Quantum Simulation of Exactly Solvable Models

1. Quantum Simulation of Exactly Solvable Models Vladimir Korepin Frank Verstraete Valentin Murg BANGALORE 2013 arXiv:1201.5636 arXiv:1201.5627

2. Quantum information vs many body quantum mechanics I will consider one dimensional models like XXX Heisenberg anti ferromagnet, Hubbard model, Bose gas with delta interaction, t-J model… There are two ways to describe dynamics of a Hamiltonian: Use mps to approximate the ground state [use virtual dimension]. Unify models into universality classes and to consider a representative Each universality class has one model solvable by Bethe Ansatz. Dynamics of solvable models is 2 body reducible. Bethe Ansatz can be reformulated similar to mps: using virtual dimension. It is quantum inverse scattering method or algebraic Bethe Ansatz. The point of our work is to reformulate mps so that the method gives exact analytical results, when available.

3. Motivation: Coordinate Bethe Ansatz Two-body reducible Bethe eq. Difficult to calcualte correlation for short lattice !

4. Motivation: Algebraic Bethe Ansatz (State with ) … Creation operator for one down-spin L operator =

5. Motivation: Algebraic Bethe Ansatz (State with )

6. Tensor Network Formulation of the Algebraic Bethe Ansatz Outline Yang-Baxter Algebra Virtual dimension Bethe equations problem Local Hamiltonian Energy

7. Tensor Network Formulation of the Algebraic Bethe Ansatz Yang-Baxter Algebra: Virtual dimension Property: R-Matrix: Also known as S matrix

8. Tensor Network Formulation of the Algebraic Bethe Ansatz Yang-Baxter Equation self consistency for 2 body reducibility

9. Tensor Network Formulation of the Algebraic Bethe Ansatz Transfer Matrix: Property:

10. Tensor Network Formulation of the Algebraic Bethe Ansatz Fundamental Representation: Yang-Baxter equation

11. Tensor Network Formulation of the Algebraic Bethe Ansatz Co-Multiplication Property:

12. Tensor Network Formulation of the Algebraic Bethe Ansatz Fundamental Models: Matrix Product Operator Representation

13. Tensor Network Formulation of the Algebraic Bethe Ansatz: Local Hamiltonian from Transfer Matrix: Regular Solution: Identical operator

14. Tensor Network Formulation of the Algebraic Bethe Ansatz: Local Hamiltonian from Transfer Matrix:

15. Tensor Network Formulation of the Algebraic Bethe Ansatz: Local Hamiltonian from Transfer Matrix:

16. Tensor Network Formulation of the Algebraic Bethe Ansatz: Local Hamiltonian from Transfer Matrix: R matrix , L-operator and Scattering matrix XXX Heisenberg Model: XXZ Model:

17. Tensor Network Formulation of the Algebraic Bethe Ansatz: Symmetries:

18. Tensor Network Formulation of the Algebraic Bethe Ansatz: Symmetries: Keeps number of down-spins constant Creates one down-spin Annihilates one down-spin

19. Tensor Network Formulation of the Algebraic Bethe Ansatz: Bethe Ansatz: Eigenvalue Problem: Difficult to solve Bethe Equations:

20. Tensor Network Formulation of the Algebraic Bethe Ansatz: Bethe-Eigenstate: State with

21. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Reflection Algebra: Reflection Equations:

22. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Transfer Matrix:

23. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Fundamental Models: Composition of a new representation out of 2 R-matrices and a known representation. Transfer Matrix:

24. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Proof of Composition-Equation:

25. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Simplest Representation:

26. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Local Hamiltonian from Transfer Matrix:

27. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Local Hamiltonian from Transfer Matrix:

28. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Local Hamiltonian from Transfer Matrix:

29. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Symmetries: Keeps number of down-spins constant Creates one down-spin Annihilates one down-spin

30. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions) Bethe Ansatz: Eigenvalue Problem: Bethe Equations

31. Tensor Network Formulation of the Algebraic Bethe Ansatz (open boundary conditions)

32. Tensor Network States ↔ Algebraic Bethe Ansatz: Condensed Matter Theory Quantum Information Theory Entanglement Renormalization Group Methods: Tensor Network States NRG, DMRG MPS, PEPS, TTNS, MERA Integrability: Bethe-Ansatz

33. Motivation: Algebraic Bethe Ansatz

34. Contraction of the Bethe-Network

35. Contraction of the Bethe-Network Approximation

36. Contraction of the Bethe-Network Approximation

37. Contraction of the Bethe-Network Approximation

38. Contraction of the Bethe-Network Approximation

39. Contraction of the Bethe-Network Order Optimization Order of the B's is arbitrary!

40. Example: Heisenberg model with periodic boundary conditions Two-spinon excited states Structure factor for ground state and 3 two-spinon excited states (N=50, D=1000, pbc) Rhs: two-spinon excited states with S=1 and Sz=1 Lhs: relative error in the energy as a function of D

41. Example: Heisenberg Model

42. Example: Heisenberg Model Ground State

43. Example: XXZ Model with open boundary conditions Ground State Structure factor for ground state if the XXZ-model (N=50, D=1000, obc) Lhs: relative error as a function of D

44. Summary • Bethe wavefunction can be • - represented as tensor network • - approximated by a MPS with low bond dimension • Makes possible the calculation of correlations for small lattice with open bc. Outlook • Nested Bethe Ansatz: tJ-Model Fermi-Hubbard Model • Two-dimensional models: