Far-from-Equilibrium and Time-Dependent Phenomena: Theory Avraham - PowerPoint PPT Presentation

Nonequilibrium: A theoretical challenge Two possible strategies to treat steady state Time-independent formulation: Time-dependent formulation Evolve the system in time Work directly at steady state by to reach steady state imposing suitable boundary conditions e.g., by constructing the many- particle scattering states Avraham Schiller / QIMP11

Brief division of theoretical approaches Steady state Avraham Schiller / QIMP11

Brief division of theoretical approaches Steady state Keldysh diagrammatics Avraham Schiller / QIMP11

Brief division of theoretical approaches Steady state Keldysh diagrammatics Scattering Bethe ansatz (Andrei et al. ) Avraham Schiller / QIMP11

Brief division of theoretical approaches Steady state Keldysh diagrammatics Scattering Bethe ansatz (Andrei et al. ) Nonequilibrium variants of perturbative RG Poor-man’s scaling (Rosch et al. ) Flow equations (Kehrein) Real-time diagrammatics (Schoeller et al .) Functional RG (Meden et al .) Avraham Schiller / QIMP11

Brief division of theoretical approaches Steady state Keldysh diagrammatics Scattering Bethe ansatz (Andrei et al. ) Nonequilibrium variants of perturbative RG Poor-man’s scaling (Rosch et al. ) Flow equations (Kehrein) Real-time diagrammatics (Schoeller et al .) Functional RG (Meden et al .) Exactly solvable models: Toulouse limit (AS & Hershfield) Extension to double dots (Sela & Affleck) Avraham Schiller / QIMP11

Brief division of theoretical approaches Evolution in time Avraham Schiller / QIMP11

Brief division of theoretical approaches Evolution in time Keldysh diagrammatics Avraham Schiller / QIMP11

Brief division of theoretical approaches Evolution in time Keldysh diagrammatics Time-dependent DMRG (White, Schollwoeck,…) Avraham Schiller / QIMP11

Brief division of theoretical approaches Evolution in time Keldysh diagrammatics Time-dependent DMRG (White, Schollwoeck,…) Keldysh Quantum Monte Carlo (Werner, Muehlbacher,…) Avraham Schiller / QIMP11

Brief division of theoretical approaches Evolution in time Keldysh diagrammatics Time-dependent DMRG (White, Schollwoeck,…) Keldysh Quantum Monte Carlo (Werner, Muehlbacher,…) Nonequilibrium variants of perturbative RG Flow equations (Kehrein) Real-time diagrammatics (Schoeller et al .) Avraham Schiller / QIMP11

Brief division of theoretical approaches Evolution in time Keldysh diagrammatics Time-dependent DMRG (White, Schollwoeck,…) Keldysh Quantum Monte Carlo (Werner, Muehlbacher,…) Nonequilibrium variants of perturbative RG Flow equations (Kehrein) Real-time diagrammatics (Schoeller et al .) Time-dependent NRG (Anders & AS) Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Incoming state: decoupled leads Scattered state: entangled system Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Incoming state: decoupled leads Scattered state: entangled system φ = φ Divide H into H 0 + H 1 , where and Η 1 contains all terms | | H E 0 that drive the system out of equilibrium. Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Incoming state: decoupled leads Scattered state: entangled system φ = φ Divide H into H 0 + H 1 , where and Η 1 contains all terms | | H E 0 that drive the system out of equilibrium. Assume approach to steady state Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Because of the approach to steady state, one can “smear” the initial time: 0 ∫ t e η − ψ = η φ i ( H E ) t | lim | dt e 0 0 + η → 0 0 − ∞ Gell-Man and Goldberger , 1953 Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Because of the approach to steady state, one can “smear” the initial time: 0 ∫ t e η − ψ = η φ i ( H E ) t | lim | dt e 0 0 + η → 0 0 − ∞ 1 ψ = φ + − φ | | ( ) | H E − + η E H i Gell-Man and Goldberger , 1953 Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Because of the approach to steady state, one can “smear” the initial time: 0 ∫ t e η − ψ = η φ i ( H E ) t | lim | dt e 0 0 + η → 0 0 − ∞ 1 ψ = φ + − φ | | ( ) | H E − + η E H i − φ = Since , we arrive at the Lippmann-Schwinger equation ( ) | 0 H E 0 1 ψ = φ + φ | | | H − + η 1 E H i Gell-Man and Goldberger , 1953 Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Important points to take note of: H and H 0 must therefore have continuous overlapping spectra, → ∞ which implies the limit L Avraham Schiller / QIMP11

The nonequilibrium steady-state density operator ∑ ρ = φ φ Starting from , where p i are typically equilibrium ˆ 0 | | p i i i i Boltzmann factors, one formally has that Avraham Schiller / QIMP11

The nonequilibrium steady-state density operator ∑ ρ = φ φ Starting from , where p i are typically equilibrium ˆ 0 | | p i i i i Boltzmann factors, one formally has that ∑ ρ → ρ = ψ ψ ˆ 0 ˆ | | p i i i i with 1 ψ = φ + φ | | | H − + η 1 i i i E H i i Avraham Schiller / QIMP11

The nonequilibrium steady-state density operator ∑ ρ = φ φ Starting from , where p i are typically equilibrium ˆ 0 | | p i i i i Boltzmann factors, one formally has that ∑ ρ → ρ = ψ ψ ˆ 0 ˆ | | p i i i i with 1 ψ = φ + φ | | | H − + η 1 i i i E H i i Assuming the approach to steady state, the form of the nonequilibrium density operator is formally known Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form Zubarev , 1960’s, Hershfield 1993, Doyon & Andrei 2006 Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form In practice, the initial density matrix generically takes the form − β − ( ) H Y e 0 0 ρ = { } ˆ − β − 0 ( ) H Y Trace e 0 0 [ ] = with , 0 H Y 0 0 Zubarev , 1960’s, Hershfield 1993, Doyon & Andrei 2006 Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form In practice, the initial density matrix generically takes the form − β − ( ) H Y e 0 0 ρ = { } ˆ − β − 0 ( ) H Y Trace e 0 0 [ ] = with , 0 H Y 0 0 Indeed, in many cases one takes ∑ ∑ = ε + µ + ( ) H c c α α α σ α σ 0 k k k α = σ , , L R k ∑ ∑ = µ + Y c k c α α σ α σ 0 k α = σ , , L R k Zubarev , 1960’s, Hershfield 1993, Doyon & Andrei 2006 Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form − β − ( ) H Y e ρ = ˆ { } − β − ( ) H Y Trace e with [ ] = η − → , ( ) 0 Y H i Y Y 0 Zubarev , 1960’s, Hershfield 1993, Doyon & Andrei 2006 Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form − β − ( ) H Y e ρ = ˆ { } − β − ( ) H Y Trace e with [ ] = η − → , ( ) 0 Y H i Y Y 0 The steady-state density operator takes an equilibrium- like form! Zubarev , 1960’s, Hershfield 1993, Doyon & Andrei 2006 Avraham Schiller / QIMP11

Generalized fermionic scattering states The steady-state density operator can be represented in terms of generalized fermionic scattering states: ∑ ∑ − = ε ψ + ψ H Y α α σ α σ k k k α = σ , , L R k with [ ] ψ + = − ε ψ + + η + − ψ + , ( ) H i c α σ α α σ α σ α σ k k k k k Hershfield 1993, Han 2007 Avraham Schiller / QIMP11

Generalized fermionic scattering states The steady-state density operator can be represented in terms of generalized fermionic scattering states: ∑ ∑ − = ε ψ + ψ H Y α α σ α σ k k k α = σ , , L R k with [ ] ψ + = − ε ψ + + η + − ψ + , ( ) H i c α σ α α σ α σ α σ k k k k k ψ + In general is a complicated many-body operator: α σ k ∑ ∑ + + α + α + + ψ = + + + k k  c A c B c c c α σ α σ k k i i ijl i j l i i , j , k Hershfield 1993, Han 2007 Avraham Schiller / QIMP11

Generalized fermionic scattering states In the absence of interactions ψ α k σ reduce to the familiar single-particle scattering states ∑ + + α + ψ = + k c A c α σ α σ k k i i i and one recovers the Landauer-Buttiker formulation Hershfield 1993 Avraham Schiller / QIMP11

Time-dependent formulation Avraham Schiller / QIMP11

Time-dependent formulation ∑ ρ = φ φ ˆ 0 | | p Starting from at time t 0 , expectation values are i i i i explicitly propagated in time: { } ˆ + ˆ = ρ ˆ ( ) Trace ( , ) ( , ) A t U t t A U t t 0 0 0 Avraham Schiller / QIMP11

Time-dependent formulation ∑ ρ = φ φ ˆ 0 | | p Starting from at time t 0 , expectation values are i i i i explicitly propagated in time: { } ˆ + ˆ = ρ ˆ ( ) Trace ( , ) ( , ) A t U t t A U t t 0 0 0 Recurrence A ( t ) Steady state value t Avraham Schiller / QIMP11

Time-dependent formulation Avraham Schiller / QIMP11

Time-dependent formulation Avraham Schiller / QIMP11

Time-dependent formulation A challenge when the system features small energy Scales such as the Kondo temperature! Avraham Schiller / QIMP11

Selected review of theoretical approaches Scattering Bethe ansatz Nonequilbrium variants of perturbative RG Time-dependent NRG Avraham Schiller / QIMP11

Selected review of theoretical approaches Scattering Bethe ansatz Nonequilbrium variants of perturbative RG Time-dependent NRG Will not addressed: Keldysh-based approaches (Hans Kroha’s talk) Theories based on Fermi-liquid theory (weak nonequilibrium) Keldysh Quantum Monte Carlo Time-dependent DMRG Avraham Schiller / QIMP11

Scattering Bethe ansatz One uses the Bethe ansatz approach, adjusted to open boundary conditions, to explicitly construct the many-particle scattering states Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz One uses the Bethe ansatz approach, adjusted to open boundary conditions, to explicitly construct the many-particle scattering states Step 1: Conversion to continuum-limit Hamiltonian Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz One uses the Bethe ansatz approach, adjusted to open boundary conditions, to explicitly construct the many-particle scattering states Step 1: Conversion to continuum-limit Hamiltonian Left lead Right lead Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz One uses the Bethe ansatz approach, adjusted to open boundary conditions, to explicitly construct the many-particle scattering states Step 1: Conversion to continuum-limit Hamiltonian Left lead Right lead Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz General form of N -electron wave function: Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz General form of N -electron wave function: Impurity states Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz General form of N -electron wave function: Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz General form of N -electron wave function: where F obeys the Schroedinger-type equation Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz General form of N -electron wave function: where F obeys the Schroedinger-type equation Within each sector where x 1 ,…, x N , and x 0 = 0 obey some fixed ordering, F has solutions in terms of plane waves Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz The Bethe ansatz then seeks solutions of the form Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz The Bethe ansatz then seeks solutions of the form Projection onto the ordering affiliated with P Mehta & Andrei, 2005 Avraham Schiller / QIMP11

Scattering Bethe ansatz The Bethe ansatz then seeks solutions of the form Mehta & Andrei, 2005 Avraham Schiller / QIMP11