Semiclassical approach for interacting fermionic systems: - PowerPoint PPT Presentation

Semiclassical approach for interacting fermionic systems: interference and echos in the Hubbard model Thomas Engl 1 , Peter Schlagheck 2 , Juan Diego Urbina 1 and Klaus Richter 1 1 Universit¨ at Regensburg 2 Universit´ e de Li` ege March 18, 2015 Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 1 / 10

Introduction & Motivation Coherent backscattering 0.004 ϕ =0 ϕ =0.125 π ϕ =0.25 π probability ϕ =0.5 π ϕ =0, classical 0 223334 232334 233243 233432 242333 322334 323243 323432 332234 332342 333242 334332 343223 423233 432323 433322 [cf. F. Jendrzejewski, K. M¨ uller, J. Richard, A. Date, final state [TE, J. Dujardin, A. Arg¨ uelles, P. Schlagheck, K. Richter T. Plisson, P. Bouyer, A. Aspect and V. Josse, PRL 109 and J. D. Urbina, PRL 112 140403 (2014)] 195302 (2012)] → talk by Peter Schlagheck (Fr. 11h) Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 2 / 10

Introduction & Motivation Coherent backscattering 0.004 ϕ =0 ϕ =0.125 π ϕ =0.25 π probability ϕ =0.5 π ϕ =0, classical 0 223334 232334 233243 233432 242333 322334 323243 323432 332234 332342 333242 334332 343223 423233 432323 433322 [cf. F. Jendrzejewski, K. M¨ uller, J. Richard, A. Date, final state [TE, J. Dujardin, A. Arg¨ uelles, P. Schlagheck, K. Richter T. Plisson, P. Bouyer, A. Aspect and V. Josse, PRL 109 and J. D. Urbina, PRL 112 140403 (2014)] 195302 (2012)] → talk by Peter Schlagheck (Fr. 11h) Echoes: ? [cf. http://en.wikipedia.org/wiki/Spin echo] Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 2 / 10

Fermionic Hubbard model Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model c † | n � = ˆ 1 ↓ | 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 � Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model c † c † | n � = ˆ 2 ↑ ˆ 2 ↓ | 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 � Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model c † | n � = − ˆ 3 ↑ | 0 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 � Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model | n � = | 0 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 1 , 0 � Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model Hamiltonian � � � � �� ˆ � c † c † c † c † c † H = ǫ j ˆ j σ ˆ c j σ − J ˆ j σ ˆ c j +1 σ + ˆ j +1 σ ˆ c j +1 σ + U ˆ j ↑ ˆ j ↓ ˆ c j ↓ ˆ c j ↑ σ = ↑ , ↓ j � � � � + κ ∗ � c † c † c † c † + κ ˆ j , ↓ ˆ c j +1 , ↑ − ˆ j +1 , ↓ ˆ c j , ↑ ˆ j +1 , ↑ ˆ c j , ↓ − ˆ j , ↑ ˆ c j +1 , ↓ Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model J J Hamiltonian � � � � �� ˆ � c † c † c † c † c † H = ǫ j ˆ j σ ˆ c j σ − J ˆ j σ ˆ c j +1 σ + ˆ j +1 σ ˆ c j +1 σ + U ˆ j ↑ ˆ j ↓ ˆ c j ↓ ˆ c j ↑ σ = ↑ , ↓ j � � � � + κ ∗ � c † c † c † c † + κ ˆ j , ↓ ˆ c j +1 , ↑ − ˆ j +1 , ↓ ˆ c j , ↑ ˆ j +1 , ↑ ˆ c j , ↓ − ˆ j , ↑ ˆ c j +1 , ↓ Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model J − κ κ J Hamiltonian � � � � �� ˆ � c † c † c † c † c † H = ǫ j ˆ j σ ˆ c j σ − J ˆ j σ ˆ c j +1 σ + ˆ j +1 σ ˆ c j +1 σ + U ˆ j ↑ ˆ j ↓ ˆ c j ↓ ˆ c j ↑ σ = ↑ , ↓ j � � � � + κ ∗ � c † c † c † c † + κ ˆ j , ↓ ˆ c j +1 , ↑ − ˆ j +1 , ↓ ˆ c j , ↑ ˆ j +1 , ↑ ˆ c j , ↓ − ˆ j , ↑ ˆ c j +1 , ↓ Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Fermionic Hubbard model J U − κ κ J Hamiltonian � � � � �� ˆ � c † c † c † c † c † H = ǫ j ˆ j σ ˆ c j σ − J ˆ j σ ˆ c j +1 σ + ˆ j +1 σ ˆ c j +1 σ + U ˆ j ↑ ˆ j ↓ ˆ c j ↓ ˆ c j ↑ σ = ↑ , ↓ j � � � � + κ ∗ � c † c † c † c † + κ ˆ j , ↓ ˆ c j +1 , ↑ − ˆ j +1 , ↓ ˆ c j , ↑ ˆ j +1 , ↑ ˆ c j , ↓ − ˆ j , ↑ ˆ c j +1 , ↓ Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 3 / 10

Semiclassical theory � Ht � � � ˆ � � e − i Propagator in Fock basis: K ( n , m , t ) = n � m � � Construct Path integral → talk by Juan Diego Urbina (Fr. 11h30) Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory � Ht � � � ˆ � � e − i Propagator in Fock basis: K ( n , m , t ) = n � m � � Construct Path integral → talk by Juan Diego Urbina (Fr. 11h30) ⇒ Classical limit: � 2 , c † � � ˆ j σ ˆ c j σ → � φ j σ ( t ) ( k ,σ ′ ) 2 −| φ k σ ′ ( t ) | 2 � 1 − 2 | φ l σ ′′ ( t ) | 2 � c † j σ ( t ) φ k σ ′ ( t ) e − | φ j σ ( t ) | c k σ ′ → φ ∗ � ˆ j σ ˆ ( l ,σ ′′ )=( j ,σ ) Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory � Ht � � � ˆ � � e − i Propagator in Fock basis: K ( n , m , t ) = n � m � � Construct Path integral → talk by Juan Diego Urbina (Fr. 11h30) ⇒ Classical limit: � 2 , c † � � ˆ j σ ˆ c j σ → � φ j σ ( t ) ( k ,σ ′ ) 2 −| φ k σ ′ ( t ) | 2 � 1 − 2 | φ l σ ′′ ( t ) | 2 � c † j σ ( t ) φ k σ ′ ( t ) e − | φ j σ ( t ) | c k σ ′ → φ ∗ � ˆ j σ ˆ ( l ,σ ′′ )=( j ,σ ) Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory � Ht � � � ˆ � � e − i Propagator in Fock basis: K ( n , m , t ) = n � m � � Construct Path integral → talk by Juan Diego Urbina (Fr. 11h30) ⇒ Classical limit: � 2 , c † � � ˆ j σ ˆ c j σ → � φ j σ ( t ) ( k ,σ ′ ) 2 −| φ k σ ′ ( t ) | 2 � 1 − 2 | φ l σ ′′ ( t ) | 2 � c † j σ ( t ) φ k σ ′ ( t ) e − | φ j σ ( t ) | c k σ ′ → φ ∗ � ˆ j σ ˆ ( l ,σ ′′ )=( j ,σ ) Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical theory � Ht � � � ˆ � � e − i Propagator in Fock basis: K ( n , m , t ) = n � m � � Construct Path integral → talk by Juan Diego Urbina (Fr. 11h30) ⇒ Classical limit: � 2 , c † � � ˆ j σ ˆ c j σ → � φ j σ ( t ) ( k ,σ ′ ) 2 −| φ k σ ′ ( t ) | 2 � 1 − 2 | φ l σ ′′ ( t ) | 2 � c † j σ ( t ) φ k σ ′ ( t ) e − | φ j σ ( t ) | c k σ ′ → φ ∗ � ˆ j σ ˆ ( l ,σ ′′ )=( j ,σ ) Stationary phase analysis ⇒ Sum over classical trajectories Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 4 / 10

Semiclassical propagator 1 � � � ˆ Ht � � � e − i i � � R γ + i µ γ π K ( m , n , t ) = m � n ≈ A γ e � � 2 γ : n → m t d t ′ � J ( t ′ ) − H ( cl ) ( ψ ∗ ( t ′ ) , ψ ( t ′ )) � � θ ( t ′ ) ˙ � Classical action: R γ = 0 1 TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133 , 1563; arXiv:1409.4196 Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10

Semiclassical propagator 1 � � � ˆ Ht � � � e − i i � � R γ + i µ γ π K ( m , n , t ) = m � n ≈ A γ e � � 2 γ : n → m Classical trajectory γ : n → m : | ψ j σ (0) | 2 = n j σ | ψ j σ ( t ) | 2 = m j σ t d t ′ � J ( t ′ ) − H ( cl ) ( ψ ∗ ( t ′ ) , ψ ( t ′ )) � � θ ( t ′ ) ˙ � Classical action: R γ = 0 1 TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133 , 1563; arXiv:1409.4196 Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10

Semiclassical propagator 1 � � � ˆ Ht � � � e − i i � � R γ + i µ γ π K ( m , n , t ) = m � n ≈ A γ e � � 2 γ : n → m Classical trajectory γ : n → m : | ψ j σ (0) | 2 = n j σ | ψ j σ ( t ) | 2 = m j σ ψ = ∂ H ( cl ) i � ˙ ∂ ψ ∗ t d t ′ � J ( t ′ ) − H ( cl ) ( ψ ∗ ( t ′ ) , ψ ( t ′ )) � � θ ( t ′ ) ˙ � Classical action: R γ = 0 1 TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133 , 1563; arXiv:1409.4196 Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10

Semiclassical propagator 1 � � � ˆ Ht � � � e − i i � � R γ + i µ γ π K ( m , n , t ) = m � n ≈ A γ e � � 2 γ : n → m Classical trajectory γ : n → m : θ ( γ =1) (0) θ ( γ =2) (0) | ψ j σ (0) | 2 = n j σ | ψ j σ ( t ) | 2 = m j σ θ ( γ =3) (0) ψ = ∂ H ( cl ) i � ˙ ∂ ψ ∗ t d t ′ � J ( t ′ ) − H ( cl ) ( ψ ∗ ( t ′ ) , ψ ( t ′ )) � � θ ( t ′ ) ˙ � Classical action: R γ = 0 1 TE, J. D. Urbina and K. Richter, Theor. Chem. Acc. 133 , 1563; arXiv:1409.4196 Thomas Engl (Uni Regensburg) Interference in the Fermi-Hubbard model March 18, 2015 5 / 10