Lecture 1: tensor network states (MPS, PEPS & iPEPS, Tree TN, MERA, 2D MERA) Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11i12 i13 i14 i15i16 i17 i18 Outline ‣ Lecture 1: tensor network states ✦ Main idea of a tensor network ansatz & area law of the entanglement entropy ✦ MPS, PEPS & iPEPS, Tree tensor networks, MERA & 2D MERA ✦ Classify tensor network ansatz according to its entanglement scaling ‣ Lecture II: tensor network algorithms (iPEPS) ✦ Contraction & Optimization ‣ Lecture III: Fermionic tensor networks ✦ Formalism & applications to the 2D Hubbard model ✦ Other recent progress Motivation: Strongly correlated quantum many-body systems High-Tc Quantum magnetism / Novel phases with superconductivity spin liquids ultra-cold atoms Challenging!tech-faq.com Typically: • No exact analytical solution Accurate and efficient • Mean-field / perturbation theory fails numerical simulations • Exact diagonalization: O(exp(N)) are essential! Quantum Monte Carlo • Main idea: Statistical sampling of the exponentially large configuration space • Computational cost is polynomial in N and not exponential Very powerful for many spin and bosonic systems Quantum Monte Carlo • Main idea: Statistical sampling of the exponentially large configuration space • Computational cost is polynomial in N and not exponential Very powerful for many spin and bosonic systems Example: The Heisenberg model . Sandvik & Evertz, PRB 82 (2010): . H = SiSj system sizes up to 256x256 i,j 65536 ⇥ Hilbert space: 2 Ground state sublattice magn. m =0.30743(1) has Néel order . Quantum Monte Carlo • Main idea: Statistical sampling of the exponentially large configuration space • Computational cost is polynomial in N and not exponential Very powerful for many spin and bosonic systems BUT Quantum Monte Carlo & the negative sign problem Bosons Fermions (e.g. 4He) (e.g electrons) Ψ (x ,x )=Ψ (x ,x ) Ψ (x ,x )= Ψ (x ,x ) B 1 2 B 2 1 F 1 2 − F 2 1 symmetric! antisymmetric! this leads to the infamous ✓ negative sign problem t (poly(N/T )) t (exp(N/T )) sim ∼ O sim ∼ O cannot solve large systems at low temperature! Strongly correlated fermionic systems ˆ 2D Hubbard model H = t cˆi†σcˆjσ + U nˆi nˆi − ↑ ⇥ i,j ,σ i • High-temperature superconductors ⇤⌅ Hopping between On-site repulsion between nearest-neighbor sites electrons with opposite spin Is it the relevant model of high-temperature superconductors? Quantum Monte Carlo & the negative sign problem Non-frustrated Frustrated spin systems spin systems frustrated ? all bonds satisfied not all bonds satisfied this leads to the infamous ✓ negative sign problem t (poly(N/T )) t (exp(N/T )) sim ∼ O sim ∼ O cannot solve large systems at low temperature! To make prog ress in strongly correlated systems it is essential to negative develop new accurate numerical methods! sign problem • DMFT / DCA • Diagrammatic Monte Carlo • Tensor network algorithms • Fixed-node Monte Carlo • Series expansion • Density Matrix Embedding Theory • Variational Monte Carlo • Functional renormalization group • Coupled-cluster methods • ... 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 Underlying ansatz of the network density-matrix renormalization ‣ correlator product states group (DMRG) method i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11i12 i13 i14 i15i16 i17 i18 ‣ ... 2D PEPS 2D MERA and more projected entangled-pair state ‣ Entangled- plaquette states ‣ 2D tree tensor network ‣ String-bond states ‣ ... Diagrammatic notation l c i v i M j i A j i B j ... k k number vector vi matrix Mij Aijk Bijkl rank-0 tensor rank-1 tensor rank-2 tensor rank-3 tensor rank-4 tensor ★ We don’t need to write down formulas with tensors with many indices! i j Example 1: M v = i u Mijvj = ui j X ★ Connected lines: sum over corresponding indices! Diagrammatic notation i j Example 2: u M v = c ★ sum over all connected indices: contraction of a tensor network uiMijvj = c ij X B j Example 3: i = T A C ★ The rank of the k resulting tensor u v u v corresponds to the number of open legs in the network AuikBijCvjk = Tuv Xijk Diagrammatic notation ★ Hard to write down with all indices... ★ We know the result is going to be a number Introduction to tensor networks ➡ Aim: Efficient representation of quantum many-body states V : local Hilbert space Lattice with .... e.g. , N sites 1 2 3 4 5 N-1 N {| ⇧ | ⇥⇧} Full Hilbert dimension 2N .... space V⊗V⊗V⊗V⊗V⊗ ⊗V⊗V grows exponentially with N Hamiltonian Hˆ = hˆij sum of local terms ij ⇥ Represent the Ψ = Ψ i i i ground state | ⇤ i1i2...iN | 1 ⇥ 2 ⇥ ···⇥ N ⇤ i i ...i 1 2 N i , k ⇤ {↑ ⇥} 2N coefficients Complexity ~exp(N) many numbers inefficient! Tensor network ansatz for a wave function 2 basis states per site: , {| "i | #i} Lattice: 26 basis states 1 2 3 4 5 6 26 coefficients State: Ψ = Ψ i i i i i i | ⇥ i1i2i3i4i5i6 | 1 ⊗ 2 ⊗ 3 ⊗ 4 ⊗ 5 ⊗ 6⇥ i i i i i i 1 23 4 5 6 Tensor/multidimensional array Tensor network: matrix product state (MPS) Why is athisb c d e Big tensor i1i2i3i4i5i6 A B C D E F D bond dimension i1 i2 i3 i4 i5 i6 i1 i2 i3 i4 i5 possible??i6 Aa BabCbcDcdEdeF e = ˜ Ψi1i2i3i4i5i6 i1 i2 i3 i4 i5 i6 i1i2i3i4i5i6 ⇡ abcdeX Efficient exp(N) many numbers vs poly(D ,N) numbers representation! “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 space ★ Area law of the entanglement entropy Splitting in the middle Left side Right side Big tensor i1i2i3i4i5i6 i1 i2 i3 i4 i5 i6 i1 i2 i3 i4 i5 i6 Singular value decomposition U s V† = UsV † skk 0 l r l, r 1,...,M l r 2 { } ≥ M =2N/2 diagonal matrix! lr = UlkskkVrk⇤ Xk = l r = UlkskkV ⇤ l r lr rk| i| i | i | i| i lr k Xlr X X Schmidt = skk uk vk | i| i decomposition Xk How many relevant singular values? M how many non-zero = s u v U s V† | i kk| ki| ki singular values? Xk l r ★ Special cases: s11 =1,skk = 0 for k>1 =1u v Product state | i | 1i| 1i 1 1 s11 = ,s22 = ,skk = 0 for k>2 p2 p2 1 1 = u1 v1 + u2 v2 Entangled state | i p2| i| i p2| i| i 1 Maximally skk = , for all k pM entangled state How many relevant singular values? bond dimension D = χ M how many relevant = s u v U s V† | i kk| ki| ki singular values? Xk skk l r χ ˜ = s u v | i⇡| i kk| ki| ki Ground state (local H) Xk random state: keeping the χ largest all singular values important singular values minimizes the error ˜ k ||| i| i|| χ M KEY IDEA OF relevant singular values truncate this part DMRG! Reduced density matrix M A B = s u v | i kk| ki| ki Xk ★ Reduced density matrix of left side: describes system on the left side 2 ⇢ =tr [⇢]=tr [ ]= λ u u λk = s probability A B B | ih | k| kih k| kk Xk ★ Entanglement entropy: S(A)= tr[⇢ log ⇢ ]= λ log λ − A A − k k Xk ‣ Product state: S(A)= 1 log 1 = 0 ... − 1 1 ‣ Maximally entangled state: S(A)= log = logM − M M Xk How large is S in a ground state? How does it scale with system size? . Area law of the entanglement entropy E . 1D 2D A E A E . L L . # relevant states Entanglement entropy S(A)= tr[⇥ log ⇥ ]= λ log λ − A A − i i χ 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 χ = const (all in 1D but not all in 2D) 2D S(L) αL ⇥ exp(αL) 1D S(L) log(L) ∼ ∼ ⇠ 2D S(L) L log(L) ⇠ 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 at most log(D) to the A E entanglement entropy L rank(⇢ ) D S(A) log(D)=const A ✓ Reproduces area-law in 1D S(L)=const MPS & PEPS 1D 2D MPS can we use Matrix-product state an MPS? Bond dimension D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) L 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) 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 Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115 ✓ Reproduces area-law in 1D ✓ Reproduces area-law in 2D S(L)=const S(L) L ∼ PEPS: Area law one “thick” bond of dimension DL DL A B ... ... L S(A) L log D L ⇠ each cut auxiliary bond can contribute (at most) log D to the entanglement entropy The number of cuts scales ✓ Reproduces area-law in 2D with the cut length S(L) L ∼ 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.