Introduction to Tensor Networks: PEPS, Fermions, and More

Transcription

1 Introduction to Tensor Networks: PEPS, Fermions, and More Román Orús Institut für Physik, Johannes Gutenberg-Universität, Mainz (Germany)! School on computational methods in quantum materials Jouvence, June

2 Some reviews J. Eisert, Modeling and Simulation 3, 520 (2013), arxiv: N. Schuch, QIP, Lecture Notes of the 44th IFF Spring School 2013, arxiv: R. Orus, arxiv: J. I. Cirac, F. Verstraete, J. Phys. A: Math. Theor. 42, (2009) F. Verstratete, J. I. Cirac, V. Murg, Adv. Phys. 57,143 (2008) J. Jordan, PhD thesis, G. Evenbly, PhD thesis, arxiv: U. Schollwöck, RMP 77, 259 (2005) U. Schollwöck, Annals of Physics 326, 96 (2011)

3 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

4 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

5 Entanglement obeys area-law

6 Entanglement 2d system E E A key resource in quantum information teleportation, quantum algorithms, quantum error correction, quantum cryptography Reduced density matrix of subsystem A Entanglement entropy (von Neumann entropy) For many ground states In d dimensions (L > ξ) Generic state S(A) ~ L d (volume) Ground states of (most) local Hamiltonians S(A) ~ L d 1 (area) Srednicki, Plenio, Eisert, Dreißig, Cramer, Wolf Locality of interactions area-law

7 Many-body Hilbert space is far too large

8 System of N spins 1/2 Everyday material, N ~ Avogadro number ~O(10 23 ) Number of basis states in the Hilbert space ~ O( ) Compare to Number atoms in the observable universe ~ O(10 80 )

9 Hilbert space is a convenient illusion Exploration time ~ O( ) sec. Most states here are not even reachable by a time evolution with a local Hamiltonian in polynomial time!!! Poulin, Qarry, Somma, Verstraete, PRL (2011) Compare to Age of the universe ~ O(10 17 ) sec. Set of relevant states (area-law, ground states of local Hamiltonians) Set of product states (mean field) We need a language to target the relevant corner of quantum states directly

10 Tensor Networks

11 A new language DNA person

12 A new language tensor quantum state Ψ Tensors are local building blocks for the quantum state (like a DNA, or LEGO)

13 Tensor network diagrams vector v matrix A matrix product AB trace of matrix product tr(abc) tensor contraction f (A,B,C,D)

14 Break the wavefunction locally Ψ = Tensor (multidimensional array of complex numbers) Ψ i1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 1 i 2 i 9 replacement O( p N ) O(poly(N, p, D)) i 1 i 2 i N p-level i's Ψ i1 i 2...i N summed ancillary bond index 1 D (carries entanglement) physical index (local basis) i 4 i 5 i 2 δ µ γ i 1 i i i B βγδµ C αγδµ systems Tensor Network i 6 β i 3 i A 9 αβ α i 9 i 8 i 9 i = A 9 αβ β β =1, 2,, D i B 8 βγδµ Efficient representation, satisfies area-law, and targets low-energy eigenstates of local Hamiltonians

15 Important families of Tensor Network states

16 Matrix Product States (MPS) 1d systems DMRG, Power Wave Function Renormalization Group, Time Evolving Block Decimation c.f., Miles Stoudenmire & Ulrich Schollwöck s talk Projected Entangled Pair States (PEPS), Tensor Product States (TPS) 2d,3d systems Tensor Product Variational Approach, PEPS, Infinite-PEPS algorithm, Tensor-Entanglement Renormalization, TRG/SRG/HOTRG/HOSRG Multiscale Entanglement Renormalization Ansatz (MERA) Entanglement Renormalization RG any-d scale invariant systems Other: MPO, MPDO, TTN, PEPO

17 This lecture Projected Entangled Pair States (PEPS), Tensor Product States (TPS) 2d,3d systems Tensor Product Variational Approach, PEPS, Infinite-PEPS algorithm, Tensor-Entanglement Renormalization, TRG/SRG/HOTRG/HOSRG Multiscale Entanglement Renormalization Ansatz (MERA) Entanglement Renormalization RG any-d scale invariant systems

18 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

19 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

20 From MPS to PEPS Matrix Product States (MPS) 1d systems But we want to go beyond 1d systems!!! Very painful for DMRG...

21 From MPS to PEPS Matrix Product States (MPS) 1d systems This lecture Projected Entangled Pair States (PEPS), Tensor Product States (TPS) and so on 2d systems 3d systems

22 Two exact examples

23 An exact example: Kitaev s Toric Code Kitaev, 1997 star operator i plaquette operator Simplest known model with topological order Ground state (and in fact all eigenstates) are PEPS with D= = 2 = 2 1 = = And another tensor rotated 90

24 Resonating Valence Bond State SU(2) singlets Anderson, Equal superposition of all possible nearest-neighbor singlet coverings of a lattice (spin liquid) Proposed to understand high-t C superconductivity It is a PEPS with D= = = 1 3 And 2 3 rotations

25 PEPS F. Verstraete, I. Cirac, cond-mat/

26 and infinite PEPS (ipeps) assuming translation invariance J. Jordan, RO, G. Vidal, F. Verstraete, I. Cirac, Phys. Rev. Lett. 101, (2008)

27 PEPS obey 2d area-law

28 1..D

29 in 1..D

30

31 in

32 in 1..D prefactor size of the boundary

33 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007) 1..D

34 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007)

35 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007)

36 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007) (double indices) 1..D 2

37 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007)

38 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007)

39 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007)

40 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007)

41 Exact contraction is inneficient N. Schuch et al, PRL 98, (2007) times computing time ~ Exponential amount of time! Mathematical statement: exact contraction of a PEPS is a #P-Hard problem (harder than NP-Complete) Applies also to expectation values of observables

42 Critical correlation functions F. Verstraete et al, PRL 96, (2006) Expectation values are those of the classical 2d Ising model It is a PEPS with D=2 (left as exercise): = = = = = + permutations At the correlation length is infinite:

43 PEPS to/from Hamiltonians Ground state of a gapped Parent Hamiltonian with local interactions World of Hamiltonians Hamiltonian with local interactions e.g. D. Perez-Garcia et al, QIC 8, 0650 (2008) M. Hastings, PRB 73, (2006) Generic PEPS with finite D World of states Thermal states can be approx. by a PEPS with finite D PEPS target the relevant corner of the Hilbert space (area-law)

44 MPS vs PEPS MPS in 1d 1d area law Exact contraction is efficient Finite correlation length to/from 1d Hamiltonians PEPS in 2d 2d area law Exact contraction is inefficient Finite and infinite correlation lengths to/from 2d Hamiltonians

45 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

46 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

47 PEPS as ansatz: variational optimization

48 Variational optimization (e.g. finite PEPS) e.g. F. Verstraete, I. Cirac, cond-mat/ Optimize over each tensor individually and sweep over the entire system (as in DMRG) Minimization of quadratic function Generalized eigenvalue problem Once and are known, we can solve this problem efficiently Approximate calculation of and

49 e.g. calculation of

50 e.g. calculation of

51 e.g. calculation of

52 e.g. calculation of

53 e.g. calculation of MPS

54 e.g. calculation of MPS MPO 1..D 2 1d problem: use a 1d method for MPS (e.g., DMRG or TEBD)

55 e.g. calculation of 1..χ MPSup

56 e.g. calculation of 1..χ MPSup MPSdown 1..χ

57 e.g. calculation of 1..χ MPSup 0d problem: exact! MPSdown 1..χ

58 e.g. calculation of Dimensional reduction 2d problem 1..χ 1d problem: use DMRG! 0d problem: exact! 1..χ 1..χ 1..χ is the environment of tensor is computed similarly, but sandwitching with the Hamiltonian Valid also for any expectation value

59 Time evolution (real, imaginary)

60 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) Divide into small time-steps Split the Hamiltonian (e.g. 2-body n.n.)

61 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) Divide into small time-steps Split the Hamiltonian (e.g. 2-body n.n.)

62 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) Divide into small time-steps Split the Hamiltonian (e.g. 2-body n.n.)

63 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) Divide into small time-steps Split the Hamiltonian (e.g. 2-body n.n.)

64 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) Divide into small time-steps Split the Hamiltonian (e.g. 2-body n.n.)

65 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) 2-body gates

66 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) 2-body gates

67 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) evolved state

68 Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, (2008) Main idea 1..D Different approaches to this problem: full update, simplified update, TPVA... Full update: Finite systems: optimize over all tensors in the PEPS (as before) Infinite systems: optimize over tensors in the PEPS unit cell (ipeps) Require calculations of environments, like the one shown before.

69 Environments with infinite PEPS

70 Infinite systems e.g. J. Jordan et al, PRL 101, (2008) R. Orús, G. Vidal, PRB (2009) Finite PEPS Infinite PEPS Unit cell of tensors is repeated periodically over the whole PEPS: translational invariance

71

72

73 A A * (double indices) Let s put it on the plane of the screen!

74 A A * (double indices) Environment calculations Contraction of this infinite lattice

75 Contracting the infinite 2d lattice 1-dim transfer matrix: dominant eigenvector? 1..χ Can be approximated using infinite MPS itebd, idmrg, PWFRG, etc

76 Contracting the infinite 2d lattice Coarse-graining approaches: TRG/SRG, HOSVD

77 Contracting the infinite 2d lattice corner transfer matrix (Baxter, 1968) (nice spectral properties) half-row transfer matrix half-column transfer matrix

78 Contracting the infinite 2d lattice Renormalized Corner Transfer Matrices Renormalization (numerical) C1 T4 T1 C2 T2 (Baxter, 1968,1978) C4 T3 C3 1..χ Directional version of the corner transfer matrix renormalization group (faster than 1d transfer matrix methods) T. Nishino, K. Okunishi, JPS Jpn. 65, 891 (1996)

79 Corner transfer matrix and ipeps R. Orús, G. Vidal, PRB (2009), R. Orús, PRB 85, (2012) Example: left move C1 T4 T1 C2 T2 1) insertion C1 T4 T1 T1 C2 T2 C4 T3 C3 C4 T3 T3 C3 Z Z Z Z C 1 T 4 C 4 T1 T3 C2 T2 C3 3) renormalization " C 1 " T 4 " C 4 Iterate in all directions until convergence T1 T3 C2 T2 C3

80 An example: Toric Code in arbitrary field S. Dusuel et al., PRL 106, (2011) x y z H= J A s J B p h x σ i h y σ i h z σ i s p i i i s i p

81 Phase diagram Critical quantum Ising points Multicritical line Multicritical point topological (deconfined) phase polarized phase bare Toric Code 2 nd order sheets, quantum Ising universality class Self-dual point 1 st order sheet

82 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

83 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

84 Fermionic 2d systems Fermionic systems are extremely interesting physical systems, e.g. the 2d fermionic Hubbard model may be the key to understand the emergence of high-t C superconductivity Unfortunately, fermionic systems are also amongst the most difficult to simulate, because of the sign problem in Quantum MonteCarlo (sampling of negative probabilities) Fermions are a NUMERICAL MONSTER for Quantum MonteCarlo because of the sign problem! but there is hope...

85 Tensor Networks + Fermions e.g., P. Corboz, R. Orús, B. Bauer, G. Vidal, PRB 81, (2010)

86 Bosons Fermions Ψ B (x 1, x 2 ) = Ψ B (x 2, x 1 ) Symmetric wavefunction b i b j = b j b i Operators commute Ψ F (x 1,x 2 ) = Ψ F (x 2, x 1 ) Antisymmetric wavefunction c i c j = c j c i Operators anticommute Crossings in a TN + _ Ignore crossings Careful!!!!

87 Tensor Network fermionization rules P P T P P P = T Use parity-preserving tensors T i1 i 2...i M = 0 if P(i 1 )P(i 2 )...P(i M ) 1 Symmetry of the Hamiltonian i 1 i 2 i 1 i X 2 i 2 i 1 j 1 j 2 Replace crossings by fermionic swap gates X i2 i 1 j 1 j 2 = δ i1 j 1 δ i2 j 2 S(P(i 1 ),P(i 2 )) # S(P(i 1 ),P(i 2 )) = 1 if P(i ) = P(i ) = $ % +1 otherwise Fermionic operators anticommute The leading order of the computational cost is the same as in the bosonic case

88 1 1 1 fermionic order ~ graphical projection of a PEPS physics is independent of the order physics is independent of graphical projection (different choices of Jordan-Wigner transformation, if mapping to a spin system)

89 Example: scalar product of 3x3 PEPS Ψ Ψ Ψ Ψ Contract as for bosons!

90 H = t An example: 2d t-j model ( c iσ c jσ + hc) + J (S i S j 1 4 n n ) µ i j i, j σ i, j P. Corboz, R. Orús, B. Bauer, G. Vidal, PRB 81, (2010) P. Corboz, S. White, G. Vidal, M. Troyer, PRB (2011) i n i

91 First simulations P. Corboz, R. Orús, B. Bauer, G. Vidal, PRB 81, (2010) Energy (t /J = 3) Compatible results

92 Some improved simulations P. Corboz, S. White, G. Vidal, M. Troyer, PRB (2011) Energy (t / J = 2.5) Promising!

93 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

94 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

95 From MPS to MERA Matrix Product States (MPS) 1d systems But we want to do critical systems!!! Also very painful for DMRG...

96 From MPS to MERA Matrix Product States (MPS) 1d systems Multiscale Entanglement Renormalization Ansatz (MERA) and so on 1d systems 2d systems

97 From MPS to MERA Matrix Product States (MPS) 1d systems Multiscale Entanglement Renormalization Ansatz (MERA) This lecture and so on 1d systems 2d systems

98 1d MERA 1..χ holographic dimension (RG) spatial dimension

99 Tensors obey constraints

100 Unitaries (disentanglers) u u + = Isommetries (coarse-grainings) w w + =

101 Reason: entanglement is built locally at all length scales

102 1 top tensor (for finite system)... length/energy scale L/8 L/4 L/2 L coarse-grain entangle locally coarse-grain entangle locally coarse-grain entangle locally Extra dimension defines an RG flow: Entanglement Renormalization

103 Entropy of 1d MERA

104 geodesic curve Entanglement as boundary in holographic geometry:

105 Entanglement as boundary in holographic geometry: Constant contribution at every layer 1d MERA can produce logarithmic violations to the area-law: (like 1d critical systems!)

106 Norm of MERA u u + = w w + =

107 Norm of MERA u u + = w w + =

108 Norm of MERA u u + = w w + =

109 Norm of MERA u u + = w w + =

110 Norm of MERA u u + = w w + =

111 Norm of MERA u u + = w w + =

112 Norm of MERA u u + = w w + =

113 Norm of MERA u u + = w w + =

114 Norm of MERA u u + = w w + =

115 Norm of MERA u u + = w w + =

116 Norm of MERA u u + = w w + =

117 Norm of MERA u u + =... The norm is just the contraction of the top tensors w w + =

118 Expectation values u u + = w w + =

119 Expectation values u u + = w w + =

120 Expectation values u u + = w w + =

121 Expectation values causal cone with bounded width Only tensors inside of the causal cone contribute to the expectation value

122 MPS vs 1d MERA MPS in 1d 1d area law Exact contraction is efficient Finite correlation length to/from 1d Hamiltonians Arbitrary tensors MERA in 1d Beyond 1d area law Exact contraction is efficient Finite and infinite correlation lengths to/from 1d Hamiltonians Constrained tensors

123 An example: 1d critical systems G. Evenbly, G. Vidal, in "Strongly Correlated Systems. Numerical Methods", Springer, Vol. 176 (2013)

124 Critical Energies Critical XX Correlators

125 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

126 Outline 1) Generalities 2) 2d PEPS 2.1) Basics 2.2) Computing 2.3) Fermions 3) 1d MERA: Basics 4) Some further topics

127 PEPS & Entanglement Hamiltonians e.g. I. Cirac et al, PRB 83, (2011), N. Schuch et al, PRL 111, (2013)

128 (double indices) 1-dim transfer matrix: dominant eigenvector? Can be approximated using infinite MPS itebd, idmrg, PWFRG, etc

129 Virtual indices of bra 1 D Virtual indices of ket 1 D Boundary virtual index 1...χ It is also hermitian and positive by construction (up to finite-χ effects) 1d Entanglement Hamiltonian ρ = exp( H ) Who is H??? E E Bulk Holographic principle Boundary Gapped 2d systems, trivial phase Critical 2d systems Gapped 2d systems, topological order 1d Hamiltonian, short-range 1d Hamiltonian, long-range Completely non-local (projector) Based on examples Contraction of 2d PEPS for gapped trivial phases could be approximated efficiently!

130 MERA & AdS/CFT e.g. B. Swingle, PRD 86, (2012), G. Evenbly,G. Vidal, JSTAT 145: (2011)

131 For a scale-invariant MERA, the tensors of a critical model with a CFT limit correspond to a gravitational description in a discretized AdS space: lattice realization of AdS/CFT correspondance Bulk is a discretized AdS space

132 branching MERA G. Evenbly, G. Vidal, arxiv:

133 RG

134 RG

135 RG 1d MERA

136 RG 1d branching MERA Decoupling of degrees of freedom along RG (e.g. spin-charge separation), and allows arbitrary scalings of the entanglement entropy In 2d, ansatz for e.g., Fermi & Bose liquids,

137 Lots of other things: Chiral PEPS Entanglement measures Time evolution Infinite systems Classical systems Corner Transfer Matrices Tensor Renormalization Group Symmetries 2d MERA 3d PEPS MERA optimization Topological systems Excited states Thermal states & dissipation Continuous tensor networks Tensor Networks & Montecarlo Practical implementations Blablabla... Ask me if interested!

138 Thank you!