1. Max Planck Institut of Quantum Optics (Garching) New perspectives on Thermalization Aspen 17-21.3.2014 (NON) THERMALIZATION OF 1D SYSTEMS: numerical studies with MPS M. C. Bañuls, A. Müller-Hermes, J. I. Cirac M. Hastings, D. Huse, H. Kim, N. Yao, M. Lukin

2. Using Tensor Network techniques (MPS) for numerical studies of thermalization

3. Tensor Network States: MPS techniques for dynamics Applications to out-of- equilibrium problems In this talk...

4. What are TNS? Context: quantum many body systems TNS = Tensor Network States interacting with each other Goal: describe equilibrium states ground, thermal states Goal: describe interesting states

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

6. Which properties characterize physically interesting states? Area law finite range gapped Hamiltonians states with little entanglement WHY SHOULD TNS BE USEFUL? MPS and PEPS satisfy the area law by construction TNS parametrize the structure of entanglement lots of theoretical progress going on

7. mps Area law by construction MPS = Matrix Product States number of parameters Bounded entanglement

8. good approximation of ground states Verstraete, Cirac, PRB 2006 Hastings J. Stat. Phys 2007 gapped finite range Hamiltonian ⇒ area law (ground state) extremely successful for GS, low energy time evolution can be simulated too MPS Verstraete, Porras, Cirac, PRL 2004 White, PRL 1992 Schollwöck, RMP 2005, Ann. Phys. 2011 Vidal, PRL 2003, PRL 2007White, Feiguin, PRL 2004Daley et al., 2004 Completar lo del TDVP!!!!! but entanglement can grow fast! little entangled

9. alternatively... time dependent observables as TN TN describe observables, not states problem is contracting the network exact contraction not possible #P complete

10. observable as tn time space

11. OBSERVABLE AS TN time space

12. different approximate contraction strategies standard (TEBD, tDMRG) OBSERVABLE AS TN time space evolved state approximated as MPS

13. different approximate contraction strategies Heisenberg picture OBSERVABLE AS TN time space evolved operator as MPO

14. different approximate contraction strategies transverse contraction, folding MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012 OBSERVABLE AS TN time space

15. different approximate contraction strategies transverse contraction, folding MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012 relevant: entanglement in the network OBSERVABLE AS TN in particular, for infinite system transverse eigenvectors as MPS

16. Toy Tensor Network model helps to understand entanglement in the network

17. toy model tn intuition: model free propagating excitations MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012

18. toy model tn MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012

19. toy model tn MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012

20. toy model tn MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012

21. toy model tn MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012

22. toy model tn MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012 entanglement also in the transverse eigenvector folding can reduce the entanglement in this case

23. MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012 observable as tn closest real case: global quench in free fermionic models XY model other problems may benefit from combined strategies folded transverse

24. toy model tn MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012 eigenstate of the evolution no entanglement created in space a second case

25. toy model tn MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012 fast growing entanglement in transverse direction folding works Ising GS folded transverse

26. GS found via iTEBD OBSERVABLE AS TN minimal TN MCB, Hastings, Verstraete, Cirac, PRL 2009 Müller-Hermes, Cirac, MCB, NJP 2012 combined techniques XY model

27. TN-MPS tools can be used to study out-of-equilibrium problems, thermalization questions

28. Application: thermalization Closed quantum system initialized out of equilibrium Does it thermalize? Local observables: do they reach thermal equilibrium values?

29. Application: thermalization MCB, Cirac, Hastings, PRL 2011 compute for small number of sites compare to the thermal state with the same energy thermalization of infinite quantum spin chain fix non-integrable Hamiltonian varying initial state can also be computed using TN

30. APPLICATION: THERMALIZATION MCB, Cirac, Hastings, PRL 2011 non-integrable regime We observed different regimes of thermalization for the same Hamiltonian parameters strong instantaneous state relaxes weak only after time average for some initial states, none of them

31. Application: thermalization instant distance

32. Application: thermalization time averaged

33. Other problems showing absence of thermalization: MBL ongoing collaboration with D. Huse, N. Yao, M. Lukin

34. many body localization Interactions and disorder more interesting scenario Many-body localization Anderson localization: single particle states localized due to disorder Basko, Aleiner, Altshuler, Ann. Phys. 2006 Gornyi, Mirlin, Polyakov, PRL 2005 Oganesyan, Huse, PRB 2007 Rigol et al PRL 2007 Znidaric, Prosen, Prelovsek, PRB 2008 Pal, Huse, PRB 2010 Gogolin, Müller, Eisert, PRL 2011 Bardarsson, Pollmann, Moore, PRL 2012 Bauer, Nayak, JStatMech 2013 Serbyn, Papic, Abanin, PRL 2013 environment destroys localization weak interactions ⇒ MBL phase highly excited states localized system will not thermalize

35. MPS and mbl Allow to study larger sizes than ED Discover TI models exhibiting MBL Oganesyan, Huse, PRB 2007 Pal, Huse, PRB 2010 states at high temperature study spin transport to decide thermalization plus small modulation of spin density MPS description of mixed states TN description of evolution MBL transition

36. MPS AND MBL Similar model with discrete valued fields polarization time 10 random realizations moderate bond dimensions needed

37. many body localization Discover TI models exhibiting MBL work in progress with J=0 produces average over ALL realizations of single chain with discrete values of radom fields Paredes, Verstraete, Cirac, PRL 2005 MBL in TI model!! phase diagram being explored

38. MPS AND MBL polarization time preliminar y (40 spins) moderate bond dimensions needed

39. MPS AND MBL polarization time preliminar y (40 spins) things will change with interactions J

40. many body localization Discover TI models exhibiting MBL work in progress Open questions phase diagram J, B characterize MBL from results accessible to finite t simulations? limitations of the mixed state MPO description of time evolution?

41. conclusions Versatile TNS tools can be used for time evolution approximations involved state more general TN contraction understanding entanglement in TN important Applications to non-equilibrium thermalization MBL in TI systems evolution of operators ongoing work with M. Hastings, H. Kim, D. Huse

42. Max Planck Institut of Quantum Optics (Garching) THANKS!