Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam PDF
Preview Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam
Introduction to iPEPS Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam Overview: tensor networks in 1D and 2D 1D MPS 1D MERA Matrix-product state Multi-scale entanglement renormalization ansatz and more 1 2 3 4 5 6 7 8 ‣ 1D tree tensor network Underlying ansatz of the density-matrix renormalization ‣ correlator group (DMRG) method product states i i i i i i i i i i i i i i i i i i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 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 ‣ ... 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 PART I: iPEPS ansatz “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 Area law of the entanglement entropy . . . E . . . 1D 2D A E A E . . . . . . L L . . . . . . # relevant states Entanglement entropy S(A) = tr[⇥ log ⇥ ] = � log � A A i i � exp(S) � � ⇠ i � General (random) state Ground state (local Hamiltonian) d d 1 S(L) L (volume) S(L) L (area law) � ⇠ ⇠ Critical ground states: 1D S(L) = const � = const (all in 1D but not all in 2D) ⇥ exp(�L) 2D S(L) �L � 1D S(L) log(L) � ⇠ S(L) L log(L) 2D ⇠ 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 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 A E entanglement entropy L S(A) log(D) = const rank(⇢ ) D A ✓ Reproduces area-law in 1D S(L) = const 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) ⇠ MPS & PEPS 1D 2D PEPS (TPS) MPS projected entangled-pair state Matrix-product state (tensor product state) D Bond dimension D Bond dimension 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 �
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