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

2017/10/27 Novel Quantum States in Condensed Matter 2017@YITP Classical analogue of finite entanglement scaling around the criticality arXiv:1709.01275 Hiroshi Ueda (RIKEN AICS)

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 𝜏 , Γ 𝜏 ∈ ℂ 𝑛×𝑛 : 𝝁 ∈ ℝ 𝑛 , Λ = diag(𝝁) : ΛΓ 𝜏 1 ⋯ ΛΓ 𝜏 𝑂 Λ 𝛽𝛾 𝑛 𝑒 |Ψ = σ 𝛽,𝛾=1 ۧ σ 𝜏 1 ,⋯,𝜏 𝑂 =1 ۧ ۧ ۧ |𝛽 ⊗ |𝜏 1 ⋯ 𝜏 𝑂 ⊗ |𝛾 𝜏 𝑂 𝜏 1 𝛽𝜏 1 ⋯ 𝜏 𝑂 𝛾 Ψ = … 𝛽 𝛾

Matrix product state ✓ Canonical form Tr Λ 2 = 1 , σ 𝜏 Γ †𝜏 Λ 2 Γ 𝜏 = σ 𝜏 Γ 𝜏 Λ 2 Γ †𝜏 = 𝐽  Ψ Ψ = 1 = 1 , = = Ψ Ψ =

Matrix product state ✓ Canonical form Tr Λ 2 = 1 , σ 𝜏 Γ †𝜏 Λ 2 Γ 𝜏 = σ 𝜏 Γ 𝜏 Λ 2 Γ †𝜏 = 𝐽  Ψ Ψ = 1 = 1 , = = Ψ Ψ =

Matrix product state ✓ Canonical form Tr Λ 2 = 1 , σ 𝜏 Γ †𝜏 Λ 2 Γ 𝜏 = σ 𝜏 Γ 𝜏 Λ 2 Γ †𝜏 = 𝐽  Ψ Ψ = 1 = 1 , = = Ψ Ψ =

Matrix product state ✓ Canonical form Tr Λ 2 = 1 , σ 𝜏 Γ †𝜏 Λ 2 Γ 𝜏 = σ 𝜏 Γ 𝜏 Λ 2 Γ †𝜏 = 𝐽  Ψ Ψ = 1 = 1 , = = Ψ Ψ =

Matrix product state ✓ Canonical form Tr Λ 2 = 1 , σ 𝜏 Γ †𝜏 Λ 2 Γ 𝜏 = σ 𝜏 Γ 𝜏 Λ 2 Γ †𝜏 = 𝐽  Ψ Ψ = 1 = 1 , = = Ψ Ψ =

Transfer matrix ✓ Local operator 𝑃 ∈ ℂ 𝑒×𝑒 : ✓ Eigenproblem = σ 𝑗 𝜂 𝑗 𝑗 𝑗 ✓ 𝐹 𝑃 ∈ ℂ 𝑛 2 ×𝑛 2 : Because of the canonical form , 𝜂 1 = 1 , = = 1 1 ✓ 𝐹 1 = 𝜂 𝑗>1 < 1 Assume MPS is not a cat state:

Correlation length of MPS … Ψ 𝑃 1 𝑃 𝑠+1 Ψ = ✓ …

Correlation length of MPS 𝑠−1 Ψ 𝑃 1 𝑃 𝑠+1 Ψ = σ 𝑗 𝜂 𝑗 ✓ 𝑗 𝑗 𝑠−1 𝐺 = σ 𝑗 𝜂 𝑗 𝑗 − 𝑠 −1 𝑓 −1 1 + σ 𝑗=2 𝜂 𝑗 𝜊𝑗 𝐺 = 𝐺 𝑗 where 𝜊 𝑗 = −ln 𝜂 𝑗

Correlation length of MPS For the power-law decay: 1) 𝐺 1 = 0 𝑠−1 Ψ 𝑃 1 𝑃 𝑠+1 Ψ = σ 𝑗 𝜂 𝑗 ✓ 𝑗 𝑗 − 𝑠 2) Infinite sum of 𝜂 𝑗 −1 𝑓 𝜊𝑗 𝐺 𝑗 MPS with finite 𝑛 : 𝑠−1 𝐺 = σ 𝑗 𝜂 𝑗 𝑗 intrinsic correlation length 𝜊 𝑛 ≔ 𝜊 2 − 𝑠 −1 𝑓 −1 1 + σ 𝑗=2 𝜂 𝑗 𝜊𝑗 𝐺 = 𝐺 𝑗 where 𝜊 𝑗 = −ln 𝜂 𝑗

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)] Fixed point: Equivalent each other ✓ TDVP [Haegeman et.al.(2011)] [ Question ] The form of 𝜊 𝑛 at criticality: 𝜊 𝑛 → ∞ → ∞

Intrinsic correlation length of MPS at criticality (Classical 2D Ising) ✓ Nishino, Okunishi, and Kikuchi, Phys. Lett. A 213 , 69 (1996). CTMRG 𝜊 𝑛 , 𝜊 𝑛, 𝑂 = 𝜊 𝑛 ℱ 𝑂 ℱ 𝑦 = ቊ 𝑦 −1 if 𝑦 ≫ 1, ( 𝜉 = 1 , 𝑒 = 2 ) if 𝑦 ≪ 1 const.

Intrinsic correlation length of MPS at criticality (1D free fermion) ✓ Andersson, Boman, and Östlund , Phys. Rev. B 59 , 10493 (1999). 𝜂 2 ≃ 1 − 𝑙𝑛 −𝛾 iDMRG 1 1 𝑙 𝑛 𝛾 𝜊 𝑛 ≃ − ln 𝜂 2 ≃ 𝜇 ≔ 𝜂 2 ( 𝛾 ≃ 1.3 , 𝑙 ∼ 0.45 )

Intrinsic correlation length of MPS at criticality (Quantum 1D) ✓ Tagliacozzo, Oliveira, Iblisdir, and Latorre, Phys. Rev. B 78 , 024410 (2008). iTEBD 𝜓 ≔ 𝑛 𝜊 𝑛 ≃ 𝑛 𝜆 𝜆 Ising ≈ 2.0 Transverse Field Ising 𝑑 = 1/2 S=1/2 Heisenberg 𝜆 HB ≈ 1.37 𝑑 = 1

Finite-entanglement scaling in quantum 1D systems at criticality ✓ Pollmann, Mukerjee, Turner, and Moore, Phys. Rev. Lett. 102 , 255701 (2009) Asymptotic theory: 6 𝜆 = 12 𝑑 𝑑 + 1 The mean # of values larger than 𝜇 : Calabrese and Lefevre, Phys. Rev. A 78 , 032329 (2008).

Motivation ✓ Finite-entanglement( 𝑛 ) scaling 6 ✓ 𝜊 𝑛 ≃ 𝑛 𝜆 , 𝜆 = at the critical point 12 𝑑 𝑑 +1 ✓ Classical analogue of the finite- 𝑛 scaling near the criticality if 𝑈 − 𝑈 ✓ 𝜊 𝑛, 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 ] Nishino, Okunishi and Kikuchi, PLA, 1996 Andersson, Boman, and Östlund , PRB 1999 + Finite- 𝑛 scaling at criticality Tagliacozzo, Oliveira, Iblisdir, and Latorre, PRB, 2008 Pollmann, Mukerjee, Turner, and Moore, PRL, 2009 Pirvu, Vidal, Verstraete, and Tagliacozzo, PRB, 2012 ✓ Scaling assumption 1 𝑐 : characteristic length scale intrinsic to the system

Finite- 𝑛 scaling near criticality ✓ Effective correlation length at the fixed point of CTMRG, iDMRG, iTEBD … 𝜂 1 and 𝜂 2 : the largest and second-largest eigenvalues of the row-to-row transfer matrix. ✓ Scaling assumption 2 ✓ 𝑐 ∼ 𝜊(𝑛, 𝑢) & Scaling assumption 1

Classical analogue of Entanglement Entropy • Quantum 1D Hamiltonian • Classical 2D Transfer matrix {𝝉} {𝝉} 𝑰 {𝝉′} {𝝉′} • Eigenvector • Ground state 𝑰 = 𝛀 = 𝐹 𝑕 𝛀 Corner transfer matrix : 𝑀 × ∞ , 𝑀 = 4

Classical analogue of Entanglement Entropy • Reduced density matrix ： 𝜍 A Ω Ω 𝑾 ∗ 𝛀 ∗ 𝑾 {𝝉 A } ′ } 𝑽 ∗ {𝝉 A } 𝑽 {𝝉 A = 𝛀 ′ } {𝝉 A 𝑽 ∗ 𝑽 Λ 𝑽 ∗ 𝑾 = 𝑾 ∗ 𝑾 𝑾 ∗ 𝑽 Λ Ω Ω 𝑽 ∗ 𝑽 ∗ = Λ 2 = Ω 4 𝑽 𝑽

Classical analogue of Entanglement Entropy CTM: 𝑴 × ∞ • Entanglement Entropy 4 × 4 log Ω 𝑗 𝑇 E = − σ 𝑗 Ω 𝑗 𝑀 ≫ 𝜊(𝑛, 𝑈) 2 × 2 log 𝜇 𝑗 𝑇 A = − σ 𝑗 𝜇 𝑗 • CTM of CTMRG 【 Nishino,Okunishi(1996) 】 ： 𝑀 × 𝑀 Same Ω 𝑗 𝑀 ≫ 𝜊(𝑛, 𝑈) 𝑛 : # of renormalized states ※ finite 𝑛 ⇒ finite 𝜊 𝑛, 𝑈 2 3 𝑀 4

Classical analogue of Entanglement entropy ✓ Definition: Near the criticality: Vidal, Latorre, Rico, and Kitaev, PRL, 2003 Calabrese and Cardy, J. Stat. Mech., 2004 ✓ Finite- 𝑛 scaling 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 6 2D Ising model: 𝑈 C = 2.269 ⋯ , 𝑑 = 1/2 , 𝜉 = 1 , 𝛾 = 1/8 , 𝜆 = 𝑑 1+ 12/𝑑

Finite- 𝑛 scaling for 𝑁 6 2D Ising model: 𝑈 C = 2.269 ⋯ , 𝑑 = 1/2 , 𝜉 = 1 , 𝛾 = 1/8 , 𝜆 = 𝑑 1+ 12/𝑑 𝑁 ≡ 2 𝜀 1,𝜏 − 1

Finite- 𝑛 scaling for 𝑇 E 6 2D Ising model: 𝑈 C = 2.269 ⋯ , 𝑑 = 1/2 , 𝜉 = 1 , 𝛾 = 1/8 , 𝜆 = 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: 4 6 8 12 20 Ising × 3 Universality 4-state Potts 2nd-order 2nd-order BKT? Class: [Wu,1982] ［ Surungan&Okabe, 2012 ］ [Patrascioiu, [Patrascioiu, et al ., 2001] et al ., 1991] MC MC MC ↓ [Surungan& ↓ Weak 1st-order Okabe, 2012] MC 2nd-order [Roman, et al ., 2016] CTMRG ［ Surungan&Okabe, 2012 ］ MC

Discretization & Universality class # of vertexes: 4 6 8 12 20 Ising × 3 Universality 4-state Potts 2nd-order 2nd-order BKT? Class: [Wu,1982] ［ Surungan&Okabe, 2012 ］ [Patrascioiu, [Patrascioiu, et al ., 2001] et al ., 1991] MC MC MC ↓ [Surungan& ↓ Weak 1st-order Okabe, 2012] MC 2nd-order [Roman, et al ., 2016] CTMRG ［ Surungan&Okabe, 2012 ］ MC