Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering - PowerPoint PPT Presentation

Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems Wojciech Dybalski 1 joint work with Sven Bachmann 2 and Pieter Naaijkens 3 1 Technical University of Munich 2 University of British Columbia 3 RWTH Aachen University ICMP, AHP Journal Prize session, 27/07/2018 Wojciech Dybalski Scattering theory for spin systems

Introduction Our observation: For a class of gapped quantum spin systems satisfying Lieb-Robinson bounds, admitting single-particle states Haag-Ruelle scattering theory can be developed in a natural, model independent manner. Wojciech Dybalski Scattering theory for spin systems

Introduction Our observation: For a class of gapped quantum spin systems satisfying Lieb-Robinson bounds, admitting single-particle states Haag-Ruelle scattering theory can be developed in a natural, model independent manner. Comparison with the literature 1 Haag-Ruelle scattering theory for Euclidean lattice quantum field theories. [Barata-Fredenhagen 91, Barata 91, 92, Auil-Barata 01,05] 2 Scattering theory for quantum spin systems relying on properties of concrete Hamiltonians. [Hepp 65, Graf-Schenker 97, Malyshev 78, Yarotsky 04... ] Wojciech Dybalski Scattering theory for spin systems

Introduction Our observation: For a class of gapped quantum spin systems satisfying Lieb-Robinson bounds, admitting single-particle states Haag-Ruelle scattering theory can be developed in a natural, model independent manner. Comparison with the literature 1 Haag-Ruelle scattering theory for Euclidean lattice quantum field theories. [Barata-Fredenhagen 91, Barata 91, 92, Auil-Barata 01,05] 2 Scattering theory for quantum spin systems relying on properties of concrete Hamiltonians. [Hepp 65, Graf-Schenker 97, Malyshev 78, Yarotsky 04... ] Wojciech Dybalski Scattering theory for spin systems

Outline Scattering in Quantum Mechanics 1 Scattering in QFT and spin systems 2 The problem of asymptotic completeness 3 Conclusions and outlook 4 Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 Hilbert space: H := L 2 ( R 3 , dx ) 2 Hamiltonian: H = − 1 2 ∆ + V ( x ) 3 Schrödinger equation: i ∂ t Ψ t = H Ψ t 4 Time evolution: Ψ t := e − i tH Ψ t = 0 Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 Hilbert space: H := L 2 ( R 3 , dx ) 2 Hamiltonian: H = − 1 2 ∆ + V ( x ) 3 Schrödinger equation: i ∂ t Ψ t = H Ψ t 4 Time evolution: Ψ t := e − i tH Ψ t = 0 y v V x v Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 Hilbert space: H := L 2 ( R 3 , dx ) 2 Hamiltonian: H = − 1 2 ∆ + V ( x ) 3 Schrödinger equation: i ∂ t Ψ t = H Ψ t 4 Time evolution: Ψ t := e − i tH Ψ t = 0 y v V x v Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 There are states Ψ out ∈ H of the particle in potential V which for large times evolve like states of the free particle. 2 For any such Ψ out there exists Ψ ∈ H s.t. 3 Def: Ψ out := lim t →∞ e i tH e − i tH 0 Ψ is the scattering state. 4 Def: W out := lim t →∞ e i tH e − i tH 0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 There are states Ψ out ∈ H of the particle in potential V which for large times evolve like states of the free particle. 2 For any such Ψ out there exists Ψ ∈ H s.t. t →∞ � e − i tH Ψ out − e − i tH 0 Ψ � = 0 lim 3 Def: Ψ out := lim t →∞ e i tH e − i tH 0 Ψ is the scattering state. 4 Def: W out := lim t →∞ e i tH e − i tH 0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 There are states Ψ out ∈ H of the particle in potential V which for large times evolve like states of the free particle. 2 For any such Ψ out there exists Ψ ∈ H s.t. t →∞ � Ψ out − e i tH e − i tH 0 Ψ � = 0 lim 3 Def: Ψ out := lim t →∞ e i tH e − i tH 0 Ψ is the scattering state. 4 Def: W out := lim t →∞ e i tH e − i tH 0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 There are states Ψ out ∈ H of the particle in potential V which for large times evolve like states of the free particle. 2 For any such Ψ out there exists Ψ ∈ H s.t. t →∞ � Ψ out − e i tH e − i tH 0 Ψ � = 0 lim 3 Def: Ψ out := lim t →∞ e i tH e − i tH 0 Ψ is the scattering state. 4 Def: W out := lim t →∞ e i tH e − i tH 0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Scattering in Quantum Mechanics 1 There are states Ψ out ∈ H of the particle in potential V which for large times evolve like states of the free particle. 2 For any such Ψ out there exists Ψ ∈ H s.t. t →∞ � Ψ out − e i tH e − i tH 0 Ψ � = 0 lim 3 Def: Ψ out := lim t →∞ e i tH e − i tH 0 Ψ is the scattering state. 4 Def: W out := lim t →∞ e i tH e − i tH 0 is the wave-operator. Wojciech Dybalski Scattering theory for spin systems

Cook’s method 1 Let Ψ t := e i tH e − i tH 0 Ψ . 2 Suppose we can show � ∂ t Ψ t � = � e i tH V e − i tH 0 Ψ � ∈ L 1 ( R , dt ) . � ∞ 3 Then lim t →∞ Ψ t = t 0 ( ∂ τ Ψ τ ) d τ + Ψ t 0 exists. Wojciech Dybalski Scattering theory for spin systems

Cook’s method 1 Let Ψ t := e i tH e − i tH 0 Ψ . 2 Suppose we can show � ∂ t Ψ t � = � e i tH V e − i tH 0 Ψ � ∈ L 1 ( R , dt ) . � ∞ 3 Then lim t →∞ Ψ t = t 0 ( ∂ τ Ψ τ ) d τ + Ψ t 0 exists. Wojciech Dybalski Scattering theory for spin systems

Cook’s method 1 Let Ψ t := e i tH e − i tH 0 Ψ . 2 Suppose we can show � ∂ t Ψ t � = � e i tH V e − i tH 0 Ψ � ∈ L 1 ( R , dt ) . � ∞ 3 Then lim t →∞ Ψ t = t 0 ( ∂ τ Ψ τ ) d τ + Ψ t 0 exists. Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems 1 Γ - the abelian group of space translations ( R d or Z d ). Γ - Pontryagin dual of Γ ( R d or S d 2 � 1 ). 3 ( A , τ ) - C ∗ -dynamical system with R × Γ ∋ ( t , x ) �→ τ ( t , x ) . 4 B ⊂ A -almost-local operators: � [ B 1 , τ ( s , vs ) ( B 2 )] � = O ( | s | −∞ ) . 5 A ⊂ B ( H ) and τ ( t , x ) ( A ) = U ( t , x ) AU ( t , x ) ∗ for A ∈ A . Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems 1 Γ - the abelian group of space translations ( R d or Z d ). Γ - Pontryagin dual of Γ ( R d or S d 2 � 1 ). 3 ( A , τ ) - C ∗ -dynamical system with R × Γ ∋ ( t , x ) �→ τ ( t , x ) . 4 B ⊂ A -almost-local operators: � [ B 1 , τ ( s , vs ) ( B 2 )] � = O ( | s | −∞ ) . 5 A ⊂ B ( H ) and τ ( t , x ) ( A ) = U ( t , x ) AU ( t , x ) ∗ for A ∈ A . Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems 1 Γ - the abelian group of space translations ( R d or Z d ). Γ - Pontryagin dual of Γ ( R d or S d 2 � 1 ). 3 ( A , τ ) - C ∗ -dynamical system with R × Γ ∋ ( t , x ) �→ τ ( t , x ) . 4 B ⊂ A -almost-local operators: � [ B 1 , τ ( s , vs ) ( B 2 )] � = O ( | s | −∞ ) . 5 A ⊂ B ( H ) and τ ( t , x ) ( A ) = U ( t , x ) AU ( t , x ) ∗ for A ∈ A . Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems 1 Γ - the abelian group of space translations ( R d or Z d ). Γ - Pontryagin dual of Γ ( R d or S d 2 � 1 ). 3 ( A , τ ) - C ∗ -dynamical system with R × Γ ∋ ( t , x ) �→ τ ( t , x ) . 4 B ⊂ A -almost-local operators: � [ B 1 , τ ( s , vs ) ( B 2 )] � = O ( | s | −∞ ) . 5 A ⊂ B ( H ) and τ ( t , x ) ( A ) = U ( t , x ) AU ( t , x ) ∗ for A ∈ A . Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems 1 Γ - the abelian group of space translations ( R d or Z d ). Γ - Pontryagin dual of Γ ( R d or S d 2 � 1 ). 3 ( A , τ ) - C ∗ -dynamical system with R × Γ ∋ ( t , x ) �→ τ ( t , x ) . 4 B ⊂ A -almost-local operators: � [ B 1 , τ ( s , vs ) ( B 2 )] � = O ( | s | −∞ ) . Lieb-Robinson bounds: � [ τ t ( A ) , B ] � ≤ C A , B e λ ( v LR t − d ( A , B )) , A , B ∈ A local. 5 A ⊂ B ( H ) and τ ( t , x ) ( A ) = U ( t , x ) AU ( t , x ) ∗ for A ∈ A . Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems 1 Γ - the abelian group of space translations ( R d or Z d ). Γ - Pontryagin dual of Γ ( R d or S d 2 � 1 ). 3 ( A , τ ) - C ∗ -dynamical system with R × Γ ∋ ( t , x ) �→ τ ( t , x ) . 4 B ⊂ A -almost-local operators: � [ B 1 , τ ( s , vs ) ( B 2 )] � = O ( | s | −∞ ) . 5 A ⊂ B ( H ) and τ ( t , x ) ( A ) = U ( t , x ) AU ( t , x ) ∗ for A ∈ A . Wojciech Dybalski Scattering theory for spin systems

Framework for QFT and spin systems 1 Γ - the abelian group of space translations ( R d or Z d ). Γ - Pontryagin dual of Γ ( R d or S d 2 � 1 ). 3 ( A , τ ) - C ∗ -dynamical system with R × Γ ∋ ( t , x ) �→ τ ( t , x ) . 4 B ⊂ A -almost-local operators: � [ B 1 , τ ( s , vs ) ( B 2 )] � = O ( | s | −∞ ) . 5 A ⊂ B ( H ) and τ ( t , x ) ( A ) = U ( t , x ) AU ( t , x ) ∗ for A ∈ A . Wojciech Dybalski Scattering theory for spin systems