

|φ[A]

α |φ[B] α

{|φ[A]

α }, {|φ[B] α }

Linearly independent (but not orthonormal)

(M [A])αβ := φ[A]

α |φ[A] β , (M [B])αβ := φ[B] α |φ[B] β

Overlap matrix

Useful fact

If and is real symmetric matrix, where are the eigenvalues of .

M [A] = M [B] = M M S = −

  • α

pα ln pα, pα = d2

α

  • α d2

α

dα M

  • J. Phys. A, 43, 255303 (2010)

Reflection symmetry

M [A] = M [B] = M

slide-7
SLIDE 7

Digest

VBS state on 2D lattice Quantum lattice gas on ladder

VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder

Volume exclusion effect

Entanglement properties of 2D quantum systems Physical properties of 1D quantum systems

Total system

Entanglement Hamiltonian

Square lattice

1D AF Heisenberg

Hexagonal lattice 1D F Heisenberg Total system

Entanglement Hamiltonian

Square ladder

2D Ising

Triangle ladder

2D 3-state Potts

slide-8
SLIDE 8

VBS (Valence-Bond-Solid) state

Valence bond = Singlet pair

AKLT (Affleck-Kennedy-Lieb-Tasaki) model

  • I. Affleck, T. Kennedy, E. Lieb, and H. Tasaki, PRL 59, 799 (1987).

H =

  • i
  • Si ·

Si+1 + 1 3

  • Si ·

Si+1 2

Ground state: VBS state

(S = 1)

Valence bond

S = 1

  • Exact unique ground state; S=1 VBS state
  • Rigorous proof of the “Haldane gap”
  • AFM correlation decays fast exponentially

(projection)

slide-9
SLIDE 9

VBS (Valence-Bond-Solid) state VBS state = Singlet-covering state

2D hexagonal lattice 2D square lattice

MBQC using VBS state T-C. Wei, I. Affleck, and R. Raussendorf, Phys. Rev. Lett.106, 070501 (2011).

  • A. Miyake, Ann. Phys. 326, 1656 (2011).
slide-10
SLIDE 10

VBS (Valence-Bond-Solid) state VBS state = Singlet-covering state

Schwinger boson representation

| = a†|vac, | = b†|vac

Valence bond solid (VBS) state

|VBS =

  • k,l
  • a†

kb† l b† ka† l

  • |vac

n(b)

k

= b†

kbk

n(a)

k

= a†

kak

1 2 3 4 1 2 3 4

S=0 1/2 1 3/2 2

a†

kak + b† kbk = 2Sk

slide-11
SLIDE 11

VBS (Valence-Bond-Solid) state

2D hexagonal lattice 2D square lattice

Subsystem B Subsystem A Subsystem B Subsystem A

Subsystem

A

Subsystem

B

Reflection symmetry

slide-12
SLIDE 12

VBS (Valence-Bond-Solid) state

Subsystem B Subsystem A

|VBS =

  • k,l
  • a†

kb† l b† ka† l

  • |vac

=

  • {α}

|φ[A]

α |φ[B] α

{α} =

  • α1, · · · , α|ΛA|
  • αi = ±1/2

Auxiliary spin: #bonds on edge: |ΛA|

  • Local gauge transformation
  • Reflection symmetry

Overlap matrix

M{α},{β} 2|ΛA| × 2|ΛA|

: matrix Each element can be obtained by Monte Carlo calculation!!

  • Phys. Rev. B, 84, 245128 (2011)

SU(N) case can be also calculated.

  • cf. H. Katsura, arXiv:1407.4262
slide-13
SLIDE 13

Entanglement properties

  • Entanglement entropy
  • Entanglement spectrum
  • Nested entanglement entropy
slide-14
SLIDE 14

Entanglement properties of 2D VBS states VBS state = Singlet-covering state

2D hexagonal lattice 2D square lattice

Subsystem B Subsystem A Subsystem B Subsystem A

Lx Ly PBC OBC

slide-15
SLIDE 15

Entanglement entropy of 2D VBS states

  • cf. Entanglement entropy of 1D VBS states

|VBS =

N

  • i=0
  • a†

ib† i+1 b† ia† i+1

S |vac

S=1 S=3 S=4 S=6 S=8 S=2

  • H. Katsura, T. Hirano, and Y. Hatsugai, PRB 76, 012401 (2007).

S = ln (# Edge states)

Subsystem B Subsystem A

slide-16
SLIDE 16

Entanglement entropy of 2D VBS states

2D hexagonal lattice 2D square lattice

Subsystem B Subsystem A Subsystem B Subsystem A

Lx Ly PBC or OBC OBC

S |ΛA| = ln 2 − σ

σ1D = 0 σsquare > σhexagonal σ ≥ 0 ξsquare > ξhexagonal

#bonds on edge: |ΛA|

slide-17
SLIDE 17

Entanglement spectra of 2D VBS states

ρA = e−HE (HE = − ln ρA) des Cloizeaux- Pearson mode Spin wave Hexagonal (Lx=5, Ly=32) Square (Lx=5, Ly=16)

Reduced density matrix Entanglement Hamiltonian

  • H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).

ρA =

  • α

e−λα|φ[A]

α φ[A] α |

1D antiferro Heisenberg 1D ferro Heisenberg

  • cf. J. I. Cirac, D. Poilbranc, N. Schuch, and F. Verstraete, Phys. Rev. B 83, 245134 (2011).
slide-18
SLIDE 18

Nested entanglement entropy

“Entanglement” ground state := g.s. of :

HE |Ψ0 HE|Ψ0 = Egs|Ψ0 ρA|Ψ0 = ρ0|Ψ0

Maximum eigenvalue

Nested reduced density matrix

() := Tr+1,··· ,L [|Ψ0Ψ0|]

Nested entanglement entropy

HE = − ln ρA S(, L) = −Tr1,··· , [() ln ()]

1D quantum critical system (periodic boundary condition)

P . Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.

B A

slide-19
SLIDE 19

Nested entanglement entropy

0.6 0.8 1 1.2 1.4 2 4 6 8 10 12 14 16 S(l,16) l

(a)

s1=0.77(4) c=1.01(7) Lx=5, Ly=16 fitting

0.5 0.6 0.7 0.8 2 4 6 8 1 S(l,16) l

(b)

a=0.393(1) c1/v=0.093(3) Lx=5, Ly=16 fitting

Square lattice (PBC) Square ladder (OBC)

c = 1

Central charge:

1D antiferromagnetic Heisenberg

VBS/CFT correspondence

des Cloizeaux-Pearson mode in ES supports this result.

S(, L) = −Tr1,··· ,[() ln ()]

B A

slide-20
SLIDE 20

Digest

VBS state on 2D lattice Quantum lattice gas on ladder

VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder

Volume exclusion effect

Entanglement properties of 2D quantum systems Physical properties of 1D quantum systems

Total system

Entanglement Hamiltonian

Square lattice

1D AF Heisenberg

Hexagonal lattice 1D F Heisenberg Total system

Entanglement Hamiltonian

Square ladder

2D Ising

Triangle ladder

2D 3-state Potts

slide-21
SLIDE 21

Digest

VBS state on 2D lattice Quantum lattice gas on ladder

VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder

Volume exclusion effect

Entanglement properties of 2D quantum systems Physical properties of 1D quantum systems

Total system

Entanglement Hamiltonian

Square lattice

1D AF Heisenberg

Hexagonal lattice 1D F Heisenberg Total system

Entanglement Hamiltonian

Square ladder

2D Ising

Triangle ladder

2D 3-state Potts

slide-22
SLIDE 22

Rydberg Atom

Max Planck Institute

Ground state Rydberg atom (excited state) Interaction

H = Ω

  • i∈Λ

σx

i + ∆

  • i∈Λ

ni + V

  • i,j

ninj |ri − rj|γ

slide-23
SLIDE 23

Quantum hard-core lattice gas model

ni = σz

i + 1

2 Construct a solvable model Hsol = −√z

  • iΛ

(σ+

i + σ i )Pi +

  • iΛ

[(1 − z)ni + z]Pi Hsol =

  • iΛ

h†

i(z)hi(z),

hi(z) := [σ

i − √z(1 − ni)]Pi

Creation/annihilation Interaction btw particles & chemical potential

1-dim chain

Transverse Ising model with constraint Pi :=

  • jGi

(1 − nj)

Hamiltonian is positive semi-definite. Eigenenergies are non-negative.

Zero-energy state (ground state)

Unique (Perron-Frobenius theorem) H =

L

  • i=1

P

  • −√zσx

i + (1 − 3z)ni + zni−1ni+1 + z

  • P

|z = 1

  • Ξ(z)
  • iΛ

exp(zσ+

i Pi) | · · ·

: Vacuum state | · · ·

slide-24
SLIDE 24

GS of the quantum hard-core lattice gas model

unnormalized ground state: |Ψ(z) :=

  • Ξ(z)|z =
  • C∈S

znC/2|C C Λ S

: classical configuration of particle on : set of classical configurations with “constraint”

C|C = δC,C ( is orthonormal basis) |C Normalization factor = Partition function of classical hard-core lattice gas model nC: number of particles in the state C : chemical potential z Ξ(z) = Ψ(z)|Ψ(z) =

  • C∈S

znC

slide-25
SLIDE 25

GS of the quantum hard-core lattice gas model

Periodic boundary condition is imposed in the leg direction. unnormalized ground state:

τ1 τ2 τL σ1 σ2 σL τ1 τ2 τL σ1 σ2 σL

a

b c

d

= w(a, b, c, d)

[T(z)]τ,σ =

L

  • i=1

z(σi+τi)/2(1 − σiτi)(1 − σiσi+1)(1 − τiτi+1)

[T(z)]τ,σ =

L

  • i=1

z(σi+τi)/2(1 − σiτi)(1 − σiσi+1)(1 − τiτi+1)(1 − τiσi+1)

Square ladder Triangle ladder

1 z1/4 z1/2 z1/4 z1/4 z1/4 z1/2 1 z1/4 z1/2 z1/4 z1/4 z1/4

Square ladder Triangle ladder

Subsystem B Subsystem A Subsystem B Subsystem A |Ψ(z) =

  • σ
  • τ

[T(z)]τ,σ|τ |σ, [T(z)]τ,σ :=

L

  • i=1

w(σi, σi+1, τi+1, τi)

slide-26
SLIDE 26

GS of the quantum hard-core lattice gas model

Periodic boundary condition is imposed in the leg direction. unnormalized ground state:

τ1 τ2 τL σ1 σ2 σL τ1 τ2 τL σ1 σ2 σL

a

b c

d

= w(a, b, c, d)

Square ladder Triangle ladder

Subsystem B Subsystem A Subsystem B Subsystem A |z = 1

  • Ξ(z)
  • σ
  • τ

[T(z)]τ,σ |τ |σ |Ψ(z) =

  • σ
  • τ

[T(z)]τ,σ|τ |σ, [T(z)]τ,σ :=

L

  • i=1

w(σi, σi+1, τi+1, τi) M = 1 Ξ(z)[T(z)]TT(z)

Overlap matrix

  • Phys. Rev. A, 86, 032326 (2012)
slide-27
SLIDE 27

Entanglement entropy

S = −Tr [M ln M] = −

  • α

pα ln pα pα (α = 1, 2, · · · , NL) 1 2 3 4 5 10 20 30 S z (a) 2 4 6 8 10 50 100 150 S z (b)

Square ladder Triangle ladder

# of states

slide-28
SLIDE 28

Estimation of zc

ξ(z) := 1 ln[p(1)(z)/p(2)(z)] p(1)(z) p(2)(z)

: the largest eigenvalue of M : the second-largest eigenvalue of M

L 1 2 2 4 6 ξ(z)/L z (e) L 1 2 5 10 15 ξ(z)/L z (f)

Finite-size scaling for correlation length Square ladder Triangle ladder

correlation length crosses at

slide-29
SLIDE 29

Finite-size scaling

ξ(z)/L = f[(z − zc)L1/ν]

0.5 1 1.5 2 2.5

  • 10

10 ξ(z)/L (z-zc)L1/ν (a)

L= 6 L= 8 L=10 L=12 L=14 L=16 L=18 L=20 L=22

0.5 1 1.5 2

  • 50

50 ξ(z)/L (z-zc)L1/ν (b)

L= 6 L= 9 L=12 L=15 L=18 L=21

Finite-size scaling relation: 2D Ising

ν = 1 ν = 5/6 Square ladder Triangle ladder

2D 3-state Potts

slide-30
SLIDE 30

Entanglement spectra at z=zc

Eigenvalues of entanglement Hamiltonian at

2 4 6 0.25 0.5 0.75 1 λ-λ0 k/2π

Square ladder

1 2 0.25 0.5 0.75 1 λ-λ0 k/2π

Triangular ladder

Primary field Descendant field

c = 1/2 (2-dim Ising) λα − λ0 = 2πv L (hL,α + hR,α)

v hL,α hR,α : Velocity : Holomorphic conformal weight : Antiholomorphic conformal weight hL,α + hR,α Scaling dimension

(2-dim 3-state Potts) c = 4/5 z = zc

  • M. Henkel “Conformal invariance and

critical phenomena” (Springer)

slide-31
SLIDE 31

Nested entanglement entropy at z=zc

0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 s(l,L) ln[g(l)] (a)

c=1/2

0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 s(l,L) ln[g(l)] (b)

c=4/5

|ψ0: Ground state of entanglement Hamiltonian ( ) () := Tr+1,··· ,L[|00|]

nested reduced density matrix:

s(, L) := −Tr1,··· ,[() ln ()]

nested entanglement entropy:

  • Phys. Rev. B 84, 245128 (2011).

Interdisciplinary Information Sciences, 19, 101 (2013)

s(, L) = c 3 ln[g()] + s1, g() = L sin

  • L
  • Triangle ladder

L=6-24 Square ladder L=6-24

B A

  • 2D Ising

2D 3-state Potts

z = zc

slide-32
SLIDE 32

Digest

VBS state on 2D lattice Quantum lattice gas on ladder

VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder

Volume exclusion effect

Entanglement properties of 2D quantum systems Physical properties of 1D quantum systems

Total system

Entanglement Hamiltonian

Square lattice

1D AF Heisenberg

Hexagonal lattice 1D F Heisenberg Total system

Entanglement Hamiltonian

Square ladder

2D Ising

Triangle ladder

2D 3-state Potts

slide-33
SLIDE 33

Conclusion

VBS state on 2D lattice Quantum lattice gas on ladder

VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder

Volume exclusion effect

Entanglement properties of 2D quantum systems Physical properties of 1D quantum systems

Total system

Entanglement Hamiltonian

Square lattice

1D AF Heisenberg

Hexagonal lattice 1D F Heisenberg Total system

Entanglement Hamiltonian

Square ladder

2D Ising

Triangle ladder

2D 3-state Potts

slide-34
SLIDE 34

Thank you for your attention!!

VBS on symmetric graphs, J. Phys. A, 43, 255303 (2010) “VBS/CFT correspondence”, Phys. Rev. B, 84, 245128 (2011) Quantum hard-square model, Phys. Rev. A, 86, 032326 (2012) Nested entanglement entropy, Interdisciplinary Information Sciences, 19, 101 (2013)