Many-body localization characterized by entanglement and occupations of natural orbitals

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1 Many-body localization characterized by entanglement and occupations of natural orbitals Jens Hjörleifur Bárðarson Max Planck Institute PKS, Dresden KTH Royal Institute of Technology, Stockholm 1

2 I: Broken symmetry II: Adiabatic continuity & renormalization

3 Fermi liquid theory Ground state Fermi Gas Fermi Liquid Quasiparticles See e.g. P. Coleman, Introduction to Many-Body Physics

4 Fermi liquid theory See e.g. P. Coleman, Introduction to Many-Body Physics

5 Many-body localization Anderson Insulator Many-body localized insulator Basko, Aleiner, Altshuler, Ann. Phys 2006 see also: Altshuler, Gefen, Kamenev, Levitov PRL 1997 Gornyi, Mirlin, Polyakov PRL 2005

6 Anderson insulator (many-body) H 0 = 1 2 LX c i+1 c i + c i c i+1 + i n i = X i " i a i a i i 2 [ W, W ] i=1 ñ i = a i a i [ñ, H 0 ]=0 Local integrals of motion ñ 1, ñ 2,...ñ L i Eigenstates are product states P. W. Anderson Phys. Rev. 1958

7 Emergent integrability in many-body localization H = X i h i n i + X ij J ij n i n j +... where [n i,h]=0 n 1,n 2,...n L i Local conserved integrals of motion All eigenstates product states c i = X j Z i ja j + X jkl f i jkla j a ka l +... Quasiparticles Basko, Aleiner and Altshuler Ann. Phys. 2006; Gornyi, Mirlin, Polyakov, PRL 2005 Serbyn, Papic, and Abanin PRL 2013; Huse, Nandkishore, and Oganesyan PRB 2014; Imbrie J. Stat. Phys Ros, Müller, Scardicchio Nucl. Phys. B 2015; Imbrie, Ros, Scardicchio arxiv: (review)

8 Every MBL eigenstate related to a Anderson insulator eigenstate via a finite depth quantum circuit U {n}i Bauer and Nayak, J. Stat. Mech 2013; Pekker & Clark PRB 2017

9 Many-body localized eigenstates have an area law entanglement Bauer and Nayak, J. Stat. Mech 2013; Kjäll, JHB, Pollmann PRL 2014

10 Mutual information directly reveals the entanglement structure of eigenstates De Tomasi, Bera, JHB, Pollmann PRL 2017

11 One particle density matrix definitions H n i = E n n i ij = h n c i c j n i i = n i n 1 n 2... n L i n Natural orbitals Occupations tr = LX =1 n = N 11 Penrose & Onsager PR 1956; Koch & Goedecker Solid State Comm. 2001; Gottlieb & Mauser PRL 2005 Nessi & Iucci PRA 2011; Gramsch & Rigol PRA 2012

12 Occupations H = t LX i=1 apple 1 2 (c i c i+1 +h.c.)+ i 1 n i 2 + V 1 1 n i n i Bera, Schomerus, Heidrich-Meisner, JHB PRL 2015

13 Many-body localization as eigenstate quantum phase transition Volume law Area law Occupation discontinuity Basko, Aleiner and Altshuler Ann. Phys. 2006; Oganesyan and Huse PRB 2007;

14 Entanglement entropy A B Eigenstate thermalization hypothesis 14 Deutsch PRA 1991, Srednicki PRE 1994 Popescu et al. Nature 2006, Rigol et al. Nature 2008

15 Many-body localization as eigenstate quantum phase transition Volume law Area law Occupation discontinuity Thermal Breakdown of ETH & statistical physics Integrability Basko, Aleiner and Altshuler Ann. Phys. 2006; Oganesyan and Huse PRB 2007;

16 16 Global quench from a charge density wave Anderson insulator

17 Entanglement vs. time starting from a product state J z /J ,L= 20 S S J z t Jt 17 JHB, Pollmann and Moore PRL 2012

18 Quasiparticle interaction leads to dephasing in dynamics Serbyn, Papic, Abanin PRL 2013

19 Entanglement vs. time starting from a product state J z /J with interactions 0.1,L= 20 S no interactions 0.1 S J z t Jt 0 19 JHB, Pollmann and Moore PRL 2012

20 Mutual information after quench from product state 20 De Tomasi, Bera, JHB, Pollmann PRL 2017

21 Charge density wave provides partial occupation of quasiparticles Lezama, Bera, Schomerus, Heidrich-Meisner, JHB arxiv:

22 Initial correlations in CDW state destroyed by dephasing (a) hn (t)i Vt Lezama, Bera, Schomerus, Heidrich-Meisner, JHB arxiv:

23 Long time occupations nonthermal with smeared discontinuity (a) hn i, hn diag i L h ni L W =8,V= ( +0.5) L Lezama, Bera, Schomerus, Heidrich-Meisner, JHB arxiv:

24 Many-body localization may be experimentally realized in cold atoms in a quasiperiodic lattice initial state Aubry-André model e o e o e o e o 0.8, U/J =4.7(1) U/J =10.3(1) /J =8 Imbalance /J =3 0 /J = Time (τ) Schreiber et al, Science 2015; see also Choi et al Science 2016 for 2D

25 Occupation imbalance similar to density imbalance, but with slower relaxation towards steady state 1.0 hiq(t)i, hi(t)i L Vt Lezama, Bera, Schomerus, Heidrich-Meisner, JHB arxiv:

26 Collaborators Talía Lezama Dresden Giuseppe De Tomasi Dresden Soumya Bera Dresden Frank Pollmann München Henning Schomerus Lancaster Fabian Heidrich-Meisner München

27 Summary JHB, Pollmann, Moore PRL 2012; Kjäll, JHB, Pollmann PRL 2014 Bera, Schomerus, Heidrich- Meisner, JHB PRL 2015; De Tomasi, Bera, JHB, Pollmann PRL 2017 Bera, Martynec, Schomerus, Heidrich-Meisner, JHB Ann. Phys. 2017; Lezama, Bera, Schomerus, Heidrich-Meisner, JHB arxiv: