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Introduction to iPEPS Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam Overview: tensor networks in 1D and 2D 1D 1D MERA MPS Matrix-product state Multi-scale entanglement renormalization ansatz and more 1D tree

  1. Introduction to iPEPS Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam

  2. Overview: tensor networks in 1D and 2D 1D 1D MERA MPS Matrix-product state Multi-scale entanglement renormalization ansatz and more ‣ 1D tree tensor 1 2 3 4 5 6 7 8 network Underlying ansatz of the ‣ correlator density-matrix renormalization product states group ( DMRG ) method i 1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 i 17 i 18 ‣ ... 2D PEPS (TPS) 2D MERA and more projected entangled-pair state (tensor product state) ‣ Entangled- plaquette states ‣ 2D tree tensor network ‣ String-bond states ‣ ...

  3. Outline of the 2 lectures ‣ Part I: iPEPS ansatz ✦ Repetition: area law of the entanglement entropy ‣ Part II: Contraction of PEPS / iPEPS ✦ MPS-MPO approach, corner-transfer-matrix (CTM) method, Tensor Renormalization Group (TRG), Tensor network renormalization (TNR) ✦ Simple examples to get started: ➡ solving the 2D classical Ising model with the CTM method ➡ simple 2D quantum case (D=2, rotational symmetric) ‣ Interlude: Example application: the Shastry-Sutherland model ‣ Part III: Optimization ‣ Part IV: Computational cost & benchmarks ‣ Part V: iPEPS applications ‣ Outlook & summary

  4. PART I: iPEPS ansatz

  5. “Corner” of the Hilbert space Ground states (local H) ★ GS of local H’s are less entangled than a Hilbert random state in the Hilbert space ★ Area law of the entanglement entropy space

  6. Area law of the entanglement entropy . . . . E . . 1D 2D A E A E . . . . . . L L . . . . . . # relevant states � Entanglement entropy S ( A ) = − tr[ ρ A log ρ A ] = − λ i log λ i χ ∼ exp( S ) i General (random) state Ground state (local Hamiltonian) S ( L ) ∼ L d ( volume ) S ( L ) ∼ L d − 1 ( area law ) Critical ground states: 1D S ( L ) = const χ = const (all in 1D but not all in 2D) χ ∼ exp( α L ) 2D S ( L ) ∼ α L 1D S ( L ) ∼ log( L ) 2D S ( L ) ∼ L log( L )

  7. MPS & PEPS 1D MPS Matrix-product state Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) ✓ Reproduces area-law in 1D S ( L ) = const

  8. MPS & PEPS 1D MPS Matrix-product state Bond dimension D ➡ One bond can contribute 1 2 3 4 5 6 7 8 at most log(D) to the entanglement entropy A E L S ( A ) ≤ log ( D ) = const rank ( ρ A ) ≤ D ✓ Reproduces area-law in 1D S ( L ) = const

  9. MPS & PEPS 1D 2D MPS can we use an MPS? Matrix-product state Bond dimension D 1 2 3 4 5 6 7 8 L Physical indices (lattices sites) S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) !!! Area-law in 2D !!! ✓ Reproduces area-law in 1D S ( L ) ∼ L S ( L ) = const D ∼ exp ( L )

  10. MPS & PEPS 1D 2D PEPS (TPS) MPS projected entangled-pair state Matrix-product state (tensor product state) Bond dimension D Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) F. Verstraete, J. I. Cirac, cond-mat/0407066 Nishino, Hieida, et al., Prog. Theor. Phys. 105 (2001). Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115 ✓ Reproduces area-law in 1D ✓ Reproduces area-law in 2D S ( L ) = const S ( L ) ∼ L

  11. PEPS: Area law D L one “thick” bond of dimension D L A B ... ... L S ( A ) ≤ L log D ∼ L each cut auxiliary bond can contribute (at most) log D to the entanglement entropy ✓ Reproduces area-law in 2D The number of cuts scales with the cut length S ( L ) ∼ L

  12. MPS & PEPS 1D 2D PEPS (TPS) MPS projected entangled-pair state Matrix-product state (tensor product state) Bond dimension D Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) F. Verstraete, J. I. Cirac, cond-mat/0407066 Nishino, Hieida, et al., Prog. Theor. Phys. 105 (2001). Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115 ✓ Reproduces area-law in 1D ✓ Reproduces area-law in 2D S ( L ) = const S ( L ) ∼ L

  13. Infinite PEPS (iPEPS) 1D 2D iMPS iPEPS infinite matrix-product state infinite projected entangled-pair state A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008) Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115 ★ Work directly in the thermodynamic limit: No finite size and boundary effects!

  14. Infinite PEPS (iPEPS) 1D 2D iMPS iPEPS infinite matrix-product state infinite projected entangled-pair state A B A B A B B A B A B A A B A B A B B A B A B A A B A B A B B A B A B A ★ Work directly in the thermodynamic limit: No finite size and boundary effects!

  15. iPEPS with arbitrary unit cells 1D 2D iMPS iPEPS infinite matrix-product state with arbitrary unit cell of tensors D A B C D A H E F G H E D B D A C A H E F G H E B D D A C A H E F G H E here: 4x2 unit cell PC, White, Vidal, Troyer, PRB 84 (2011) ★ Run simulations with different unit cell sizes and compare variational energies

  16. Overview: Tensor network algorithms (ground state) MPS PEPS 2D MERA TN ansatz (variational) 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)

  17. PART II: Contraction

  18. Contracting a tensor network (repetition)

  19. Pairwise contractions...

  20. Pairwise contractions...

  21. Pairwise contractions...

  22. Pairwise contractions...

  23. Pairwise contractions...

  24. Pairwise contractions... done! the order of contraction matters for the computational cost!!!

  25. Contracting a tensor network ★ Reshape tensors into matrices and multiply them with optimized routines (BLAS) dimension D u u i cost D 5 w = A B T w j v v ( ij ) A ( uv ) B T ( uv ) = w w dimension D 2 ★ Computational cost: multiply the dimensions of all legs (connected legs only once)

  26. Contracting an MPS � Ψ | Ψ ⇥ BAD! = � Ψ | Ψ ⇥ Good! =

  27. Contracting the PEPS � Ψ | Ψ ⇥ reduced tensors D 2

  28. Contracting the PEPS dimension D 2 Problem: how do we contract this?? no matter how we contract, we will get intermediate tensors with O(L) legs number of coefficients D 2L Exponentially increasing with L! NOT EFFICIENT

  29. Contracting the PEPS ★ Exact contraction of an PEPS is exponentially hard! use controlled approximate contraction scheme TRG Corner transfer MPS-based Tensor Renormalization Group matrix method approach (+HOTRG, SRG, HOSRG) Murg,Verstraete,Cirac, PRA75 ’07 Levin, Nave, PRL99 (2007) Nishino, Okunishi, JPSJ65 (1996) Jordan,et al. PRL79 (2008) Xie et al. PRL 103, (2009) Orus, Vidal, PRB 80 (2009) ★ Accuracy of the approximate contraction is controlled by “boundary dimension” χ TNR Tensor Network Renormalization ★ Convergence in needs to be carefully checked χ Evenbly & Vidal, PRL 115 (2015) ★ Overall cost: with χ ∼ D 2 Loop-TNR: O ( D 10 ... 14 ) Yang, Gu & Wen, arXiv:1512.04938

  30. Contracting the PEPS Example: 2D Heisenberg model (CTM) ★ Fast convergence − 0.6685 ★ Effect of finite D is much larger! E s − 0.669 ★ Be careful with D=4 “variational” energy!!! D=5 D=6 − 0.6695 0 0.02 0.04 0.06 0.08 0.1 1/ χ

  31. Contracting the PEPS ★ Exact contraction of an PEPS is exponentially hard! use controlled approximate contraction scheme TRG Corner transfer MPS-based Tensor Renormalization Group matrix method approach (+HOTRG, SRG, HOSRG) Murg,Verstraete,Cirac, PRA75 ’07 Nishino, Okunishi, JPSJ65 (1996) Levin, Nave, PRL99 (2007) Jordan,et al. PRL79 (2008) Orus, Vidal, PRB 80 (2009) Xie et al. PRL 103, (2009) ★ Accuracy of the approximate contraction is controlled by “boundary dimension” χ TNR Tensor Network Renormalization ★ Convergence in needs to be carefully checked χ Evenbly & Vidal, PRL 115 (2015 ★ Overall cost: with χ ∼ D 2 Loop-TNR: O ( D 10 ... 14 ) Yang, Gu & Wen, arXiv:1512.04938

  32. Contracting the PEPS using an MPS Verstraete, Murg, Cirac, Adv. in Phys. 57, 143 (2008) dimension D 2 this is an MPS this is an MPO (matrix product operator)

  33. Contracting the PEPS using an MPS Verstraete, Murg, Cirac, Adv. in Phys. 57, 143 (2008) dimension D 2 xD 2 this is an MPS with bond dimension D 2 x D 2 truncate the bonds to χ there are different techniques for the efficient MPO-MPS multiplication (SVD, variational optimization, zip-up algorithm...) Schollwöck, Annals of Physics 326, 96 (2011) Stoudenmire, White, New J. of Phys. 12, 055026 (2010).

  34. Contracting the PEPS using an MPS Verstraete, Murg, Cirac, Adv. in Phys. 57, 143 (2008) dimension χ proceed... ★ We can do this from several directions ★ Similar procedure when computing an expectation value

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