Quantum Many Body Systems and Tensor Networks

Transcription

1 Quantum Many Body Systems and Tensor Networks Aditya Jain International Institute of Information Technology, Hyderabad July 30, 2015 Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

2 Introduction Quantum Many Body Systems Quantum lattice systems considered, described by G = {V,E} Vertices associated with a quantum degree of freedom each Edges correspond to neighbourhood relations Hilbert space is given by H = (C d ) n Number of parameters needed for exact description grow exponentially with n. Can we still understand such systems? Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

3 Introduction We can understand naturally occurring systems with constant or polynomial number of degrees of freedom Theories and methods from the past : Mean-field, BCS, density functional, DMRG Most of these methods approximate the full state by a tensor product ρ = ρ 1 ρ 2... ρ N Tensor-networks: a new language to describe many-body entanglement which enables us to describe very complicated entanglement structures using only poly(n) parameters. Tensor networks can be used to describe ground states of gapped local Hamiltonians Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

4 Outline 1 Tensor Networks 2 Gapped Ground States 3 1D systems 4 Problem with 2D systems 5 Proposed Solution 6 Results 7 Summary and Future work Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

5 What is a Tensor? It is an object that consists of a vertex with a constant number of edges flowing out of it Each edge is labelled from a finite label set {0,1,2,..,d-1} If all edges are labelled, the value of a tensor is a number Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

6 Interpretation of Tensors Let H = C d have basis 0, 1,..., d 1 As a vector m H 5 : m = m i1,i 2,..i 5 i 1,..., i 5 As linear map, eg : H 2 H 3, with m( i 1 i 2 ) = m i1,i 2,..i 5 i 3 i 4 i 5 Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

7 Composition of Tensors 1 Tensor Product When all free edges are labelled, the value of tensor is a product of value of n and m. Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

8 Composition of Tensors 1 Tensor Product When all free edges are labelled, the value of tensor is a product of value of n and m. Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

9 Composition of Tensors 1 Tensor Product When all free edges are labelled, the value of tensor is a product of value of n and m. 2 Contraction When all free edges are labelled, value of tensor is the sum over labelling of internal edge of the product value of n and m. Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

10 Composition of Tensors 1 Tensor Product When all free edges are labelled, the value of tensor is a product of value of n and m. 2 Contraction When all free edges are labelled, value of tensor is the sum over labelling of internal edge of the product value of n and m. Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

11 Small things to figure out Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

12 Small things to figure out Inner Product Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

13 Small things to figure out Inner Product Matrix Product Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

14 Small things to figure out Inner Product Matrix Product Trace Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

15 Evaluating tensor Networks - Bubble Method Approximating the value of a closed Tensor Network with a given ordering Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

16 Evaluating tensor Networks - Bubble Method Approximating the value of a closed Tensor Network with a given ordering A can be interpreted as a linear map from 0 dimension to H 3 Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

17 Evaluating tensor Networks - Bubble Method Approximating the value of a closed Tensor Network with a given ordering B can be interpreted as a linear map from H to H Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

18 Local Hamiltonian and Spectral Gap Hamiltonian with finite-ranged interaction given by : H = h j j V Each h j is non-trivially supported only on finitely many sites in V Geometrically k-local : each h j is supported on a V j with max a,b Vj dist(a, b) = k 1 Ground space : The space formed by the lowest energy eigenvectors of the Hamiltonian that minimize ψ H ψ Spectral Gap ( E) : It is the energy gap from the ground space to the first excited state. E = 0 : Gapless or critical E > 0 : Gapped Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

19 Clustering of Correlation and Area Laws For gapped models, correlation functions always decay exponentially with the distance : < O A O B > < O A >< O B > Ce dist(a,b) E (2v) O A O B Entropy of Entanglement : For some lattice G = (V,E), let V be partitioned into two disjoint sets A and B. Entropy of Entanglement is given by S(ρ A ) = tr(ρ A log(ρ A )) How does S(ρ A ) scale with the size A of the region A if G represents the ground state of a local Hamiltonian? Area law : The entropy scales as boundary area of A, so S(ρ A ) = O( δa ) The boundary δa of region A is define as δa := {j A : k B with dist(j, k) = 1} Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

20 Tensor Networks for Gapped Ground States Matrix Product States (MPS) : It is clear that MPS satisfy area law for a constant D because Schmidt Rank across any cut is D. Hence the entanglement entropy is O(log D) which is constant in n. Number of parameters : O(ndD 2 ) - linear in n as opposed to general vector defined by O(d n ) parameters Projected Entangled Pair States (PEPS) Satisfies area law Area law for 2D still a conjecture Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

21 Operators Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

22 Finding ground states for 1D Local Hamiltonians Steps : Start with a random MPS Sweep left to right and right to left optimizing energy at each site until convergence Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

23 Difficulty with 2D systems 1D 2D Efficient bubbling for PEPS? Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

24 Approach Problem Statement : Approximate the expectation value of an operator acting on the ground state of a Gapped Local Hamiltonian. Theorem (M. B. Hastings; 2006) If Ω is unique and gapped, then l > 0 and for every local operator B, an operator KB l such that : K B Ω = Ω B Ω Ω + δ where δ < e O(l) Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

25 Approach (Contd) Schmidt decomposition : Ω = D δl α=1 λ α O α I α { O α } - Orthonormal basis δ = λ α O α δ α ; where { δ α } is Not Orthonormal α δ 2 = λ 2 α δα 2 e O(l) where λ 2 α = 1 α α Substituting Ω back into the main theorem we get, λ α O α K B I α = Ω B Ω λ α O α I α + λ α O α δ α α α α=1 = α, K B I α = Ω B Ω I α + δ α Let { V α } be the vectors contained within the boundary for α = 1, 2,.., D δl V = sp{ V α } = sp{ I α } dim(v ) = D δl = D O(l) Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

26 Approach (Contd) dims = d O(l2 ) dimv = D O(l) dims = at least one eigenvector of K B which is almost inside V. Construction of K B Hasting : K l B = c l c l t 2 c dt e 2 e ith l Be ith l Chebyshev : K l B = l i=0 b i[h, [H,..., [H, B]]] ( Commutator upto i th level) Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

27 Results F (z) = min ψ V ( K B ψ z ψ 2 ) Expectation : F(z) should minimize at z Ω B Ω 1 Ising Model : H = ( N 1 σi z σi+1 z + h N σi x) i=1 B = P x = 0.5 (σ x + I) < B > from graph : < B > actual : i=1 Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

28 Results(Contd) B = P x P x < B > from graph : < B > actual : Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

29 Results(Contd) 2 XY model H = ( N 1 0.5(1 α)σi x σi+1 x + 0.5(1 + α)σy i σ y i+1 + γ N σi z) i=1 B = P x < B > from graph : 0.5 < B > actual : 0.5 i=1 Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

30 Summary and Future Work Summary Tensor Networks - An important tool to study the Entanglement Structure in Many Body Systems Gapped Ground States do not require exponential number of parameters parameters for their description Variational Algorithms (DMRG) : Revolutionised the understanding of 1D systems 1D area law : Justifies the working of DMRG in some sense Future Work : Simulate 2D gapped systems and find the expectation value of local operators using the described approach. Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

31 References M. B. Hastings: Solving Gapped Hamiltonians Locally, 2006; arxiv:cond-mat/ Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24

32 Thank you! Aditya Jain (IIIT-H) Quantum Hamiltonian Complexity July 30, / 24