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Entanglement and dynamics in many-body localized systems Frank Pollmann! Max-Planck-Institut für Physik komplexer Systeme, Dresden, Germany! Together with: Jonas Kjäll! ! Kjäll, Bárðarson, FP, PRL 113, 107204 (2014) Singh, Bárðarson, FP (in progress) Jens Bárðarson! Rajeev Singh! TENSOR NETWORK STATES: ALGORITHMS AND APPLICATIONS Beijing, Dec. 2 2014! Many-body localization t t Anderson (1958) Basko, Aleiner, Altshuler (2006)! Oganesyan and Huse (2007) ! Pal and Huse (2010)! Bauer and Nayak (2013) … Many-body localization t µ t Anderson (1958) Basko, Aleiner, Altshuler (2006)! Oganesyan and Huse (2007) ! Pal and Huse (2010)! Bauer and Nayak (2013) … Many-body localization | t µ ni = † † † † c↵ c c c |0i t Anderson (1958) Basko, Aleiner, Altshuler (2006)! Oganesyan and Huse (2007) ! Pal and Huse (2010)! Bauer and Nayak (2013) … Many-body localization Interactions | t µ ? ni = † † † † c↵ c c c |0i t Anderson (1958) Basko, Aleiner, Altshuler (2006)! Oganesyan and Huse (2007) ! Pal and Huse (2010)! Bauer and Nayak (2013) … Many-body localization ✏ > ✏0 Extended >0 MBL =0 Volume law Area law ETH ETH breaks down ⌘ (disorder) Transition is hard to detect! Band insulator! A ⇠ exp( B /kT ) Anderson (1958) Basko, Aleiner, Altshuler (2006)! Oganesyan and Huse (2007) ! Pal and Huse (2010)! Bauer and Nayak (2013) … Overview (1) Many-body localization (MBL) in a disordered quantum Ising chain Detection of the MBL transition! Localization protected quantum order - (2) Quenches in disordered systems - Efficient time evolution using MPS! Entanglement and bipartite fluctuations Extended Disordered quantum Ising chain • Model system: Z2 symmetric 1D quantum Ising model ( |h| ⌧ J = 1) X X X z z x z z H= (1 + Ji ) i i+1 + h i + J2 i i+2 i • • FM ground states: | Excitations are domain walls: h | i,! i hJ2 Interactions i i Localization-protected quantum order Huse, Nandkishore, Oganesyan, Pal, Sondhi (2013) Many-body localization transition • Localized and extended phase: AREA vs. VOLUME law (Exact Diagonalization) (Random state) ... ... A E=0 B S = ln 2 | """i ± | ###i Bauer and Nayak (2013) Page (1993) Many-body localization transition • Localized and extended phase: AREA vs. VOLUME law ➡ Variance of S diverges at the transition point and goes to zero for small/large disorder strength ✏=1 s s J = L(1 = [(L dL 1 )f [( J 2) ln 2 2 1] Jc )Lb ] f [( J Jc )Lb ] Many-body localization transition • Repeating the scaling for various energy densities yields the phase diagram Extended MBL Spin-glass transition Extended Disordered XXZ model • • Model system: Anisotropic Heisenberg S=1/2 chain! Equivalent to spinless fermions • } } } H = J y y x x z z z +J (Si Si+1 + Si Si+1 ) + S hi Si z i Si+1 i i i hopping random potential interaction with hi 2 [ , ] All single particle states localized for ⌘ 6= 0 Anderson (1958) Dynamics of entanglement entropy • Start from an unentangled product state ( S = 0 )! | ! • 0⇤ = | ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥⇤ Measure the entanglement after quench and the time itH evolution with U (t) = e using TEBD ![Vidal 2003] ! B A • Clean system:! Disordered system (?):! localization SS ? ⇠ Ssaturation tJ tJ?? tJtJ ?? x Lieb and Robinson (1972) P. Calabrese and J. Cardy (2006) 0.4 Jz /J? 0.0 0.01 0.1 0.2 Dynamics of entanglement entropy 0.1, L = 20 Time evolution of S in the disordered XXZ model S 0.2 Jz /J 0.0 0.01 0.1 0.3 0.14 /J? = 5 0.2 0.1, L = 20 S 0.4 0.1 0.2 Logarithmic growth 0.07 0.1 1 Rapid expansion! S 0.1 0 product state! 0 0.1 1 10 Jt 0.07 0 0.14 0 S • 0.3 0.1 1 10 100 Jz t 100 10 J? t 0.1 1 10 100 Jz t 100 1000 S ⇠ ⇠ ln(Jz t) Serbyn, Papic, Abanin (2013) R.Vosk & E. Altmann (2012) 1000 Bárðarson, FP, Moore, PRL 109, 017202 (2012)! M. Znidaric, T. Prosen, and P. Prelovsek (2008) Dynamics of entanglement entropy • Bipartite fluctuations of the XXZ chain z 2 F = h |(SA ) | i with z SA = PL/2 z S i=1 i z h |SA | i2 ... ... A B ⌘=1 ⌘ = 10 H. F. Song, S. Rachel, and K. Le Hur (2010) M. Endres et al (2011) Dynamics of the bipartite fluctuations Summary Extended Kjäll, Bárðarson, FP, PRL 113, 107204 (2014) Thank You! Singh, Bárðarson, FP (in progress)!