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Tensor networks for dynamical observables in 1D systems

Tensor networks for dynamical observables in 1D systems. Mari-Carmen Bañuls. Tensor network techniques and dynamics An application to experimental situation Limitations, advances. Introduction. Approximate methods are fundamental for the numerical study of many body problems.

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Tensor networks for dynamical observables in 1D systems

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  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

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