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

Exclusion processes and quantum phase transitions in XXZ spin chains. 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) – London 1 / 29 Marc Cheneau 1 Juan P. Garrahan 2 , Frédéric van Wijland 3 Cécile Appert-Rolland 4 , Bernard Derrida 5 , Alberto Imparato 6 1 Institut d’Optique, Palaiseau 2 Nottingham University 3 MSC, Paris 4 LPT, Orsay 5 LPS, ENS, Paris 6 Aarhus University 18 th December 2014

Introduction Perspectives opened ; questions raised 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) (I will ask questions to you .) import/export techniques from/to stat. mech. large/small scale spectrum fjnite-size efgects 2 / 29 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 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) (I will ask questions to you .) Perspectives opened ; questions raised 2 / 29 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 · import/export techniques from/to stat. mech.

Exclusion Process b 18/12/2014 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 / 29 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 18/12/2014 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 / 29 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 18/12/2014 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 / 29 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 18/12/2014 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 / 29 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 18/12/2014 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 / 29 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 Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 18/12/2014 6 / 29 Example 1 : exclusion process on a ring

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 18/12/2014 7 / 29 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 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) Ring geometry 7 / 29 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 Periodic Boundary Conditions 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) with 7 / 29 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

Exclusion Process Periodic Boundary Conditions 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) 8 / 29 SSEP Quantum Spin Chain Classical/Quantum dictionnary local occupation number n k ( 1 ≤ k ≤ L ) local spin σ z k ( 1 ≤ k ≤ L ) n k = 0 , 1 ≡ ◦ , • σ z k = 1 , − 1 ≡ ↑ , ↓ (fjxed) total occupation N 0 ≡ ρ 0 L (fjxed) total magnetization M ≡ m 0 L (mesoscopic) density ρ ( x ) ( 0 ≤ x ≤ 1 ) (mesoscopic) magnet. m ( x ) ( 0 ≤ x ≤ 1 )

Exclusion Process evolution operator 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) ground state energy cumulant generating function Periodic Boundary Conditions J xy 9 / 29 Quantum Spin Chain SSEP Classical/Quantum dictionnary local occupation number n k ( 1 ≤ k ≤ L ) local spin σ z k ( 1 ≤ k ≤ L ) n k = 0 , 1 ≡ ◦ , • σ z k = 1 , − 1 ≡ ↑ , ↓ (fjxed) total occupation N 0 ≡ ρ 0 L (fjxed) total magnetization M ≡ m 0 L (mesoscopic) density ρ ( x ) ( 0 ≤ x ≤ 1 ) (mesoscopic) magnet. m ( x ) ( 0 ≤ x ≤ 1 ) ferromagnetic XXZ Hamiltonian ( J xy = − 1 ) L − 1 W s = L − 1 − e − s [ ] ∑ ( ) 2 H ∆ H ∆ = σ x k σ x k +1 + σ y k σ y + J z σ z k σ z k +1 k +1 2 k =1 L − 1 [ ] ∑ = − σ x k σ x k +1 + σ y k σ y k +1 + ∆ σ z k σ z k +1 k =1 counting factor ∆ = e s of the activity K anisotropy ∆ = − J z along direction Z ψ ( s ) = max Sp W s = L − 1 − e − s 2 E L ( s ) E L ( s ) = min Sp H ∆ 2

Exclusion Process Microscopic solution Bethe Ansatz [Appert, Derrida, VL, van Wijland, PRE 78 021122] Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 18/12/2014 10 / 29

Exclusion Process eigenvalue 18/12/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA – Paris VI-VII) Bethe equations Microscopic solution 10 / 29 eigenvector of components Coordinate Bethe Ansatz: Bethe Ansatz [Appert, Derrida, VL, van Wijland, PRE 78 021122] Integrability known from long ; diffjculty: L → ∞ N 0 ∑ ∏ [ ] x i A ( P ) ζ P ( i ) P i =1 [ 1 ] + . . . + 1 ψ ( s ) = − 2 N 0 + e − s [ ] − e − s ζ 1 + . . . + ζ N 0 ζ 1 ζ N 0 [ ] N 0 − 1 − 2 e s ζ i + ζ i ζ j ∏ ζ L i = 1 − 2 e s ζ j + ζ i ζ j j =1 j ̸ = i