Dynamics of Entanglement in the Heisenberg Model Simone Montangero, Gabriele De Chiara, Davide Rossini, Matteo Rizzi, Rosario Fazio Scuola Normale Superiore Pisa
Outline Ground state entanglement in Spin Chains Dynamics of entanglement in the Ising model Numerical method: DMRG, t-dmrg Entanglement in critical Heisenberg model Entanglement dynamics in the Heisenberg model I.Latorre, E.Rico, G.Vidal, Quant. Inf. and Comp. 4, (2004) P. Calabrese, J. Cardy, JSTAT 0504 (2005) WORK IN PROGRESS
Entropy of Entanglement Ground State: Ψ GS ρ L = tr N L ( Ψ GS Ψ GS ) S(ρ L ) tr(ρ L logρ L ) L sites B A N sites
Heisenberg Model H = i J i ( σ x i σ x i+1 + σ y i σy i+1 + σz i σ z i+1 ) J i = J constant couplings anisotropy [ 1 : 1] Critical J i [0, J] Random HM
Entropy Scaling I N=18 < 1 1.8 = 2 S L 1.6 1.4 =2.5 1.2 1 2 4 6 8 L Bethe Ansatz, integration of non-linear equations I.Latorre, E.Rico, G.Vidal, Quant. Inf. and Comp. 4, (2004)
Ising Model H = i ( σ x i σ x i+1 + λσ z i ) For λ = 1 the system is critical Correlations diverge in the system ground state System can be solved via Jordan-Wigner + Fourier + Bogoliubov transformation
Entropy Scaling II Critical 1.8 1.6 λ 1 S L 1.4 1.2 1 0.8 0 50 100 150 200 L S L = c 3 log 2L + a central charge c=1/2 I.Latorre, E.Rico, G.Vidal, Quant. Inf. and Comp. 4, (2004)
Evolution of Entanglement I Instantaneous Quench λ 0 = λ 1 t=0 GS: separable state vt = L/2 P. Calabrese, J. Cardy, JSTAT 0504 (2005)
Evolution of Entanglement II Instantaneous Quench λ 0 λ 1 = 1 P. Calabrese, J. Cardy, JSTAT 0504 (2005)
Physical Interpretation Simple Model λ λ 0 1 S L t S L L t < t t > t P. Calabrese, J. Cardy, JSTAT 0504 (2005)
Half Time Summary Static: critical scaling Dynamic: S L c 3 log 2L Entropy increase is proportional to quench Entropy saturates at t t depends on L and velocity
Numerical Simulation DMRG, White PRA (1992) t-dmrg, White, Feigun, PRL (2004) Approximate method to study many-body quantum system (ground state properties, time evolution) Open boundary conditions Finite size scaling
H SB = H E + H E + H int DMRG scheme ψ G = ψ αabβ H B (2) = O 1 2 H E O 2 1 ρ L = T r R Ψ G Ψ G m states
t-dmrg scheme Time evolution operator Trotter expansion H = even F i,i+1 + odd G i,i+1 exp( ıht) = (e ıf dt/2 e ıgdt e ıf dt/2) = exp( ıfi,i+1 dt/2) exp( ıg i,i+1 dt) exp( ıf i,i+1 dt/2) F, G even/odd Hamiltonan operator Ψ = O l l+1 O N l 3 N l 2 Ψ
DMRG Parameters N sistem size m size of truncated basis P discarded dt Trotter approx (second order).
Entropy and CFT Entropy of a spin block in a critical infinite chain: S L c 3 log 2L Entropy of a block L in a critical chain of size N SL B = c [ ( )] L πl 6 log 2 π sin + a N N sites L sites B P. Calabrese, J. Cardy, JSTAT 1 (2004)
Numerical Results N=200 Critical Static scaling - critical, non critical, analytical Non-critical prediction, finite size effects. N=1000.
Central Charge N=1000 Numerical evaluated central charge. N=1000, convergenza.
Finite Fixed Quench N=50 L=20 m=50 dt=10^-3 0 1 = 1 0 1 = 1.5 v( ) = 2Jπ sinθ θ cosθ = 1 Eggert et.al. PRL 73 (1994)
Finite Quench N=50 L=20 m=50 dt=10^-3 0 = 1.5 Δ 1 0 vt = L
Random Heisenberg Model Nr=1200 m=50 c=ln 2 c=1 CFT: G. Refael, J.E.Moore, PRL 93, 260602 (2004) XX Model: N.Laflorencie, cond-mat/0504446
Time Evolution in RHM N=50 L=20 m=50 dt=10^-3 Nr=250
Conclusions and Outlook Static scaling in HM confirmed. Central charge can be fitted from numerical simulations. Time evolution scheme holds in different models. Central charge in random Heisenberg model Time evolution in RH under investigation