Tensor network renormalization

Transcription

1 Walter Burke Institute for Theoretical Physics INAUGURAL CELEBRATION AND SYMPOSIUM Caltech, Feb 23-24, 2015 Tensor network renormalization Guifre Vidal Sherman Fairchild Prize Postdoctoral Fellow ( )

2 Renormalization group Tensor networks Tensor network renormalization (TNR) TNR = Renormalization group + Tensor networks

3 Renormalization group Tensor networks In collaboration with GLEN EVENBLY IQIM Caltech UC Irvine TNR = Renormalization group + Tensor Networks

4 Emergent phenomena in many-body systems quantum criticality metal topological order insulator fractional quantum Hall effect superconductor superfluid spin liquid

5 The problem we have we want H local Hamiltonian Ψ ground state Ψ o x, t o(0,0) Ψ low energy, collective excitations, Z = tr e βh Euclidean path integral

6 Example: 1d quantum Ising model 2d classical Ising model H q 1d = i σ i x σ j x + λ σ i z i H 2d cl = <i,j> S i S j Z λ = tr e β H q 1d Z(T) = s e 1 T H 2d cl β 1d quantum ~ L y 2d classical L L x σ x S λ T λ crit T crit Other examples: material science, quantum chemistry, QCD,

7 The Renormalization Group H k H[k(s)] k = (k 1, k 2, k 3, ) H[J, λ ] = J i σ i x σ j x H[k 1, k 2, 0,0, ] = k 1 + λ σ i z i i σ i x σ j x + k 2 σ i z i RG flow in the space of Hamiltonians Leo Kadanoff Kenneth Wilson Phase A Phase B stable fixed point A k 4 stable fixed point B k 5 critical fixed point

8 Leo Kadanof block spin + some rule: majority vote, etc Change of scale? coarse-graining transformation Z = Dφe H φ,k p Λ e H φ,k = Dφe H φ,k Λ Λ p Λ p Λ Kenneth Wilson Z = Dφe H φ,k p Λ 0 exact renormalization group equation (ERGE)

9 We would like (wish list) Non-perturbative RG approach H H H Universal coarse-graining rules valid for a generic system Solve QCD! How far did we get? (over the last 10 years) Reformulated the RG using quantum information tools/concepts (quantum circuits, entanglement) Efficient representation of ground state wave-functions (MERA) universal, non-perturbative, real-space RG approach! Key ingredient: removal of short-rage entanglement Ψ (but we have not yet solved QCD, sorry )

10 Condensed matter (Frustrated magnets, interacting fermions, quantum criticality) Classification of gapped phases Classical statistical mechanics TENSOR NETWORKS Verstraete Levin Cirac Wen White Pollmann Read Chen Gu Qi Swingle Hastings Nave Schuch Eisert Nishino and many more Orus Tagliacozzo Singh Evenbly Corboz Material science Xiang Renormalization group Quantum chemistry Holography

11 Many-body wave-function of N spins Ψ = Ψ i1 i 2 i N i 1 i 2 i N i 1,i 2,,i N Ψ i 1 i 2 i N = 2 N parameters tensor network i 1 i 2 i N graphical notation i i j α 1 α2 α N a b i c ij d α1 α 2 α N i = j i k j = T ij = R ik S kj a = y M x tr(abcd) k why bother? ijklmnop A ijk B jlm C nko D kmr x i y l z n v r

12 Many-body wave-function of N spins Ψ = Ψ i1 i 2 i N i 1,i 2,,i N i 1 i 2 i N i 1 i 2 Ψ 2 N parameters i N inefficient = i 1 i 2 2 N parameters tensor network i N O(N2χ 2 ) parameters efficient α = 1,2,, χ generic state tensor network states ground states of local Hamiltonians H (N) χ = 1 χ = 2 χ = 3 H (N)

13 Example of tensor network Vidal, 2006 Multi-scale entanglement renormalization ansatz (MERA) Variational ansatz for 1d systems, which extends in space and scale Variational parameters for different scales It is secretly a quantum circuit and an RG transformation

14 Example of tensor network Vidal, 2006 Multi-scale entanglement renormalization ansatz (MERA) 0 disentangler two-body gate isometry = also a two-body gate

15 Multi-scale entanglement renormalization ansatz (MERA) time quantum circuit U ground state Ψ = U Entanglement is introduced by the gates at different times (=scales)

16 L L RG Transformation Kadanoff (1966) + White (1992) blocking variational optimization Entanglement renormalization (2005) L L

17 L RG Transformation Kadanoff (1966) + White (1992) blocking variational optimization failure to remove some short-range entanglement! L L L Entanglement renormalization (2005) removal of all short-range entanglement

18 MERA -> RG flow in the space of ground state wave-functions Ψ Ψ Ψ Ψ fp

19 MERA -> RG flow in the space of ground state wave-functions Ψ Ψ Ψ Ψ fp fixed-point wave-function topological order (2+1) quantum criticality (1+1) MERA -> RG flow in the space of Hamiltonians H H H H fp = local operators are mapped into local operators!

20 Summary so far MERA Variational parameters for different length scales It is secretly a quantum circuit and an RG transformation entanglement at different length scales removes short-range entanglement Ψ Ψ H H preservation of locality blah, blah, blah Does it work? Optimize variational parameters by energy minimization

21 Example: Critical quantum Ising model Pfeifer, Evenbly, Vidal (2008) Scaling dimensions of primary fields scaling dimension (exact ) scaling dimension (MERA) error identity % % % <10 8 % <10 8 % Operator product expansion (OPE) coefficients C εσσ = 1 2 spin energy density disorder fermions I σ ε C εμμ = 1 2 iπ C ψμσ = e 4 C 2 ψμσ = eiπ 4 C εψψ = i C εψψ = i (± ) 2 scale-invariant MERA conformal data of a CFT: central charge c scaling dimensions Δ α h α + h α conformal spin s α h α h α OPE C αβγ

22 MERA and HOLOGRAPHY s scale x space entanglement entropy S L log (L) parallel to area of minimal surface in Ryu-Takayanagi two-point correlations C L L 2Δ geodesic distance D log (L) as in hyperbolic space C L e D = e 2Δlog(L) = L 2Δ L log(l)

23 Swingle 2009 t x t s x Qi Hartman, Maldacena CFT 1+1 AdS 2+1 Haegeman, Osborne, Verschelde, Verstraete Sully, Czech Ryu, Takayanagi x Harlow, Yoshida, Pastawki, Preskill

24 So, MERA seems to work! Great! However variational optimization is expensive; local minima. do we get the correct ground state? Euclidean path integrals / classical partition functions? Tensor Network Renormalization Evenbly, Vidal

25 Euclidean path integral Statistical partition function Z λ = tr e βh q 1d Z(T) = e 1 T H 2d cl s β 1d quantum ~ L y 2d classical L L x as a tensor network Z = A

26 Tensor Renormalization Group (TRG) Levin, Nave 2006 isometry! A A

27 Fixed-point of TRG Levin, Nave 2006 isometry! CDL Tensor (zero correlation length) = A some short-range entanglement has not been removed A

28 Tensor Network Renormalization (TNR) Evenbly, Vidal isometry disentangler! A A

29 Tensor Network Renormalization (TNR) isometry disentangler! CDL Tensor (zero correlation length) = A removal of all short-range entanglement [for CDL tensors, see also Gu and Wen 2009 Tensor Entanglement Filtering Renormalization (TEFR)] A

30 TNR -> proper RG flow Example: 2D classical Ising A A A A fp A 1 A 2 A 3 A 4 below critical T = 0.9 T c ordered (Z2) fixed point critical T = T c critical (scale-invariant) fixed point above critical T = 1.1 T c disordered (trivial) fixed point

31 TNR yields MERA Evenbly, Vidal, 2015

32 MERA = variational ansatz MERA = by-product of TNR energy optimization transform tensor network into MERA energy minimization 1000s of iterations over scale local minima correct ground? TNR -> MERA single iteration over scale rewrite tensor network for ground state certificate of accuracy

33 Summary Reformulation of the RG using quantum information tools/concepts (quantum circuits, entanglement) Efficient representation of ground states (MERA) -> toy model for holography universal, non-perturbative, real-space RG approach 3 RG flows Tensors Ground states Hamiltonians A A A Ψ Ψ Ψ H H H Key ingredient: removal of short-rage entanglement Very accurate in 1+1 dimensions (Ising model, etc) What about 2+1, 3+1? (and QCD?)

34 Entanglement renormalization MERA IQI, 2005 Sherman Fairchild Prize Postdoctoral Fellow ( ) Tensor network renormalization IQIM, 2014 GLEN EVENBLY Sherman Fairchild Prize Postdoctoral Fellow ( )

35 THANK YOU!