Outline Green functions in mathematics Green functions for many-body systems in equilibrium Non-equilibrium Green functions Introduction to Green functions Stefan Kurth 1. Universidad del Pa´ ıs Vasco UPV/EHU, San Sebasti´ an, Spain 2. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain 3. European Theoretical Spectroscopy Facility (ETSF), www.etsf.eu Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Green functions for many-body systems in equilibrium Non-equilibrium Green functions Outline Green functions in mathematics Many-particle Green functions in equilibrium Zero temperature formalism One-particle Green function Response functions and two-particle Green functions Finite temperature formalism Non-equilibrium Green functions Keldysh contour and Kadanoff-Baym equations Summary Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Green functions for many-body systems in equilibrium Non-equilibrium Green functions Green functions in mathematics consider inhomogeneous differential equation (1D for simplicity) ˆ D x y ( x ) = f ( x ) where ˆ D x is linear differential operator in x . Example: damped harmonic oscillator ˆ d 2 d x 2 + γ d d x + ω 2 D x = general solution of inhomogeneous equation: y ( x ) = y hom ( x ) + y spec ( x ) where y hom is solution of the homogeneous eqn. ˆ D x y hom ( x ) = 0 and y spec ( x ) is any special solution of the inhomogeneous equation. Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Green functions for many-body systems in equilibrium Non-equilibrium Green functions Green functions in mathematics (cont.) how to obtain a special solution of the inhomogeneous equation for any inhomogeneity f ( x ) ? first find the solution of the following equation D x G ( x, x ′ ) = δ ( x − x ′ ) ˆ This defines the Green function G ( x, x ′ ) corresponding to the operator ˆ D x . Once G ( x, x ′ ) is found, a special solution can be constructed by � d x ′ G ( x, x ′ ) f ( x ′ ) y spec ( x ) = d x ′ G ( x, x ′ ) f ( x ′ ) = d x ′ δ ( x − x ′ ) f ( x ′ ) = f ( x ) check: ˆ � � D x Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions Hamiltonian of interacting electrons consider system of interacting electrons in static external potential v ext ( r ) described by Hamiltonian ˆ H −∇ 2 � � � H = ˆ ˆ T + ˆ V ext + ˆ d 3 x ˆ ˆ ψ † ( x ) W = 2 + v ext ( r ) ψ ( x ) +1 � � 1 d 3 x ′ ˆ ψ † ( x ) ˆ ψ ( x ′ ) ˆ ˆ d 3 x ψ † ( x ′ ) ψ ( x ) | r − r ′ | 2 x = ( r , σ ) : space-spin coordinate ψ † ( x ) , ˆ ˆ ψ ( x ) : electron creation and annihilation operators Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions Hamiltonian of interacting electrons consider system of interacting electrons in static external potential v ext ( r ) described by Hamiltonian ˆ H −∇ 2 � � � H = ˆ ˆ T + ˆ V ext + ˆ d 3 x ˆ ˆ ψ † ( x ) W = 2 + v ext ( r ) ψ ( x ) +1 � � 1 d 3 x ′ ˆ ψ † ( x ) ˆ ψ ( x ′ ) ˆ ˆ d 3 x ψ † ( x ′ ) ψ ( x ) | r − r ′ | 2 x = ( r , σ ) : space-spin coordinate ψ † ( x ) , ˆ ˆ ψ ( x ) : electron creation and annihilation operators Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions One-particle Green functions at zero temperature Time-ordered 1-particle Green function at zero temperature 0 | ˆ T [ ˆ ψ ( x , t ) H ˆ iG ( x , t ; x ′ , t ′ ) = � Ψ N ψ † ( x ′ , t ′ ) H ] | Ψ N 0 � � Ψ N 0 | Ψ N 0 � 0 � : N -particle ground state of ˆ H : ˆ | Ψ N H | Ψ N 0 � = E N 0 | Ψ N 0 � ψ ( x , t ) H = exp( i ˆ ˆ Ht ) ˆ ψ ( x ) exp( − i ˆ Ht ) : electron annihilation operator in Heisenberg picture ˆ T : time-ordering operator T [ ˆ ˆ ψ ( x , t ) H ˆ ψ ( x ′ , t ′ ) † H ] = H − θ ( t ′ − t ) ˆ θ ( t − t ′ ) ˆ ψ ( x , t ) H ˆ ψ ( x ′ , t ′ ) † ψ ( x ′ , t ′ ) † H ˆ ψ ( x , t ) H Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions One-particle Green functions at zero temperature Time-ordered 1-particle Green function at zero temperature 0 | ˆ T [ ˆ ψ ( x , t ) H ˆ iG ( x , t ; x ′ , t ′ ) = � Ψ N ψ † ( x ′ , t ′ ) H ] | Ψ N 0 � � Ψ N 0 | Ψ N 0 � 0 � : N -particle ground state of ˆ H : ˆ | Ψ N H | Ψ N 0 � = E N 0 | Ψ N 0 � ψ ( x , t ) H = exp( i ˆ ˆ Ht ) ˆ ψ ( x ) exp( − i ˆ Ht ) : electron annihilation operator in Heisenberg picture ˆ T : time-ordering operator T [ ˆ ˆ ψ ( x , t ) H ˆ ψ ( x ′ , t ′ ) † H ] = H − θ ( t ′ − t ) ˆ θ ( t − t ′ ) ˆ ψ ( x , t ) H ˆ ψ ( x ′ , t ′ ) † ψ ( x ′ , t ′ ) † H ˆ ψ ( x , t ) H Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions One-particle Green functions at zero temperature Time-ordered 1-particle Green function at zero temperature 0 | ˆ T [ ˆ ψ ( x , t ) H ˆ iG ( x , t ; x ′ , t ′ ) = � Ψ N ψ † ( x ′ , t ′ ) H ] | Ψ N 0 � � Ψ N 0 | Ψ N 0 � 0 � : N -particle ground state of ˆ H : ˆ | Ψ N H | Ψ N 0 � = E N 0 | Ψ N 0 � ψ ( x , t ) H = exp( i ˆ ˆ Ht ) ˆ ψ ( x ) exp( − i ˆ Ht ) : electron annihilation operator in Heisenberg picture ˆ T : time-ordering operator T [ ˆ ˆ ψ ( x , t ) H ˆ ψ ( x ′ , t ′ ) † H ] = H − θ ( t ′ − t ) ˆ θ ( t − t ′ ) ˆ ψ ( x , t ) H ˆ ψ ( x ′ , t ′ ) † ψ ( x ′ , t ′ ) † H ˆ ψ ( x , t ) H Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions Green functions as propagator t 1 < t 2 t 2 < t 1 annihilate electron (create hole) at create electron at time t 2 at position r 2 and propagate; time t 1 at position r 1 ; then create electron (annihilate then annihilate electron at time t 1 at position r 1 hole) at time t 2 at position r 2 Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions Observables from Green functions Information which can be extracted from Green functions ground-state expectation values of any single-particle operator ˆ d 3 x ˆ ψ † ( x ) o ( x ) ˆ � O = ψ ( x ) σ ˆ ψ † ( r σ ) ˆ e.g., density operator ˆ n ( r ) = � ψ ( r σ ) ground-state energy of the system Galitski-Migdal formula ∂t − ∇ 2 0 = − i � � i ∂ � E N d 3 x lim G ( r σ, t ; r ′ σ, t ′ ) t ′ → t + lim 2 2 r ′ → r spectrum of system: direct photoemission, inverse photoemission Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions Other kind of Green functions Retarded and advanced Green functions iG R ( x , t ; x ′ , t ′ ) = θ ( t − t ′ ) � Ψ N 0 |{ ˆ ψ ( x , t ) H , ˆ ψ † ( x ′ , t ′ ) † H }| Ψ N 0 � iG A ( x , t ; x ′ , t ′ ) = − θ ( t ′ − t ) � Ψ N 0 |{ ˆ ψ ( x , t ) H , ˆ ψ † ( x ′ , t ′ ) H }| Ψ N 0 � Benasque 2012: S. Kurth Introduction to Green functions
Outline Green functions in mathematics Zero-temperature formalism Green functions for many-body systems in equilibrium Finite-temperature formalism Non-equilibrium Green functions Spectral (Lehmann) representation of Green function N,k | Ψ N k �� Ψ N use completeness relation 1 = � k | − → iG ( x , t ; x ′ , t ′ ) � � = θ ( t − t ′ ) � i ( E N 0 − E N +1 )( t − t ′ ) g k ( x ) g ∗ k ( x ′ ) exp k k − θ ( t ′ − t ) )( t ′ − t ) � � � i ( E N 0 − E N − 1 f k ( x ′ ) f ∗ exp k ( x ) k k with quasiparticle amplitudes f k ( x ) = � Ψ N − 1 | ˆ ψ ( x ) | Ψ N 0 � k g k ( x ) = � Ψ N 0 | ˆ ψ ( x ) | Ψ N +1 � k note: G depends only on t − t ′ − → Fourier transform w.r.t. t − t ′ Benasque 2012: S. Kurth Introduction to Green functions
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