Report

Tensor Network States: Algorithms and Applications December 1-5, 2014 Beijing, China The multi-scale Entanglement Renormalization Ansatz MERA -- a pedagogical introduction-- Guifre Vidal outline MERA • Definition • Efficiency • Structural properties: correlations and entropy The Renormalization Group • Goals • RG by isometries (TTN): why is it “wrong”? TRG • RG by isometries and disentanglers (MERA) (Zhiyuan’s lecture) TNR (Glen’s talk) MERA: definition ⊗ Ψ ∈ (ℂ ) complex numbers Multi-scale entanglement renormalization ansatz (MERA) Matrix product state (MPS) MERA also MERA ! Efficiency ⊗ Ψ ∈ (ℂ ) + 1 1 + + … = 1+ + +⋯ 2 4 2 4 ≤ 2 complex numbers Multi-scale entanglement renormalization ansatz (MERA) Matrix product state (MPS) spins ⇒ tensors ⇒ () parameters log() spins ⇒ log() tensors ? 2 tensors ⇒ () parameters Q1 efficiency Matrix product state (MPS) Q2 〈Ψ Ψ cost () 〈Ψ| Ψ cost () efficiency |Ψ isometric tensors! † 〈Ψ|Ψ = † † =1 = = Q3 cost = 0 !!! efficiency 〈Ψ|Ψ = isometric tensors! † † † =1 = = efficiency 〈Ψ|Ψ = isometric tensors! † † † =1 = = efficiency 〈Ψ|Ψ = isometric tensors! † † † =1 = = efficiency 〈Ψ|Ψ = isometric tensors! † † † =1 = = efficiency 〈Ψ|Ψ = isometric tensors! † † † =1 = = efficiency 〈Ψ|Ψ = 1 isometric tensors! † † † =1 = = 〈Ψ| Ψ = isometric tensors! † † † =1 = = = (1) = 〈Ψ| Ψ = (2) = (3) cost O(log ) = ΨΨ Structural properties ⊗ Ψ ∈ (ℂ ) • Decay of correlations • Scaling of entanglement complex numbers ⟨Ψ| 0 ()|Ψ MPS = = −1 = ≈ −1 = −/ ≡− ⇒ Exponential decay of correlations 1 log Ψ 0 Ψ ⋯ ⋯ ⋯ = MERA ⋯ ⋯ ⋯ Ψ 0 Ψ ⋯ MERA ⋯ = ≈ = log3 log3 () = 2 log3() = 2 log3() = − log3() = log3() ≡ −2 log 3 () ⇒ Polynomial decay of correlations ⋯ ⋯ Correlations: summary and interpretation matrix product state (MPS) multi-scale entanglement renormalization ansatz (MERA) log() structure of geodesics: ⟨ 0 ≈ −/ exponential structure of geodesics: ⟨ 0 ≈ − power-law Entanglement entropy matrix product state multi-scale entanglement renormalization ansatz (MPS) (MERA) log() connectivity: () ≈ boundary law! Q4 connectivity: () ≈ log logarithmic correction! Q5 ⋮ ⋯ ⋯ () ≈ log Example: operator content of quantum Ising model ⊗ +1 +ℎ = for ℎ = ℎ = 1 scaling dimension (exact ) scaling operators/dimensions: identity spin energy disorder fermions 0 scaling dimension (MERA) 0 error ---- 0.125 0.124997 0.003% 0.99993 0.007% 0.1250002 0.0002% 0.5 <10−8 % 0.5 <10−8 % 1 0.125 0.5 0.5 OPE for local & non-local primary fields C 1 / 2 C i C 1 / 2 C i C e i / 4 C e i / 4 / fusion rules 2 4 ( 6 10 ) / 2 I I+ I I { I, , , , , } local and semi-local subalgebras { I, } { I, , } I { I, , } { I, , , } ... MERA and HOLOGRAPHY t t s x CFT1+1 x x AdS2+1 outline MERA • Definition • Efficiency • Structural properties: correlations and entropy The Renormalization Group • Goals • RG by isometries: why is it wrong? • RG by isometries and disentanglers The Renormalization Group: goals Given a local Hamiltonian = ℎ,+1 on sites (Hilbert space dimension ) Two type of questions: 1) Low energy, large distance, UNIVERSAL behavior: e.g. disordered/symmetry-breaking phase, topological order (S,T modular matrices), quantum criticality (scaling operators, CFT data) fixed point → ′ → ′′ → ⋯ → (⋆) Ψ |Ψ〉 for − → ∞ 2) Low energy, short distance, detailed MICROSCOPIC properties e.g. 〈Ψ| |Ψ〉, Ψ |Ψ〉, for all , The Renormalization Group: goals Example: given the (transverse field) Ising Hamiltonian ⊗ +1 +ℎ = (ℎ) spontaneous magnetization 0 ℎ ℎ magnetic field 1) Low energy, large distance, UNIVERSAL behavior? Is the spontaneous magnetization m(h) ≡ Ψ Ψ ≠ 0, or = 0? ordered phase disordered phase 2) Low energy, short distance, detailed MICROSCOPIC properties? How much is the spontaneous magnetization m(ℎ) ≡ 〈Ψ| |Ψ〉 as a function on ℎ? The Renormalization Group on Hamiltonians: on ground state wave-functions: on classical partition functions: → ′ → ′′ → ⋯ → (⋆) |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 → ′ → ′′ → ⋯ → (∗) fixed point Hamiltonian fixed point ground state fixed point partition function Two types of Renormalization Group transformations: • Type 1 is only required to preserve UNIVERSAL properties • Type 2 is also required to preserve MICROSCOPIC properties → ′ → ′′ → ⋯ → (⋆) such that (for instance, = in quantum Ising model) Ψ Ψ = Ψ ′ |o′|Ψ ′ = ′′ ′′ ′′ = ⋯ = (∗) (∗) (∗) For instance, TRG, TTN, MERA, are of type 2 The Renormalization Group • Type 1 is only required to preserve UNIVERSAL properties • Type 2 is also required to preserve MICROSCOPIC properties? → ′ → ′′ → ⋯ → (⋆) (for instance, = in quantum Ising model) such that Ψ Ψ = Ψ ′ |o′|Ψ ′ = ′′ ′′ ′′ = ⋯ = (∗) (∗) (∗) Types 2A and 2B!! Type 2A: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 e.g. TTN, Zhiyuan’s lecture on TRG fixed point: mixture of UNIVERSAL and MICROSCOPIC properties Type 2B: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 e.g. MERA, Glen’s talk on TNR fixed point: only UNIVERSAL properties The Renormalization Group Type 2A: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 fixed point: mixture of UNIVERSAL and MICROSCOPIC properties example: TTN ′ = /3 sites ℒ′ † ℒ = sites coarse-graining transformation = ⊗ ⊗ ⋯⊗ Q6 = A’ B’ C’ D’ E’ F’ A’ B’ C’ D’ E’ F’ The Renormalization Group Type 2A: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 example: TTN fixed point: mixture of UNIVERSAL and MICROSCOPIC properties coarse-graining transformation = ⊗ ⊗ ⋯⊗ Ψ → Ψ ′ = † |Ψ〉 = Ψ † A’ B’ C’ D’ F’ E’ Ψ′ A’ B’ C’ D’ E’ F’ → ′ = † = A’ B’ C’ D’ E’ F’ † = A’ B’ C’ D’ E’ F’ A’ B’ C’ D’ E’ F’ ′ The Renormalization Group Type 2A: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 example: TTN fixed point: mixture of UNIVERSAL and MICROSCOPIC properties coarse-graining transformation = ⊗ ⊗ ⋯⊗ Ψ → Ψ ′ = † |Ψ〉 Ψ′ Ψ = † Already a fixed point wave-function! It contains short-range entanglement = MICROSCOPIC details The Renormalization Group Type 2A: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 example: TTN fixed point: mixture of UNIVERSAL and MICROSCOPIC properties coarse-graining transformation = ⊗ ⊗ ⋯⊗ |Ψ ′ 〉 retains short-range entanglement = MICROSCOPIC details Ψ Ψ = Ψ ′ |o′|Ψ ′ ⇒ results are still accurate provided that we use a sufficiently large bond dimension Different fixed-point wave-function for the same phase! (ℎ) (⋆) |Ψh1 〉 → |Ψh1 ′〉 → |Ψℎ1 ′′〉 → ⋯ → |Ψℎ1 〉 spontaneous magnetization (⋆) (⋆) |Ψℎ1 〉 |Ψℎ2 〉 0 (⋆) |Ψh2 〉 → |Ψh2 ′〉 → |Ψℎ2 ′′〉 → ⋯ → |Ψℎ2 〉 (⋆) (⋆) |Ψℎ1 〉 and |Ψℎ2 〉 have the same UNIVERSAL information, mixed with different MICROSCOPIC details ℎ ℎ1 ℎ2 magnetic field The Renormalization Group Type 2A: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 example: TTN fixed point: mixture of UNIVERSAL and MICROSCOPIC properties coarse-graining transformation = ⊗ ⊗ ⋯⊗ |Ψ ′ 〉 retains short-range entanglement = MICROSCOPIC details Ψ Ψ = Ψ ′ |o′|Ψ ′ ⇒ results are still accurate provided that we use a sufficiently large bond dimension (⋆) (⋆) |Ψℎ1 〉 and |Ψℎ2 〉 have the same UNIVERSAL information, mixed with different MICROSCOPIC details Two problems: • Computational: RG scheme is more expensive (⋆) • Conceptual: How do we separate UNIVERSAL from MICROSCOPIC in |Ψℎ 〉 ? The Renormalization Group Type 2B: Proper RG transformation: fixed point: only UNIVERSAL properties |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 example: MERA ′ = /3 sites ℒ′ † ℒ sites coarse-graining transformation = (⋯ ⊗ ⊗ ⊗ ⋯ )(⋯ ⊗ ⊗ ⊗ ⋯) Q7 = A’ B’ C’ D’ E’ F’ A’ B’ C’ D’ E’ F’ † = = The Renormalization Group Type 2B: example: MERA Proper RG transformation: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 fixed point: only UNIVERSAL properties coarse-graining transformation = (⋯ ⊗ ⊗ ⊗ ⋯ )(⋯ ⊗ ⊗ ⊗ ⋯) Ψ → Ψ ′ = † |Ψ〉 = Ψ † Ψ′ A’ B’ C’ D’ E’ F’ → ′ = † = A’ B’ C’ D’ E’ F’ = A’ B’ C’ D’ E’ F’ A’ B’ C’ D’ E’ F’ ′ The Renormalization Group Type 2B: example: MERA Proper RG transformation: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 fixed point: only UNIVERSAL properties coarse-graining transformation = (⋯ ⊗ ⊗ ⊗ ⋯ )(⋯ ⊗ ⊗ ⊗ ⋯) Ψ → Ψ ′ = † |Ψ〉 Ψ † Product state wave-function! It contains no MICROSCOPIC details = = Ψ′ A’ B’ C’ D’ E’ F’ = = The Renormalization Group Type 2B: example: MERA Proper RG transformation: fixed point: only UNIVERSAL properties |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 coarse-graining transformation = (⋯ ⊗ ⊗ ⊗ ⋯ )(⋯ ⊗ ⊗ ⊗ ⋯) Same fixed-point wave-function for the same phase! |Ψ 〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) |Ψ ℎ=0 (⋆) |Ψ ℎ for ℎ < ℎ_ 〉 for ℎ = ℎ_ 〉 (⋆) ℎ=∞ for ℎ > ℎ_ 〉 (ℎ) magnetic field h spontaneous magnetization ℎ=0 ℎ′ |Ψ (⋆) ℎ=0 〉 ℎ ℎ |Ψ (⋆ ) ℎ 〉 ℎ′ ℎ=∞ |Ψ (⋆) ℎ=∞ 〉 The Renormalization Group Type 2B: example: MERA Proper RG transformation: |Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉 fixed point: only UNIVERSAL properties coarse-graining transformation = (⋯ ⊗ ⊗ ⊗ ⋯ )(⋯ ⊗ ⊗ ⊗ ⋯) Same fixed-point wave-function for the same phase! |Ψ 〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) |Ψ ℎ=0 (⋆) |Ψ ℎ for ℎ < ℎ_ 〉 for ℎ = ℎ_ 〉 (⋆) ℎ=∞ 〉 for ℎ > ℎ_ With MERA, we have solved the two problems of TTN: • Computational: RG scheme is now scalable (⋆) • Conceptual: |Ψ 〉 only contains UNIVERSAL information (it can be more easily extracted) summary MERA • Definition • Efficiency • Structural properties: correlations and entropy ⟨ 0 ≈ − () ≈ log The Renormalization Group • Goals ground state wave-function |Ψ〉 classical partition function • RG by isometries: why is it “wrong”? TTN TRG • RG by isometries and disentanglers MERA TNR (Zhiyuan’s lecture) (Glen’s talk)