Quantum Kibble-Zurek mechanism: scaling hypothesis in the Ising - PowerPoint PPT Presentation

Quantum Kibble-Zurek mechanism: 
 scaling hypothesis 
 in the Ising and Bose-Hubbard models Anna Francuz & Jacek Dziarmaga @ Jagiellonian U. Bartek Gardas @ U. of Silesia & Los Alamos Wojciech H. Ż urek @ Los Alamos coming soon review on quantum KZM: JD, Adv. in Phys. 59, 1063 (2010)

Quantum Ising Chain N ( ) z x x H g ∑ = − σ + σ σ n n n 1 + n 1 = Strong transverse field g>>1 ... →→→→→ Energy gap 0 Δ → Quantum phase transition at g=1 Correlatio n length ξ → ∞ Ferromagnetic states at g=0 ... or ... ↑↑↑↑↑↑↑↑ ↓↓↓↓↓↓↓↓

Ideal Adiabatic Quantum State Preparation (or Adiabatic Quantum Computation) Simple H i Adiabatic H f Interesting

Real Adiabatic Quantum State Preparation Simple H i Simple Interestin g ≠ BEC ⇓ Quantum Phase Transition Non- adiabatic Mott H f Interesting

Quantum Ising Chain N ( ) z x x H g ∑ = − σ + σ σ n n n 1 + n 1 = ``Simple’’ ... →→→→→ Adiabatic Non- adiabatic ... ↑↑↑↑↑↑↑↑ or ... ↑↑↑↑↑↓↓↓↓↓ ↓↑↑↑↑↓↓↓↓↓ ↑↑ ... Excited ↓↓↓↓↓↓↓↓ ˆ = ? ξ ``Interesting’’

Quantum Kibble-Zurek mechanism (KZM) distance from the critical point g − g c ε = g ˆ ˆ c ξ ξ z ν Non-adiabatic Δ ∝ ε energy gap Adiabatic Adiabatic (impulse) − ν ξ ∝ ε correlation length linear(ized) quench ˆ ˆ GS GS − ( t ) GS − ( t ) t t ε = ˆ ˆ 0 t t − + τ Q transition rate ˆ d / dt 1 RATE GAP at t t ε = = − = t ε ˆ z ˆ ν ν where t and ∝ τ 1 z ξ ∝ τ 1 z + ν + ν Q Q Quantum Ising chain ... ↑↑↑↑↑↓↓↓↓↓ ↓↑↑↑↑↓↓↓↓↓ ↑↑ ˆ ˆ and t ξ ∝ τ ∝ τ Final excited state Q Q ˆ ξ

K-Z scaling hypothesis rescaled time rescaled distance ˆ t ˆ / t x / ξ scaling dimension ˆ ˆ ˆ ˆ ( t ) O ( x ) ( t ) F ( x / , t / t ) − Δ ψ ψ = ξ O ξ O scaling function

Jordan-Wigner transformation x z σ , c σ → n n n quadratic correlator c c + α = R n R n + before: at: after: ˆ ˆ ˆ 1 F ( R / , t / t ) − α = ξ ξ R α ˆ ˆ , t ξ = τ = τ Q Q

Jordan-Wigner transformation x z σ , c σ → anomalous correlator n n n c + c β = R n R n at after before SAME ˆ 1 ˆ ˆ F ( R / , t / t ) ξ β = ξ R ˆ β ˆ , t ξ = τ = τ Q Q

ferromagnetic correlator x x x x C xx R ( ) = σ σ − σ σ n R n n R n + + before: at: after: SAME ˆ ˆ ˆ 1 / 4 C ( R ) F ( R / , t / t ) ξ = ξ xx xx ˆ ˆ , t ξ = τ = τ see also M. Kolodrubetz Q Q

more correlation functions a b a b C ab R ( ) = σ σ − σ σ n R n n R n + + ˆ ˆ ˆ 9 / 4 C ( R ) F ( R / , t / t ) ξ = ξ yy yy ˆ ˆ SAME , t ξ = τ = τ ˆ ˆ Q Q ˆ 5 / 4 C ( R ) F ( R / , t / t ) ξ = ξ mutual information xy xy & ˆ 1 ˆ ˆ C ( R ) F ( R / , t / t ) ξ = ξ quantum discord zz zz

entanglement entropy: block of L spins S Tr log = − ρ ρ L L L before: at: after: S ( t ) ˆ ˆ SAME L F ( L / , t / t ) ξ = ξ ˆ S ˆ S ( t / t ) log 1 + 0 3 ˆ ˆ , t ξ = τ = τ Q Q

entanglement gap: block of L spins Δ λ = λ − λ 1 2 before: at: after: SAME ˆ ˆ ˆ 1 / 8 F ( L / , t / t ) ξ Δ λ = ξ Δ λ ˆ ˆ , t ξ = τ = τ Q Q

1D Bose-Hubbard model: Mott -> superfluid transition 1 M M ( ) H J a a a a a a a a + + + + ∑ ∑ = − + + s s 1 s 1 s s s s s + + 2 s 1 s 1 = = ....... L sites MPS J J 0 at t 0 = = t J ( t ) J = cr τ Q ( 0 ) 1 , 1 , 1 , 1 , 1 ,...., 1 , 1 ψ = t

1D Bose-Hubbard model: correlation function C a a + = R s R s + ˆ t / = t 0 ˆ 1 / 4 L 50 C = ξ R L 100 is coming soon = ˆ R / ξ Kosterlitz-Thouless previous simulations & an experiment ˆ 0 . 7 10 ξ ∝ τ τ << Q Q

HOMEWORK: 1) For theorists: ˆ ˆ ˆ 1 / 4 C ( R ) F ( R / , t / t ) ξ = ξ xx xx renormalization group in both space and time KZ scaling hypothesis 2) For experimentalists: CO 2 CO 2 CO 2 quantum simulator classical supercomputer