(NON) THERMALIZATION OF 1D SYSTEMS: numerical studies with MPS

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

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

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

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

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

10. space time observable as tn

11. time space OBSERVABLE AS TN

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

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

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

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

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

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

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

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

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

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

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

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 Anderson localization: single particle states localized due to disorder environment destroys localization Interactions and disorder more interesting scenario Basko, Aleiner, Altshuler, Ann. Phys. 2006 Gornyi, Mirlin, Polyakov, PRL 2005 weak interactions ⇒ MBL phase Many-body localization 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 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 plus small modulation of spin density study spin transport to decide thermalization MBL transition MPS description of mixed states TN description of evolution

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

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

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

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

40. many body localization Discover TI models exhibiting MBL 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? work in progress

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

42. THANKS!