Introduction to iPEPS (second lecture) Philippe Corboz, Institute - PowerPoint PPT Presentation

Introduction to iPEPS (second lecture) Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam

Outline ‣ Part I: iPEPS ansatz ‣ Part II: Contraction of PEPS / iPEPS ‣ Interlude: Example application: the Shastry-Sutherland model ‣ Part III: Optimization ✦ Imaginary time evolution ✦ Variational optimization (iterative energy minimization) ‣ Part IV: Computational cost & benchmarks ✦ How large does D need to be? ✦ Comparison with 2D DMRG ✦ Comments on extrapolations ‣ Part V: Advanced tensor network applications ✦ SU(N) Heisenberg models ‣ Outlook & summary

PART III: Optimization

Summary: Tensor network algorithm for ground state MPS PEPS ✓ 2D MERA Structure Variational ansatz 1D MERA ✓ Find the best Compute (ground) state observables | ˜ � ˜ Ψ | O | ˜ Ψ � Ψ ⇥ Contraction of the iterative optimization tensor network imaginary time of individual tensors exact / approximate evolution (energy minimization)

Optimization via imaginary time evolution β → ∞ • Idea: exp( � β ˆ | Ψ GS i H ) | Ψ i i τ = β /n ! n ! n Y Trotter-Suzuki exp( − β ˆ X ˆ X ˆ exp( − τ ˆ H ) = exp( − β H b ) = exp( − τ H b ) H b ) ≈ decomposition: b b b • 1D: exp( − τ ˆ H b ) ... ... • At each step: apply a two-site operator to a bond and truncate bond back to D √ √ V † U U s ˜ sV ˜ s SVD Keep D largest singular values T ime E volving B lock D ecimation (TEBD) algorithm Note: MPS needs to be in canonical form

Optimization via imaginary time evolution exp( − τ ˆ H b ) • 2D: same idea: apply to a bond and truncate bond back to D • However , SVD update is not optimal (because of loops in PEPS)! simple update (SVD) full update Jiang et al, PRL 101 (2008) Jordan et al, PRL 101 (2008) ★ “local” update like in TEBD ★ Take the full wave function into account for truncation ★ Cheap, but not optimal (e.g. overestimates magnetization ★ optimal, but computationally more in S=1/2 Heisenberg model) expensive ★ Fast-full update [Phien et al, PRB 92 (2015)] Cluster update Wang, Verstraete, arXiv:1110.4362 (2011)

Optimization: simple update Jiang, et al., PRL 101, 090603 (2008) • iPEPS with “weights” on the bonds (takes environment effectively into account) ..." ..." ..." 1/2 λ 6 λ 8 λ 6 ..." A 1/2 Γ A λ 1 λ 1 λ 3 Γ A Γ B λ 3 A B = 1/2 1/2 λ 2 λ 2 λ 4 λ 3 C D Γ Γ D λ 7 λ 5 λ 7 C ..." ..." ..." ..." λ 8 λ 6 • Update works like in 1D with iTEBD (infinite time-evolving block decimation) G. Vidal, PRL 91, 147902 (2003) λ 6 λ 8 Θ λ 3 Γ B λ 3 Γ A λ 1 = − 1 λ 6 λ 4 ˜ − 1 Γ ' A λ 3 Γ λ 2 g A = − 1 λ 2 SVD$ − 1 λ 8 ˜ ˜ ˜ Θ ˜ Γ ' B Γ − 1 Γ Γ λ λ 3 A A B 1 = = − 1 λ 4 keep only D largest singular values

Trick to make it cheaper • Idea: Split off the part of the tensor which is updated U T sV = λ 6 λ 8 λ 3 Γ B λ 3 Γ A λ 1 = = λ 4 λ 2 g g g SVD$ ˜ ˜ ˜ Γ Γ λ A B 1 = = g keep only D largest singular − 1 − 1 λ 8 λ 6 values ˜ ˜ − 1 Γ ' B Γ ' A Γ λ 3 − 1 Γ λ 3 B A = = − 1 λ 2 − 1 λ 4

Optimization: full update Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008) Corboz, Orus, Bauer, Vidal, PRB 81, 165104 (2010) • Approximate old PEPS + gate with a new PEPS with bond dimension D Environment Environment B B ' A A ' ≈ g | ˜ | Ψ 0 i Ψ i = g | Ψ i ≈ Ψ i � | Ψ 0 i || 2 = h ˜ || | ˜ Ψ | ˜ Ψ i + h Ψ 0 | Ψ 0 i � h ˜ Ψ | Ψ 0 i � h Ψ 0 | ˜ • Minimize Ψ i • Iteratively / CG / Newton / ... • The full wave function is taken into account for the truncation! • Environment has to be computed: expensive... but optimal!

Full-update: details (sV) U p • Split off the part of the tensor A X B q Y = = which is updated p q Environment of p and q = tensors p † q † 2 | ˜ find new p’, and q’ to minimize: || | ˜ Ψ i � | Ψ 0 i || | Ψ 0 ( p 0 , q 0 ) i Ψ i = g | Ψ ( p, q ) i ≈ d ( p 0 , q 0 ) = h ˜ Ψ | ˜ Ψ i + h Ψ 0 | Ψ 0 i � h ˜ Ψ | Ψ 0 i � h Ψ 0 | ˜ Ψ i “Cost-function” p 0 q 0 p 0 q 0 p q p q g g + g † - - p † q †

Finding p’ and q’ through sweeping p 0 0 q 0 SVD p q 0 • Initial guess with SVD: = g ∂ • Keep q’ fixed and optimize with respect to p’ ∂ p 0⇤ d ( p 0 , q 0 ) = 0 p 0 [ [ ∂ = 0 + - - ∂ p 0⇤ p 0 † p 0 † p 0 = new p’ Mp 0 b • Solve linear system: =

Finding p’ and q’ through sweeping p 0 0 q 0 SVD p q 0 • Initial guess with SVD: = g ∂ • Keep q’ fixed and optimize with respect to p’: ∂ p 0⇤ d ( p 0 , q 0 ) = 0 new p’ Mp 0 b • Solve linear system: = ∂ • Keep p’ fixed and optimize with respect to q’: ∂ q 0⇤ d ( p 0 , q 0 ) = 0 ˜ ˜ Mq 0 new q’ b • Solve linear system: = d ( p 0 , q 0 ) • Repeat above until convergence in p ' Y X B ' A ' q ' • Retrieve full tensors again: = =

Optimization: full update Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008) Corboz, Orus, Bauer, Vidal, PRB 81, 165104 (2010) • Approximate old PEPS + gate with a new PEPS with bond dimension D Environment Environment B B ' A A ' ≈ g | ˜ | Ψ 0 i Ψ i = g | Ψ i ≈ Ψ i � | Ψ 0 i || 2 = h ˜ || | ˜ Ψ | ˜ Ψ i + h Ψ 0 | Ψ 0 i � h ˜ Ψ | Ψ 0 i � h Ψ 0 | ˜ • Minimize Ψ i • Iteratively / CG / Newton / ... • The full wave function is taken into account for the truncation! • At each step the environment has to be computed! expensive... but optimal!

Optimization: simple vs full update Example: 2D Heisenberg model simple update 0.42 ★ “local” update like in TEBD Simple update 0.4 Full update ★ Cheap, but not optimal (e.g. overestimates magnetization 0.38 in S=1/2 Heisenberg model) m [%] 0.36 full update 0.34 ★ Take the full wave function into 0.32 account for truncation exact result ★ optimal, but computationally more 0.3 0 0.1 0.2 0.3 0.4 0.5 expensive 1/D • Combine the two: Use simple update to get an initial state for the full update • Don’t compute environment from scratch but recycle previous one fast full update Phien, Bengua, Tuan, PC, Orus, PRB 92 (2015)

Variational optimization for PEPS Verstraete, Murg, Cirac, Adv. Phys. 57 (2008) 1. Select one of the PEPS tensors A 2. Optimize tensor A (keeping all the others fixed) by minimizing the energy: tensor network including tensor network from norm term all Hamiltonian terms E = h Ψ | H | Ψ i minimize tensor A reshaped as a vector H x = E N x h Ψ | Ψ i solve generalized eigenvalue problem H = N = in 1D: ˆ H

Variational optimization for PEPS Verstraete, Murg, Cirac, Adv. Phys. 57 (2008) 1. Select one of the PEPS tensors A 2. Optimize tensor A (keeping all the others fixed) by minimizing the energy: tensor network including tensor network from norm term all Hamiltonian terms E = h Ψ | H | Ψ i minimize tensor A reshaped as a vector H x = E N x h Ψ | Ψ i solve generalized eigenvalue problem 3. Take the next tensor and optimize (keeping other tensors fixed) 4. Repeat 2-3 iteratively until convergence is reached

) iPEPS Variational optimization for iPEPS D A B C D A H E F G H E D A B C D A H H E F G E Main challenges: A B D A D C H E F G H E 1. Need to take into account infinitely many Hamiltonian contributions ✦ Solution: use corner-transfer matrix method [PC, arXiv:1605.03006] ✦ Alternative: use “channel-environments” [Vanderstraeten et al, PRB 92 , arxiv:1606.09170] ✦ Or: Use PEPO (similar to 3D classical) [cf. Nishino et al. Prog. Theor. Phys 105 (2001)] 2. Tensor A appears infinitely many times! (Min. problem highly non-linear) ✦ Take adaptive linear combination of old and new tensor [PC, arXiv:1605.03006] [see also Nishino et al. Prog. Theor. Phys 105 (2001), Gendiar et al. PTR 110 (2003)] ✦ Alternative: use CG approach [Vanderstraeten, Haegeman, PC, Verstraete, arXiv:1606.09170] tensor network including tensor network from norm term all Hamiltonian terms E = h Ψ | H | Ψ i minimize tensor A reshaped as a vector H x = E N x h Ψ | Ψ i

H-environment ✓ tensor network including tensor network from norm all Hamiltonian terms C 1 T 1 C 2 E = h Ψ | H | Ψ i minimize H x = E N x h Ψ | Ψ i T 4 T 2 a But how about H ?? C 4 T 3 C 3 ‣ Need additional H -environment tensors: ˜ C 1 T 1 C 2 T 4 T 2 a ... ... ... ... ˜ ... ... ... ... C 1 + . . . + + + = ... ... ... ... ... ... ... ... C 4 T 3 C 3 ➡ taking into account all Hamiltonian contributions in the infinite upper left corner

H-environment h Ψ | ˆ H | Ψ i = Corner terms ... ... ... ... ˜ ... ... ... ... + + + + . . . = = + + + = ... ... ... ... ... ... ... ... ˜ + + + . . . = = ... ... ... + + + + + + + = + + + . . . = = Terms between a corner = = and an edge tensor + + + + ... ... ... ... ... ... + + + . . ... ... ... ... ... ... + + + + ... ... ... = = ... ... ... ... ... ... + Local terms + + + + + + + + + + +