The adaptive time-dependent DMRG applied to transport and non-equilibrium phenomena

Transcription

1 The adaptive time-dependent DMRG applied to transport and non-equilibrium phenomena Diffusive spin dynamics in spin ladders Langer, HM, Gemmer, McCulloch, Schollwöck PRB 2009

2 Outline: (t-dep.) DMRG and applications quasi-1d materials 3)diffusive & ballistic dynamics optical lattices Potok et al. Nature 2006 nano-structures w/ interactions 2)finite-bias transport Bloch, Dalibard, Zwerger RMP )Non equilibrium: sudden expansion 1) DMRG: ground-state, dynamics,..., time-dependence in 1D (lattice) models White PRL 1992 & PRB 1993 Schollwöck RMP 2005

3 1) Ground-state DMRG full system: system + environment = super-block System Environment xm Hm { } L 1 Henv, L 1 { j } Suppose you can keep only m states α> to represent m SB ~ SB = 2 minimize SB m SB m, j=1 a, j j by varying:a, j, optimal choice : =m first eigenvectorsof reduced density matrix m m =tr env [ SB SB ] White PRL 1992 & PRB 1993

4 Ground-state DMRG: Algorithm Environment System xm Hm L xm Hm L 1 Build super-block Hamiltonian HSB (system+2 sites+environment) Diagonalize H SB (Lanczos) SB Build reduced density matrix Henv, L m L 1=tr env [ m ] SB SB Diagonalize L 1 Transform H into eigen-basis of L+1 L 1 and truncate White PRL 1992 & PRB 1993 Example: spin-1/2 chain S H= S i i i 1 single site dimension d=2

5 ... DMRG and Matrix Product States (MPS)... A, [si ]: H m x m H d x d H mxm xm Hm L site i tensor product { i 1 } { si } x dm Hdm L 1 projection onto eigenbasis of ρ i = s 1 { i } { i 1 si } DMRG generates transformations A xm Hm L 1, [si ]:MPS, A [s ]... A [s ] si,si 1... si 0 i 1,..., s, i Östlund PRL 1995, PRB 1997 DMRG: variational method on MPS Rommer, McCulloch JstatMech 2007 TEBD: apply exp(-βh) to MPS Vidal PRL 2004

6 Why it works in 1D... area law SL~Ld-1 Bipartition H= i S S h.c. /2 i i 1 z S z S i i 1 A (length L) B Von Neumann entropy S A,L = tr [ A log2 A ] Gapped systems S A,L ~const Kaulke, Peschel EPJB 1998 & LNP 1999 Control parameter: #states kept m m discarded weight =1 i wi Critical systems S A,L ~k log2 L const Latorre, Rico, Vidal Quant. Inf Comp 2004 Area laws, Eisert, Cramer, Plenio RMP 2009

7 Adaptive time-dependent DMRG Environment System HL Henv,L ' Hij Build super-block Hamiltonian HSB (system+2 sites+environment) using a Trotter-Suzuki SB =exp ihij SB breakup of exp(-ihτ) (other schemes possible!) Build reduced density matrix L 1 =tr env [ SB SB ] Diagonalize L 1 Rotate and truncate H L+1 in eigen-basis of L 1 White, Feiguin PRL 2004 Vidal PRL 2004 Daley, Kollath, Schollwöck, Vidal JSTatM 2004

8 The adaptive time-dependent DMRG Find ground-state with DMRG for H0 in a truncated but optimized basis H0 H at time τ=0+ (quantum quench) Perform time-evolution under dynamics of H Measure observables, e.g. Szi White, Feiguin PRL 2004 Daley et al. JStatMech 2004 Vidal PRL 2004 Gobert et al PRE 2005 Control parameters: time step δτ, discarded weight δρ But: entanglement growth

9 Entanglement growth: global vs local quenches L 1 H= i=1 c c h.c. i Ui ni, ni, i, i 1, Global quench: Ui U'i vs. local quench: Ui=i' U'i=i' De Chiara et al. JstatM 2006, Calabrese, Cardy JStatM 2007, JstatM 2007a Cazalilla PRL 2006,...

10 2) Non-equilibrium transport in the single-impurity Anderson model Kondo effect in q-dots van der Wiel et al. Science 2000 Goldhaber-Gordon et al. Nature 1998 Grobis et al PRL 2008 :Computational methods: t-dependent NRG (Anders PRL 2008) Real-time QMC (Werner, Oka, Millis PRB 2009; Schiro & Fabrizio PRB 2009) Iterative real-time path integral approach (Weiss, Eckel, Thorwart, Egger PRB 2008)...

11 Transport through a q-dot: Model and set-up =2t '2 Single-impurity Anderson model H=Hqdot Hhy Hleads Hqdot =Unqdot, n qdot, Unqdot /2 Hhy = t ' c qdot, c qdot±1, h.c. Hleads= i t i c i, c i 1, h.c. SIAM: Al-Hassanieh et al. PRB 2006, Dias da Silva, HM et al PRB 2008, Kirino et al PRB 2008 IRLM: Boulat, Saleur, Schmitteckert PRL 2008, Schneider, Schmitteckert 2006

12 Current vs time at particle-hole symmetry <ndot>=1 (Quasi-)steady state reachable whenever τtrans< τrevival τtransdecreases as bias V increases! HM, Feiguin, Dagotto PRB 2009

13 IV characteristics at particle-hole symmetry Very good agreement with frg double-minimum doesn't exist (as expected) HM, Feiguin, Dagotto PRB 2009; frg: Jakobs, Pletyukhov, Schoeller th order: Fujii, Ueda PRB 2003

14 3) Ballistic & diffusive dynamics in spin systems spin-½ moments on Cu sites Large magnon thermal conductivity Hess, HM et al. PRB 2001 Sologubenko et al PRL 2000 Theory: reviews Zotos, Prelovsek 2003, HM, Honecker, Brenig EPJST 2007

15 Real-time simulations of ballistic and diffusive spin dynamics in chains and ladders c-axis: ladders! Our agenda : Prepare inhomogeneous initial states Compute the spatial variance of spin and energy density Spin dynamics, T=0: non-equilibrium Spin dynamics, T>0 Heat dynamics in chains & ladders top view: ladder plane Otter et al JMMM 2009, thermal transport: Hess, HM et al PRB 2001

16 Spin transport and dynamics Spin-1/2 Heisenberg chain: Linear response: =Ds regular Real-time simulations (T=0) 2 Bi=B0 exp i i0 2 / B 2 M t ~ i Siz t 1/2 i i0 2 Shastry & Sutherland PRL M t ~t 2 :ballistic 2 M t ~t : diffusive

17 Spin-½ chain with anisotropic interactions T=0 real-time DMRG data: ballistic - diffusive Langer, HM,Gemmer,McCulloch, Schollwöck PRB 2009

18 Diffusive transport in spin ladders at T=0 variance ~ t at long times: diffusive Langer, HM,Gemmer,McCulloch, Schollwöck PRB 2009

19 4) The sudden expansion of interacting fermions in 1D optical lattices c i, c i 1, h.c. i H= J U i ni, ni, HM, Rigol, Muramatsu, Feiguin, Dagotto PRA 2008 & w/ Manmana arxiv: D: Rosch et al PRL 2009 Experiments: Bloch's group

20 Initial states with many double occupancies: Dynamical formation of Fock states... Initial condition: 1<n<2 and U>bandwidth Radius of doublons R d~ i i2 di di =ni, ni,

21 ... and reduction of entanglement entropy HM, Manmana, Rigol, Muramatsu, Feiguin, Dagotto arxiv:

22 ... and reduction of entanglement entropy Formation of metastable Fock state HM, Manmana, Rigol, Muramatsu, Feiguin, Dagotto arxiv:

23 Summary DMRG method Applications of DMRG Algorithm, DMRG & MPS Non-equilibrium transport in the SIAM Entanglement entropy Adaptive-time dependent DMRG Ballistic vs diffusive spin dynamics in chains & ladders: non-equilibrium Outlook: Real-time evolution at finite T Sudden expansion of fermions in optical lattices Barthel, Schollwöck, White PRB 2009 Sirker, Klümper PRB 2005 Feiguin, White PRB 2005 Verstraete, Garcia-Ripoll, Cirac PRL 2004 Zwolak, Vidal PRL 2004 Ladders: diffusive

24 Thanks: Adrian Feiguin (U Wyoming) Stephan Langer (RWTH Aachen) Elbio Dagotto (U Tennessee & ORNL) Jochen Gemmer (U Osnabrück) Marcos Rigol (U Georgetown) Ian McCulloch (U of Queensland) Alejandro Muramatsu (IFW Dresden) Uli Schollwöck (LMU München) Salvatore Manmana (Lausanne) FOR 912

25 ... and why 2D is tough Area laws in d dimensions Eisert, Cramer, Plenio RMP 2009 S A,L ~L d 1 mind criticalsystems, though 2D systems encode more quantum information Corrections to the area law? Ansatz states ensuring manageable entanglement scaling: generalized MPS Tensor networks PEPS Verstraete, Cirac 2004 MERA Vidal PRL N Heisenberg ladders Kallin et al. 2009