Tensor networks for dynamical observables in 1D systems

1. Tensor networks for dynamical observables in 1D systems Mari-Carmen Bañuls

2. Tensor network techniques and dynamics An application to experimental situation Limitations, advances

3. Introduction • Approximate methods are fundamental for the numerical study of many body problems

4. Introduction • Efficient representations of many body systems • Tensor Network States • states with little entanglement are easy to describe • We also need efficient ways of computing with them

5. What are TNS? • TNS = Tensor Network States A general state of the N-body Hilbert space has exponentially many coefficients N-legged tensor A TNS has only a polynomial number of parameters

6. What are TNS? • TNS = Tensor Network States • A particular example Mean field approximation product state! Can still produce good results in some cases

7. Introduction • Successful history regarding static properties

8. Introduction • In particular in 1D • DMRG methods • Matrix Product State (MPS) representation of physical states White, PRL 1992 Schollwöck, RMP 2005 Verstraete et al, PRL 2004

9. Gu, Levin, Wen 2008 Jordan et al PRL 2008 Jiang, Weng, Xiang, 2008 Wang, Verstraete, 2011 Zhao et al., 2010 Corboz et al 2011, 2012 Introduction • In 2D • MPS generalized by PEPS • much higher computational cost • recent developments • Tensor Renormalization • iPEPS Verstraete, Cirac, 2004

10. Introduction • Dynamics is a more difficult challenge even in 1D

11. Introduction • with many potential applications

12. Introduction non-equilibrium dynamics theoretical with many potential applications transport problems applied predict experiments

13. What can we say about dynamics with MPS?

14. The tool: MPS

15. Matrix Product States

16. number of parameters Matrix Product States

17. Matrix Product States • MPS good at states with small entanglement • controlled by parameter D

18. Matrix Product States • Works great for ground state properties... • finite chains → • infinite chains → White, PRL 1992 Schollwöck, RMP 2005 Östlund, Rommer, PRL 1995 Vidal, PRL 2007

19. ...because of entanglement • (Most) ground states satisfy an area law MPS are a good ansatz!

20. Matrix Product States • Can also do time evolution • finite chains • infinite (TI) chains ⇒ iTEBD • but... Vidal, PRL 2003 White, Feiguin, PRL 2004 Daley et al., 2004 TEBD t-DMRG Vidal, PRL 2007

21. Under time evolution entanglement can grow fast !

22. Matrix Product States • Entropy of evolved state may grow linearly Osborne, PRL 2006 Schuch et al., NJP 2008 required bond for fixed precision bond dim time

23. But not completely hopeless...

24. Matrix Product States • Will work for short times • For states close to the ground state Used to simulate adiabatic processes Predictions at short times Imaginary time (Euclidean) evolution → ground states

25. Simulating adiabatic dynamics for the experiment A particular application

26. Adiabatic preparation of Heisenberg antiferromagnet with ultracold fermions

27. limit t-J model Adiabatic Heisenberg AFM • Fermi-Hubbard model describing fermions in an optical lattice interaction hopping

28. Adiabatic Heisenberg AFM Fermi-Hubbard model describing fermions in an optical lattice exchange interaction hopping

29. hopping Adiabatic Heisenberg AFM Fermi-Hubbard model describing fermions in an optical lattice exchange interaction Heisenberg model

30. Simulation of dynamics in OL experiments • Fermionic Hubbard model realized in OL Jördens et al., Nature 2008 Observed Mott insulator, band insulating phases Schneider et al., Science 2008

31. Simulation of dynamics in OL experiments • Challenge: prepare long-range antiferromagnetic order • Problem: low entropy required beyond direct preparation e.g. t-J at half filling Jördens et al., PRL 2010

32. Adiabatic Heisenberg AFM • Adiabatic protocol • initial state with low S • tune interactions to how long does it take? what if there are defects?

33. Adiabatic Heisenberg AFM Feasible proposal Band insulator Product of singlets big gap non-interacting second OL Lubasch, Murg, Schneider, Cirac, MCB PRL 107, 165301 (2011)

34. Adiabatic Heisenberg AFM Feasible proposal Product of singlets Trotzky et al., PRL 2010 Lower barriers Lubasch, Murg, Schneider, Cirac, MCB PRL 107, 165301 (2011)

35. Adiabatic Heisenberg AFM Feasible proposal Final Hamiltonian Lubasch, Murg, Schneider, Cirac, MCB PRL 107, 165301 (2011)

36. Adiabatic Heisenberg AFM We find: Feasible time scales Fraction of magnetization

37. Large system ⇒ longer time Adiabatic Heisenberg AFM We find: Local adiabaticity antiferromagnetic state on a sublattice

38. Adiabatic Heisenberg AFM We find: Local adiabaticity Fraction of magnetization

39. Adiabatic Heisenberg AFM • Experiments at finite T • holes expected

40. Adiabatic Heisenberg AFM Holes destroy magnetic order 2 holes simplified picture: free particle

41. Adiabatic Heisenberg AFM Hole dynamics

42. Control holes with harmonic trap

43. Adiabatic Heisenberg AFM

44. Adiabatic Heisenberg AFM

45. Adiabatic Heisenberg AFM Harmonic trap can control the effect of holes

46. We found • feasible proposal for adiabatic preparation (time scales) • local adiabaticity: • AFM in a sublattice faster • holes can be controlled by harmonic trap • generalize to 2D system M. Lubasch, V. Murg, U. Schneider, J.I. Cirac, MCB PRL 107, 165301 (2011)

47. Is this all we can do with MPS techniques?

48. Not really

49. In some cases, longer times attainable with new tricks

50. Key: observables as contracted tensor network