Scale invariance on the lattice

Transcription

1 Coogee'16 Sydney Quantum Information Theory Workshop Feb 2 nd - 5 th, 2016 Scale invariance on the lattice Guifre Vidal

2 Coogee'16 Sydney Quantum Information Theory Workshop Feb 2 nd - 5 th, 2016 Scale invariance on the lattice Guifre Vidal work with GLEN EVENBLY (UC Irvine)

3 Motivation: Field theory space-time symmetries e.g. translation invariance x x = x + ε at criticality scale invariance x x = 1 + ε x (augmented to global/local conformal group?) On the lattice (genuine physics or UV cut-off) translation? x x = x + a a lattice spacing discrete scale transformation?

4 Goals: 1) Define a global scale transformation on the lattice Hamiltonian (Hilbert space) Lagrangian (Euclidean path integral) Entanglement renormalization / MERA Tensor network renormalization PRL 2007 (arxiv:cond-mat/ ) PRL 2008 (arxiv:quant-ph/ ) Evenbly, Vidal PRL 2015 (arxiv: ) Evenbly, arxiv: ) Define a local scale transformation on the lattice Hamiltonian (Hilbert space) Czech, Evenbly, Lamprou, McCandlish, Qi, Sully, Vidal arxiv: Entanglement renormalization / MERA on-going discussions with Tobias Osborne and Vaughan Jones Tensor network renormalization Lagrangian (Euclidean path integral) Evenbly, Vidal, PRL 2015 arxiv: Evenbly, Vidal, PRL 2015 arxiv:

5 Outline: wave-functions / Hamiltonians global scale transformation (RG transformation) local scale transformations Euclidean path integrals / classical partition functions global scale transformation (RG transformation) local scale transformations

6 Claim 1: global scale transformation Claim 2: local scale transformation

7 wave-functions / Hamiltonians global scale transformation (RG transformation) local scale transformations Euclidean path integrals / classical partition functions global scale transformation (RG transformation) local scale transformations

8 How do we define a translation on the lattice? unitary map V N V N How do we define a scale transformation on the lattice? many possibilities here, isometric map V N/2 V N variational locality entanglement

9 natural requirement: consistency with the QFT / continuum field theory scale transformation x e s x field theory discretization lattice model discrete scale transformation x 1/2x lattice model reproduce expected RG flow, including explicit scale invariance at RG fixed-points

10 Entanglement renormalization PRL 2007 (arxiv:cond-mat/ ) L w u V N/2 L V N 1 spin blocking (as Kadanoff 1966) Leo Kadanoff

11 Entanglement renormalization PRL 2007 (arxiv:cond-mat/ ) L w u V N/2 L V N 1 spin blocking (as Kadanoff 1966) 2 variational optimization (as DMRG, White 1992) only structural ansatz ask the Hamiltonian H! (low energy) W: V N/2 V N H H W HW

12 Entanglement renormalization PRL 2007 (arxiv:cond-mat/ ) L w u V N/2 L V N 3 entanglement failure to remove some short-range entanglement! removal of all short-range entanglement!

13 Entanglement renormalization PRL 2007 (arxiv:cond-mat/ ) L w u V N/2 L V N 4 locality u w = u = w

14 Entanglement renormalization PRL 2007 (arxiv:cond-mat/ ) L w u V N/2 L V N 4 locality u w = u = w

15 Entanglement renormalization PRL 2007 (arxiv:cond-mat/ ) L w u V N/2 L V N 4 locality u w = u = w local operator on L scale transf. local operator on L

16 Claim: input Entanglement renormalization defines a proper scale transformation on the lattice 1D quantum Hamiltonian on the lattice at a critical point 1 - optimization output Numerical extraction of conformal data of underlying CFT: central charge c Explicit scale invariance at criticality! scaling dimensions Δ α h α + h α and conformal spins s α h α h α 2 - diagonalization OPE coefficients C αβγ scaling operators e.g. critical Ising model (approx. an hour on your laptop) Pfeifer, Evenbly, Vidal 08 (Δ I = 0) Δ σ Δ ε Δ μ Δ ψ Δ ψ C εσσ = 1 2 C εψ ψ = i iπ C ψμσ = e 4 2 C εμμ = 1 2 C ε ψψ = i C ψμσ = eiπ 4 2 (± )

17 multi-scale entanglement renormalization ansatz (MERA) PRL 2008 (arxiv:quant-ph/ ) Ψ

18 multi-scale entanglement renormalization ansatz (MERA) PRL 2008 (arxiv:quant-ph/ ) Ψ

19 multi-scale entanglement renormalization ansatz (MERA) PRL 2008 (arxiv:quant-ph/ ) Ψ

20 multi-scale entanglement renormalization ansatz (MERA) PRL 2008 (arxiv:quant-ph/ ) Ψ

21 MERA defines an RG flow in the space of wave-functions PRL 2008 (arxiv:quant-ph/ ) Ψ Ψ Ψ Ψ and in the space of Hamiltonians L L L L stable fixed point A Phase A H H H H Phase B critical fixed point stable fixed point B

22 wave-functions / Hamiltonians global scale transformation (RG transformation) local scale transformations Euclidean path integrals / classical partition functions global scale transformation (RG transformation) local scale transformations

23 Claim 1: MERA defines a global scale transformation on the lattice discrete RG flow with expected structure, including scale invariance at criticality Claim 2: MERA also defines local scale transformations on the lattice Czech, Evenbly, Lamprou, McCandlish, Qi, Sully, Vidal arxiv: Expected behaviour from continuum (CFT), including local scale invariance (covariance) at criticality

24 Example of local scale transformation on the lattice: Czech, Evenbly, Lamprou, McCandlish, Qi, Sully, Vidal arxiv: (involving both coarse-graining and fine-graining) ground state Ψ? on infinite (discrete) line

25 Test: compare with CFT in the continuum Czech, Evenbly, Lamprou, McCandlish, Qi, Sully, Vidal arxiv: conformal transformation z w = β π log(z) Euclidean path integral on half upper plane prepares the ground state Ψ on the infinite line Euclidean path integral on infinite strip prepares thermal state ρ β on the infinite line [*comment to experts: TFD] Can we do the same on the lattice?

26 Czech, Evenbly, Lamprou, McCandlish, Qi, Sully, Vidal arxiv: conformal transformation z w = β π log(z) ground state Ψ on infinite (discrete) line thermal state ρ β? [*comment to experts: TFD] on infinite (discrete) line Yes! (but first, let us put it in a finite geometry)

27 conformal transformation z z w = β π log(z) iβ w 1 2π 2 L e β 0 2πL quotient: half upper plane / scaling = topological cylinder quotient: infinite strip / translation = flat cylinder thermal state ρ β on finite circle Luckily, this scaling is an exact symmetry of the MERA

28 Scaling on a regular MERA without local scale transformation

29 Scaling on a MERA after a local scale transformation

30 Scaling on a MERA after a local scale transformation

31 scaling quotient Thermal state on (discrete) infinite line Thermal state on (discrete) finite circle

32 k = 1 Thermal spectrum of Ising CFT k = 16 k = k 2πΔ α log λ α λ 0 k = 3 8 π log(2) k

33 conformal transformation z w = β π log(z) Czech, Evenbly, Lamprou, McCandlish, Qi, Sully, Vidal arxiv: ground state Ψ on infinite (discrete) line thermal state ρ β on infinite (discrete) line Thus, under our local scale transformation on the lattice, the ground state transformed as it would under a local scale transformation in the continuum. This is evidence that disentanglers and isometries in MERA local scale transformations on the lattice

34 Official theme of Coogee 16: This year, the workshop will have a special focus on the connections between topology, quantum many-body physics, and quantum information... Illegal sub-theme of Coogee 16: semi-continuous limit of tensor networks / discrete analogue of conformal group Tobias Osborne: Effective conformal field theories for tensor network states Vaughan Jones: Quantum spin chains, block spin renormalization, scale invariance and Thompson s groups F and T

35 What local scale transformations can we implement naturally?

36 What local scale transformations can we implement naturally? b = 2 b = 3 a = 0 a = 1 a = 2 a = 3 a = 4 b = 4 dyadic rational a 2 b site: standard dyadic interval a a + 1, 2b 2 b

37 What local scale transformations can we implement naturally? dyadic subdivision

38 What local scale transformations can we implement naturally? dyadic rearrangement

39 (Perhaps): Local scale transformations in MERA are Thompson s group F + translations: Thompson s group T

40 wave-functions / Hamiltonians global scale transformation (RG transformation) local scale transformations Euclidean path integrals / classical partition functions global scale transformation (RG transformation) local scale transformations

41 Euclidean path integral Z λ = tr e βh q 1d Z(T) = Classical partition function s e 1 T H 2d cl β 1d quantum ~ L y 2d classical L L x as a tensor network A αβγδ Z = TRG Levin, Nave (2006) SRG Xiang et al (2009) TEFR Gu, Wen (2009) Loop-TNR Yang, Gu, Wen (Dec2015)

42 Euclidean path integral Classical partition function Z λ = tr e βh q 1d Z(T) = e 1 T H 2d cl s H q 1d = i (σ z i + σ x i σ x i+1 ) H 2d cl = σ i σ j <i,j> A A σ j A A A σ i A A σ l σ k A A A ijkl = e (σ iσ j +σ j σ k +σ k σ l +σ l σ i )/T

43 Euclidean path integral Statistical partition function Z λ = tr e βh q 1d Z(T) = s e 1 T H 2d cl as a tensor network A αβγδ Z =

44 How do we define a global scale transformation on the lattice? A 0 A 1 A 2

45 Tensor Network Renormalization (TNR) Evenbly, Vidal PRL 2015 (arxiv: ) Evenbly, arxiv:

46 Tensor Network Renormalization (TNR) Evenbly, Vidal PRL 2015 (arxiv: ) Evenbly, arxiv:

47 Tensor Network Renormalization (TNR) defines a proper RG flow in the space of tensors A A A A fp Example: 2D classical Ising model A 1 A 2 A 3 A 4 below critical T = 0.9 T c ordered (Z2) fixed point also TEFR of Gu, Wen (2009) critical T = T c critical fixed point above critical T = 1.1 T c disordered (trivial) fixed point also TEFR of Gu, Wen (2009)

48 Without disentanglers: (TRG, Levin Nave 2006) A 1 A 2 A 3 A 4 T = 1.1 T c different fixed-points for every T in same phase T = 2.0 T c With disentanglers: (TNR 2014) T = 1.1 T c unique fixed-point for any T in same phase T = 2.0 T c

49 at criticality, approximate fixed-point critical point: T = T c A 1 A 2 A 3 A 4 TNR with bond dimension: χ = 4 A 8 A 7 A 6 A 5 A 9 A 10 A 11 A 12

50 near criticality more difficult! T = T c A 1 A 2 A 3 A 4 TNR with bond dimension: χ = 4 A 8 A 7 A 6 A 5 A 9 A 10 A 11 A 12

51 wave-functions / Hamiltonians global scale transformation (RG transformation) local scale transformations Euclidean path integrals / classical partition functions global scale transformation (RG transformation) local scale transformations

52 Z = A A y x global scale transformation w = λz local scale transformation w = log (z) s Conformal symmetry: translation operator on both cylinders has same spectrum ~ Δ α y identification x + 2πr ~ x θ x Can we do the same on the lattice?

53 Plane to cylinder Evenbly, Vidal arxiv:

54 Plane to cylinder Evenbly, Vidal arxiv: y x plane radial quantization in CFT z x + iy conformal transformation z w = log(z) w s + iθ (plane) (cylinder) s log x 2 + y 2 s θ cylinder Extraction of scaling dimensions, OPE

55 Evenbly, Vidal arxiv: ε I σ μ ψ ψ

56 Evenbly, Vidal arxiv:

57 Euclidean path integral Z λ = tr e βh q 1d β 1d quantum torus L Euclidean time evolution on different geometries upper half-plane infinite strip Ψ ρ β β ground state thermal state

58 Upper half plane Evenbly, Vidal, PRL 2015 arxiv: Ψ ~e τh φ 0

59 Upper half plane Evenbly, Vidal, PRL 2015 arxiv: Ψ ~e τh φ 0 TNR produces MERA

60 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

61 ρ β ~e βh infinite strip of finite width β Evenbly, Vidal, PRL 2015 arxiv:

62 Summary: global scale transformation (RG transformation) local scale transformation on the lattice! Frontier: What about 2+1 dimensions? (3+1?) (QCD?) What about diffeomorphism invariance on the lattice? (quantum gravity?) What about tensor networks in the continuum? (space-time symmetries)