Exclusion processes and quantum phase transitions in XXZ spin - PowerPoint PPT Presentation

Exclusion processes and quantum phase transitions in XXZ spin chains. 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) – Trieste 1 / 25 Marc Cheneau 1 Juan P. Garrahan 2 , Frédéric van Wijland 3 Cécile Appert-Rolland 4 , Bernard Derrida 5 , Frédéric van Wijland 3 1 Institut d’Optique, Palaiseau 2 Nottingham University 3 MSC, Paris 4 LPT, Orsay 5 LPS, ENS, Paris 15 th July 2014

Introduction Perspectives opened / questions raised 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) (I will ask questions to you .) large/small scale spectrum fjnite-size efgects phases across a Quantum Phase Transition Motivations regimes of large deviations of dynamical observables Use: dictionnary between (Well known at least in the stat. mech. community.) Correspondence Classical and quantum systems 2 / 25 · evolution operator for stochastic classical system [particles hopping] · Hamiltonian of quantum XXZ chain

Introduction Perspectives opened / questions raised 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) (I will ask questions to you .) large/small scale spectrum fjnite-size efgects 2 / 25 Motivations Use: dictionnary between (Well known at least in the stat. mech. community.) Correspondence Classical and quantum systems · evolution operator for stochastic classical system [particles hopping] · Hamiltonian of quantum XXZ chain · regimes of large deviations of dynamical observables · phases across a Quantum Phase Transition

Introduction Motivations 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) (I will ask questions to you .) Perspectives opened / questions raised 2 / 25 Use: dictionnary between (Well known at least in the stat. mech. community.) Correspondence Classical and quantum systems · evolution operator for stochastic classical system [particles hopping] · Hamiltonian of quantum XXZ chain · regimes of large deviations of dynamical observables · phases across a Quantum Phase Transition · fjnite-size efgects · large/small scale spectrum

Exclusion Process A Large deviation function of time-integrated observables A A total current Q on time window t # jumps jumps total activity K on time window System t # jumps jumps Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 i 3 / 25 b b b b b b b b b b b Exclusion Processes – generic settings maximal occupation N 1 1 1 1 1 1 α β ρ 0 ρ 1 L 1 γ δ Confjgurations: occupation numbers { n i } Exclusion rule: 0 ≤ n i ≤ N Markov evolution for the probability P ( { n i } , t ) ∑ [ ] W ( n ′ i → n i ) P ( { n ′ i } , t ) − W ( n i → n ′ ∂ t P ( { n i } , t ) = i ) P ( { n i } , t ) n ′ ⟨ e − sA ⟩ ∼ e t ψ ( s ) ( ⇔ determining P ( A , t ) )

Exclusion Process A Large deviation function of time-integrated observables A A total current Q on time window t # jumps jumps total activity K on time window System t # jumps jumps Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 i 3 / 25 b b b b b b b b b b b Exclusion Processes – generic settings maximal occupation N 1 1 1 1 1 1 α β ρ 0 ρ 1 L 1 γ δ Confjgurations: occupation numbers { n i } Exclusion rule: 0 ≤ n i ≤ N Markov evolution for the probability P ( { n i } , t ) ∑ [ ] W ( n ′ i → n i ) P ( { n ′ i } , t ) − W ( n i → n ′ ∂ t P ( { n i } , t ) = i ) P ( { n i } , t ) n ′ ⟨ e − sA ⟩ ∼ e t ψ ( s ) ( ⇔ determining P ( A , t ) )

Exclusion Process A Large deviation function of time-integrated observables A A total current Q on time window t # jumps jumps total activity K on time window System t # jumps jumps Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 i 3 / 25 b b b b b b b b b b b Exclusion Processes – generic settings maximal occupation N 1 1 1 1 1 1 α β ρ 0 ρ 1 L 1 γ δ Confjgurations: occupation numbers { n i } Exclusion rule: 0 ≤ n i ≤ N Markov evolution for the probability P ( { n i } , t ) ∑ [ ] W ( n ′ i → n i ) P ( { n ′ i } , t ) − W ( n i → n ′ ∂ t P ( { n i } , t ) = i ) P ( { n i } , t ) n ′ ⟨ e − sA ⟩ ∼ e t ψ ( s ) ( ⇔ determining P ( A , t ) )

Exclusion Process b 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) umps umps Large deviation function of time-integrated observables A i System b b 3 / 25 b b Exclusion Processes – generic settings b b b b b b maximal occupation N 1 1 1 1 1 1 α β ρ 0 ρ 1 L 1 γ δ Confjgurations: occupation numbers { n i } Exclusion rule: 0 ≤ n i ≤ N Markov evolution for the probability P ( { n i } , t ) ∑ [ ] W ( n ′ i → n i ) P ( { n ′ i } , t ) − W ( n i → n ′ ∂ t P ( { n i } , t ) = i ) P ( { n i } , t ) n ′ ⟨ e − sA ⟩ ∼ e t ψ ( s ) ( ⇔ determining P ( A , t ) ) = # j − − → umps − j ← − − A = total current Q on time window [0 , t ] = # j − − → umps + j ← − − A = total activity K on time window [0 , t ]

Exclusion Process b 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) XXX spin chain Hamiltonian (up to boundary terms and constants). n L System instead of the wave function but eq. for the probability similar to Schrödinger eq. Evolution of probability vector P : b 4 / 25 b b b b [Schütz & Sandow PRE 49 2726] Operator representation b b b b b maximal occupation N 1 1 1 1 1 1 α β ρ 0 ρ 1 L 1 γ δ ∂ t P = W P ∑ [ ] σ + k σ − k +1 + σ − k σ + W = k +1 − ˆ n k (1 − ˆ n k +1 ) − ˆ n k +1 (1 − ˆ n k ) 1 ≤ k ≤ L − 1 [ ] [ ] σ + σ − + α 1 − (1 − ˆ n 1 ) + γ 1 − ˆ n 1 [ ] [ ] σ + σ − + δ L − (1 − ˆ n L ) + β L − ˆ σ ± = σ x ± i σ − and σ z = ˆ n − N 2 are spin operators (with j = N 2 )

Exclusion Process b 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) XXX spin chain Hamiltonian (up to boundary terms and constants). n L System instead of the wave function but eq. for the probability similar to Schrödinger eq. Evolution of probability vector P : b 4 / 25 b b b b [Schütz & Sandow PRE 49 2726] Operator representation b b b b b maximal occupation N 1 1 1 1 1 1 α β ρ 0 ρ 1 L 1 γ δ ∂ t P = W P ∑ [ ] σ + k σ − k +1 + σ − k σ + W = k +1 − ˆ n k (1 − ˆ n k +1 ) − ˆ n k +1 (1 − ˆ n k ) 1 ≤ k ≤ L − 1 [ ] [ ] σ + σ − + α 1 − (1 − ˆ n 1 ) + γ 1 − ˆ n 1 [ ] [ ] σ + σ − + δ L − (1 − ˆ n L ) + β L − ˆ σ ± = σ x ± i σ − and σ z = ˆ n − N 2 are spin operators (with j = N 2 )

Exclusion Process b 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) XXZ spin chain Hamiltonian n L Large deviations with b b 5 / 25 b b b b Operator representation for large deviations b b b b maximal occupation N 1 1 1 1 1 1 α β ρ 0 ρ 1 L 1 γ δ ⟨ e − sK ⟩ ∼ e t ψ ( s ) ψ ( s ) = max Sp W s ∑ [ ] e − s σ + k σ − k +1 + e − s σ − k σ + k +1 − ˆ n k (1 − ˆ n k +1 ) − ˆ n k +1 (1 − ˆ W s = n k ) 1 ≤ k ≤ L − 1 [ ] [ ] e − s σ + e − s σ − 1 − (1 − ˆ 1 − ˆ + α n 1 ) + γ n 1 [ ] [ ] e − s σ + e − s σ − + δ L − (1 − ˆ n L ) + β L − ˆ

Exclusion Process Periodic Boundary Conditions Focus on a simple situation Periodic boundary conditions Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 6 / 25 Simple exclusion process (SSEP): maximal occupation N = 1 density: ρ 0 = N 0 / L Fixed total particle number N 0 1 1 1 1 1

Exclusion Process Periodic Boundary Conditions 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) Ring geometry 6 / 25 Periodic boundary conditions Focus on a simple situation Simple exclusion process (SSEP): maximal occupation N = 1 density: ρ 0 = N 0 / L Fixed total particle number N 0 1 1 1 1 1 . . . 2 1 L ≡ 0 . . .

Exclusion Process k e s L k x k x k y y Periodic Boundary Conditions k z k z k with e s Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 L 6 / 25 Periodic boundary conditions Focus on a simple situation Simple exclusion process (SSEP): maximal occupation N = 1 density: ρ 0 = N 0 / L Fixed total particle number N 0 1 1 1 1 1 L − 1 [ ] ∑ e − s ( ) σ + k σ − k +1 + σ − k σ + W s = − ˆ n k (1 − ˆ n k +1 ) − (1 − ˆ n k )ˆ n k +1 k +1 k =1

Exclusion Process Periodic Boundary Conditions 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) with 6 / 25 Focus on a simple situation Periodic boundary conditions Simple exclusion process (SSEP): maximal occupation N = 1 density: ρ 0 = N 0 / L Fixed total particle number N 0 1 1 1 1 1 L − 1 [ ] ∑ e − s ( ) σ + k σ − k +1 + σ − k σ + W s = − ˆ n k (1 − ˆ n k +1 ) − (1 − ˆ n k )ˆ n k +1 k +1 k =1 − e − s = L − 1 2 H ∆ 2 L − 1 ∑ [ ] H ∆ = − σ x k σ x k +1 + σ y k σ y k +1 + ∆ σ z k σ z ∆ = e s k +1 k =1