scaling around the criticality arXiv:1709.01275 Hiroshi Ueda (RIKEN - - PowerPoint PPT Presentation

Classical analogue of finite entanglement scaling around the criticality

Hiroshi Ueda (RIKEN AICS)

2017/10/27 Novel Quantum States in Condensed Matter 2017@YITP arXiv:1709.01275

Outline

✓ Matrix product state & Intrinsic correlation length ✓ History of Finite-entanglement(𝑛) scaling at the criticality ✓ Finite-𝑛 scaling near the criticality ✓ Demonstration: 2D Ising model ✓ Discretized Heisenberg model: Icosahedron model ✓ Summary & Future issues

Matrix product state

✓ Uniform canonical MPS with infinite boundary condition

ۧ |Ψ = σ𝛽,𝛾=1

𝑛

σ𝜏1,⋯,𝜏𝑂=1

𝑒

ΛΓ𝜏1 ⋯ ΛΓ𝜏𝑂Λ 𝛽𝛾 ۧ |𝛽 ⊗ ۧ |𝜏1 ⋯ 𝜏𝑂 ⊗ ۧ |𝛾 𝝁 ∈ ℝ𝑛, Λ = diag(𝝁) : , Γ𝜏 ∈ ℂ𝑛×𝑛: 𝛽𝜏1 ⋯ 𝜏𝑂𝛾 Ψ =

…

𝛽 𝜏1 𝜏𝑂 𝛾 𝜏

Matrix product state

✓ Canonical form

= = = 1, Tr Λ2 = 1, σ𝜏 Γ†𝜏Λ2Γ𝜏 = σ𝜏 Γ𝜏Λ2Γ†𝜏 = 𝐽  Ψ Ψ = 1 Ψ Ψ =

Matrix product state

✓ Canonical form

= = = 1, Tr Λ2 = 1, σ𝜏 Γ†𝜏Λ2Γ𝜏 = σ𝜏 Γ𝜏Λ2Γ†𝜏 = 𝐽  Ψ Ψ = 1 Ψ Ψ =

Matrix product state

✓ Canonical form

= = = 1, Tr Λ2 = 1, σ𝜏 Γ†𝜏Λ2Γ𝜏 = σ𝜏 Γ𝜏Λ2Γ†𝜏 = 𝐽  Ψ Ψ = 1 Ψ Ψ =

Matrix product state

✓ Canonical form

= = = 1, Tr Λ2 = 1, σ𝜏 Γ†𝜏Λ2Γ𝜏 = σ𝜏 Γ𝜏Λ2Γ†𝜏 = 𝐽  Ψ Ψ = 1 Ψ Ψ =

Matrix product state

✓ Canonical form

= = = 1, Tr Λ2 = 1, σ𝜏 Γ†𝜏Λ2Γ𝜏 = σ𝜏 Γ𝜏Λ2Γ†𝜏 = 𝐽  Ψ Ψ = 1 Ψ Ψ =

Transfer matrix

✓ Local operator 𝑃 ∈ ℂ𝑒×𝑒: ✓ 𝐹 𝑃 ∈ ℂ𝑛2×𝑛2: ✓ 𝐹 1 = ✓ Eigenproblem

= σ𝑗 𝜂𝑗

𝑗 𝑗

Because of the canonical form 𝜂1 = 1,

1

=

, 1

= 𝜂𝑗>1 < 1 Assume MPS is not a cat state:

Correlation length of MPS

✓

Ψ 𝑃1𝑃𝑠+1 Ψ = … …

Correlation length of MPS

✓

Ψ 𝑃1𝑃𝑠+1 Ψ = σ𝑗 𝜂𝑗

𝑠−1

= σ𝑗 𝜂𝑗

𝑠−1𝐺 𝑗

= 𝐺

1 + σ𝑗=2 𝜂𝑗 −1𝑓 − 𝑠

𝜊𝑗𝐺

𝑗 where 𝜊𝑗 = −ln 𝜂𝑗 −1 𝑗 𝑗

Correlation length of MPS

✓

Ψ 𝑃1𝑃𝑠+1 Ψ = σ𝑗 𝜂𝑗

𝑠−1

= σ𝑗 𝜂𝑗

𝑠−1𝐺 𝑗

= 𝐺

1 + σ𝑗=2 𝜂𝑗 −1𝑓 − 𝑠

𝜊𝑗𝐺

𝑗 where 𝜊𝑗 = −ln 𝜂𝑗 −1 𝑗 𝑗

For the power-law decay: 1) 𝐺

1 = 0

2) Infinite sum of 𝜂𝑗

−1𝑓 − 𝑠

𝜊𝑗𝐺

𝑗

MPS with finite 𝑛: intrinsic correlation length 𝜊 𝑛 ≔ 𝜊2

Outline

✓ Matrix product state & Intrinsic correlation length ✓ History of Finite-entanglement(𝑛) scaling at the criticality ✓ Finite-𝑛 scaling near the criticality ✓ Demonstration: 2D Ising model ✓ Discretized Heisenberg model: Icosahedron model ✓ Summary & Future issues

Optimization method of iMPS

✓ iDMRG [1D Quantum: White (1992), McCulloch (2008), 2D Classical: Nishino (1995)] ✓ iTEBD [Vidal (2007)] ✓ TDVP [Haegeman et.al.(2011)]

Fixed point: Equivalent each other [ Question ]

The form of 𝜊 𝑛 at criticality: 𝜊 𝑛 → ∞ → ∞

( 𝜉 = 1, 𝑒 = 2 )

Intrinsic correlation length of MPS at criticality (Classical 2D Ising)

✓ Nishino, Okunishi, and Kikuchi, Phys. Lett. A 213, 69 (1996).

𝜊 𝑛, 𝑂 = 𝜊 𝑛 ℱ

𝜊 𝑛 𝑂

, ℱ 𝑦 = ቊ 𝑦−1 if 𝑦 ≫ 1, const. if 𝑦 ≪ 1

CTMRG

Intrinsic correlation length of MPS at criticality (1D free fermion)

✓ Andersson, Boman, and Östlund, Phys. Rev. B 59, 10493 (1999).

𝜇 ≔ 𝜂2 𝜂2 ≃ 1 − 𝑙𝑛−𝛾 𝜊 𝑛 ≃ −

1 ln 𝜂2 ≃ 1 𝑙 𝑛𝛾

( 𝛾 ≃ 1.3, 𝑙 ∼ 0.45 )

iDMRG

Intrinsic correlation length of MPS at criticality (Quantum 1D)

✓ Tagliacozzo, Oliveira, Iblisdir, and Latorre, Phys. Rev. B 78, 024410 (2008).

S=1/2 Heisenberg 𝑑 = 1 Transverse Field Ising 𝑑 = 1/2

𝜊 𝑛 ≃ 𝑛𝜆 𝜆Ising ≈ 2.0 𝜆HB ≈ 1.37

iTEBD 𝜓 ≔ 𝑛

Finite-entanglement scaling in quantum 1D systems at criticality

✓ Pollmann, Mukerjee, Turner, and Moore, Phys. Rev. Lett. 102, 255701 (2009)

Asymptotic theory: 𝜆 = 6 𝑑 12 𝑑 + 1

Calabrese and Lefevre,

• Phys. Rev. A 78,

032329 (2008). The mean # of values larger than 𝜇:

Motivation

✓ Finite-entanglement(𝑛) scaling ✓ 𝜊 𝑛 ≃ 𝑛𝜆, 𝜆 =

6 𝑑

12 𝑑 +1

at the critical point

✓ Classical analogue of the finite-𝑛 scaling near the criticality ✓ 𝜊 𝑛, 𝑈

if 𝑈 − 𝑈

c ≪ 1 ✓ Demonstration: Ising model (𝑑 = 1/2), Icosahedron model (𝑑 ∼ 2)

Outline

✓ Matrix product state & Intrinsic correlation length ✓ History of Finite-entanglement(𝑛) scaling at the criticality ✓ Finite-𝑛 scaling near the criticality ✓ Demonstration: 2D Ising model ✓ Discretized Heisenberg model: Icosahedron model ✓ Summary & Future issues

Finite-𝑛 scaling near criticality

✓ Finite size scaling [Fisher and Barber, 1972, 1983]

+ Finite-𝑛 scaling at criticality

✓ Scaling assumption 1

Nishino, Okunishi and Kikuchi, PLA, 1996 Andersson, Boman, and Östlund, PRB 1999 Tagliacozzo, Oliveira, Iblisdir, and Latorre, PRB, 2008 Pollmann, Mukerjee, Turner, and Moore, PRL, 2009 Pirvu, Vidal, Verstraete, and Tagliacozzo, PRB, 2012

𝑐: characteristic length scale

intrinsic to the system

Finite-𝑛 scaling near criticality

✓ Effective correlation length at the fixed point of CTMRG, iDMRG, iTEBD… ✓ Scaling assumption 2 ✓ 𝑐 ∼ 𝜊(𝑛, 𝑢) & Scaling assumption 1

𝜂1 and 𝜂2: the largest and second-largest eigenvalues of the row-to-row transfer matrix.

Classical analogue of Entanglement Entropy

𝛀

• Ground state
• Eigenvector
• Quantum 1D Hamiltonian
• Classical 2D Transfer matrix

𝑰

{𝝉} {𝝉′}

𝑰

= 𝐹𝑕

𝛀

{𝝉} {𝝉′}

=

Corner transfer matrix : 𝑀 × ∞, 𝑀 = 4

Classical analogue of Entanglement Entropy

• Reduced density matrix：𝜍A

{𝝉A} {𝝉A

′ } 𝛀 𝛀∗

=

𝑽 𝑽∗ Λ2

=

𝑽 𝑽∗ 𝑾∗ 𝑾 Λ Λ

{𝝉A} {𝝉A

′ }

=

𝑾∗ 𝑾 𝑽 𝑽∗ 𝑾 𝑾∗ 𝑽∗ 𝑽 Ω Ω Ω Ω

=

𝑽 𝑽∗ Ω4

Classical analogue of Entanglement Entropy

Same Ω𝑗

• Entanglement Entropy

𝑇A = − σ𝑗 𝜇𝑗

2 × 2 log 𝜇𝑗

𝑇E = − σ𝑗 Ω𝑗

4 × 4 log Ω𝑗

• CTM of CTMRG 【Nishino,Okunishi(1996)】：𝑀 × 𝑀

𝑀 2 3 4

𝑀 ≫ 𝜊(𝑛, 𝑈) 𝑀 ≫ 𝜊(𝑛, 𝑈)

𝑛: # of renormalized states ※finite 𝑛 ⇒ finite 𝜊 𝑛, 𝑈

CTM:𝑴 × ∞

Classical analogue of Entanglement entropy

✓ Definition:

Near the criticality:

✓ Finite-𝑛 scaling

Vidal, Latorre, Rico, and Kitaev, PRL, 2003 Calabrese and Cardy, J. Stat. Mech., 2004

𝑏: non-universal constant 𝑑: central charge

Outline

✓ Matrix product state & Intrinsic correlation length ✓ History of Finite-entanglement(𝑛) scaling at the criticality ✓ Finite-𝑛 scaling near the criticality ✓ Demonstration: 2D Ising model ✓ Discretized Heisenberg model: Icosahedron model ✓ Summary & Future issues

Finite-𝑛 scaling for 𝜊

2D Ising model: 𝑈C = 2.269 ⋯, 𝑑 = 1/2, 𝜉 = 1, 𝛾 = 1/8, 𝜆 =

6 𝑑 1+ 12/𝑑

Finite-𝑛 scaling for 𝑁

𝑁 ≡ 2 𝜀1,𝜏 − 1

2D Ising model: 𝑈C = 2.269 ⋯, 𝑑 = 1/2, 𝜉 = 1, 𝛾 = 1/8, 𝜆 =

6 𝑑 1+ 12/𝑑

Finite-𝑛 scaling for 𝑇E

2D Ising model: 𝑈C = 2.269 ⋯, 𝑑 = 1/2, 𝜉 = 1, 𝛾 = 1/8, 𝜆 =

6 𝑑 1+ 12/𝑑

Outline

✓ Matrix product state & Intrinsic correlation length ✓ History of Finite-entanglement(𝑛) scaling at the criticality ✓ Finite-𝑛 scaling near the criticality ✓ Demonstration: 2D Ising model ✓ Discretized Heisenberg model: Icosahedron model ✓ Summary & Future issues

Discretized classical Heisenberg model

Tetrahedron model：4 states

Discretized classical Heisenberg model

Cube model：8 states

Discretized classical Heisenberg model

Octahedron model：6 states

Discretized classical Heisenberg model

Dodecahedron model：20 states

Discretized classical Heisenberg model

Icosahedron model：12 states

Discretization & Universality class

# of vertexes: Universality Class: 4 4-state Potts

[Wu,1982]

6 2nd-order

［Surungan&Okabe, 2012］ ↓

Weak 1st-order

[Roman,et al., 2016]

8 Ising × 3 12 2nd-order

[Patrascioiu, et al., 2001] [Surungan& Okabe, 2012]

20 BKT?

[Patrascioiu, et al., 1991] ↓

2nd-order

［Surungan&Okabe, 2012］

MC MC MC MC MC CTMRG

Discretization & Universality class

# of vertexes: Universality Class: 4 4-state Potts

[Wu,1982]

8 Ising × 3 12 2nd-order

[Patrascioiu, et al., 2001] [Surungan& Okabe, 2012]

20 BKT?

[Patrascioiu, et al., 1991] ↓

2nd-order

［Surungan&Okabe, 2012］

MC MC MC MC 6 2nd-order

［Surungan&Okabe, 2012］ ↓

Weak 1st-order

[Roman,et al., 2016]

MC CTMRG

6 2nd-order

［Surungan&Okabe, 2012］ ↓

Weak 1st-order

[Roman,et al., 2016]

MC CTMRG

Discretization & Universality class

# of vertexes: Universality Class: 4 4-state Potts

[Wu,1982]

8 Ising × 3 20 BKT?

[Patrascioiu, et al., 1991] ↓

2nd-order

［Surungan&Okabe, 2012］

MC MC 12 2nd-order

[Patrascioiu, et al., 2001] [Surungan& Okabe, 2012]

MC MC

N TC n g h

320&FSS 0.555 1 1.7−0.1

+0.3

3.0−0.1

+0.5

− 256&FSS 0.555 1 1.30 1 − 0.199 1

2nd-order

[Patrascioiu, et al., 2001] [Surungan& Okabe, 2012]

MC MC

N TC n g h

320&FSS 0.555 1 1.7−0.1

+0.3

3.0−0.1

+0.5

0.25−0.44

+0.30

256&FSS 0.555 1 1.30 1 2.34 2 0.199 1

Motivation & Conclusion

m TC n b c

500&FmS 0.5550 1 1.62 2 0.12 1 1.90 2

（Using the scaling law）

This work CTMRG

Icosahedron model

✓ Icosahedral symmetry

• Centers of edges (two-fold)
• Two opposite vertexes (five-fold)
• Centers of faces (three-fold)

Icosahedron model

✓ Vertex representation

Icosahedron model

✓ Vertex representation

Finite-𝑛 scaling for 𝜊

✓ Bayesian inference

[Harada, PRE, 2011]

✓ Scaling parameters

𝑈

c =0.5550

𝜉 =1.617, 𝜆 =0.898

TC n

Patrascioiu, et al., 2001 0.555 1 1.7−0.1

+0.3

Surungan& Okabe, 2012 0.555 1 1.30 1

Finite-𝑛 scaling for 𝑁

✓ Scaling parameter

𝛾 = 0.129 𝑈

c =0.5550

𝜉 =1.617 𝜆 =0.898

Shoulder-like structure disappears at 𝑛 → ∞ Single order-disorder phase transition occurs.

Finite-𝑛 scaling for 𝑇E

✓ Scaling parameter

𝑑 = 1.894

✓ Entanglement scaling

𝜆 =

6 𝑑 12/𝑑+1

This work:

6 𝑑 12/𝑑+1 − 𝜆 = 0.009

𝑈

c =0.5550

𝜉 =1.617 𝜆 =0.898

[ Pollmann, Mukerjee, Turner, and Moore, PRL, 2009 ]

Outline

✓ Matrix product state & Intrinsic correlation length ✓ History of Finite-entanglement(𝑛) scaling at the criticality ✓ Finite-𝑛 scaling near the criticality ✓ Demonstration: 2D Ising model ✓ Discretized Heisenberg model: Icosahedron model ✓ Summary & Future issues

Summary

✓ Classical analogue of

Finite-entanglement(𝑛) scaling near the criticality

✓ Target

✓ 2D Ising model on the square lattice ✓ 2D Discretized classical Heisenberg model: Icosahedron model

Summary

✓ Icosahedron model:

• Single order-disorder phase transition
• Ordered phase:5-fold rotational symmetry

✓ Critical temperature and exponents

𝑈𝐃 𝜉 𝜆 𝛾 𝑑

6 𝑑 12/𝑑 + 1 − 𝜆 0.5550(1) 1.62(2) 0.89(2) 0.12(1) 1.90(2) ~0.009

cannot be explained by the minimal series of conformal field theory.

✓ Future issue：anisotropy effect，Dodecahedron model，etc.