Electron systems out of equilibrium: Nonequilibrium Green's function approach

Špička, Václav; Velický, B.; Kalvová, A.

doi:10.1142/s0217979214300138

Cited by 24 publications

(32 citation statements)

References 220 publications

(151 reference statements)

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“…We use the Keldysh real time Non‐Equilibrium Green's Functions , but instead of the standard Keldysh triplet

G^{R}, G^{A}, K

we employ the equivalent choice

G^{R}, G^{A}, G^{<}

, where

G^{<}

is the “less” particle correlation function (Langreth‐Wilkins (LW) convention ). These Green's functions obey their Dyson equations, whose integro‐differential form is (with

ℏ = 1

, and

σ = ↑, ↓

; opposite spin denoted by

\bar{σ}

)

\begin{matrix} true \\ right normali \partial_{t} G_{σ}^{R} (t, t^{'}) = ε_{b} G_{σ}^{R} (t, t^{'}) + \int_{t^{'}}^{t} Σ_{σ}^{R} G_{σ}^{R}, \end{matrix}

for the propagators (only the equation for the retarded

G^{R}

is shown), while for

G^{<}

we have:

\begin{matrix} true \\ right normali \partial_{t} G_{σ}^{<} (t, t^{'}) = ε_{b} G_{σ}^{<} (t, t^{'}) + \int_{t_{0}}^{t} Σ_{σ}^{R} G_{σ}^{<} + \int_{t_{0}}^{t^{'}} Σ_{σ}^{<} G_{σ}^{A} . \end{matrix}

The propagators are insensitive to the initial conditions and obey the equal time boundary condition

...

…”

Section: Formalism: Non‐equilibrium Green's Functionsmentioning

confidence: 99%

“…First, the Ansatz itself consists in an approximate factorization of the particle correlation function, the GKBA decoupling ,

G_{σ}^{<} false(t, t^{'} false) = - G_{σ}^{R} false(t, t^{'} false) n_{σ} false(t^{'} false) + n_{σ} false(t false) G_{σ}^{A} false(t, t^{'} false)

The Ansatz simplifies to a few scalar relations here, to be compared with the general case where

G^{<} = - G^{R} ρ + ρ G^{A}

with ρ denoting the one‐particle density matrix. Second, the theory is assumed to be self‐consistent, i.e., the self‐energy matrix is assumed to be a functional of the Green's function,

Σ = Σ [G]

.…”

Section: Formalism: Generalized Master Equations (Gme)mentioning

confidence: 99%

“…In the simplest form, it is assumed that there exists a time

τ^{★}

, such that

\begin{matrix} true \\ right Σ^{♮} (t, t^{'}) & \approx & left 0 for | t - t^{'} | > τ^{★}, ♮ = R, A, < \end{matrix}

This condition should not be taken too literally, as, e.g., band edges lead to weak power law tails, but it captures, in a simple manner, the essential feature of a typical self‐energy, to be concentrated along the time diagonal

t = t^{'}

. It is in agreement with the Bogolyubov concept of the decay of correlations . We will use for convenience and simplicity and call

τ^{★}

by the conventional name of collision duration time.…”

Section: Formalism: Generalized Master Equations (Gme)mentioning

confidence: 99%

“…Here, we will deal with the form of Fluctuation‐Dissipation Theorem (FDT), which is formulated within the formalism of Non‐Equilibrium Green Functions (NEGF) . The FDT in equilibrium is equivalent with the Kubo‐Martin‐Schwinger (KMS) boundary condition, which permits to write all components of the equilibrium one particle NEGF in terms of the spectral density and a thermal factor .…”

Section: Introductionmentioning

confidence: 99%

“…This Fluctuation‐Dissipation (FD) relation between the correlation and the retarded components of NEGF depends on the existence of temperature and is lost out of equilibrium where such reduction is not possible and NEGF has two independent components, typically the Kadanof‐Baym pair

G^{<}

and

G^{>}

. Still, the components of the NEGF retain some inner interconnection which may be termed FD structure out of equilibrium .…”

Section: Introductionmentioning

confidence: 99%

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Non‐equilibrium dynamics of open systems and fluctuation‐dissipation theorems

Špička

Velický

Kalvová

2017

Fortschritte der Physik

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The Non‐Equilibrium Fluctuation‐Dissipation Theorem (NE FDT) is formulated within the Non‐Equilibrium Green Function (NEGF) formalism. The relation of this theorem to a simplified kinetic theory of non‐equilibrium dynamics of electrons in open quantum systems is discussed. The possibility of such a simplified description is first discussed on non‐equilibrium dynamics of the molecular bridge model represented by calculations of transient magnetic currents between two ferromagnetic electrodes linked by tunneling junctions to a molecular size island of an Anderson local center type. This model can be treated by using the full set of equations for NEGF, which can be solved numerically. This provides a reference framework for testing the possibility of a simpler and physically more transparent solution based on a Non‐Markovian Generalized Master Equation (GME) as an approximation of the full set of NEGF equations. The advantages and limitations of the use of the Generalized Kadanoff‐Baym Ansatz (GKBA) for this simplified description is demonstrated. The basic question is the range of applicability of this approximation. It turns out that for our specific model the decisive feature is the spectral structure of the tunneling functions of both electrodes and their positioning with respect to the island level depending on the bias and the exchange splitting. Finally, the relation of this simplified description to non‐equilibrium generalization of FDT is shown independently on the chosen model. The connection between the NEGF reconstruction equations and the non‐equilibrium version of FDT is introduced and the possible approximations of this general scheme are discussed.

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“…We use the Keldysh real time Non‐Equilibrium Green's Functions , but instead of the standard Keldysh triplet

G^{R}, G^{A}, K

we employ the equivalent choice

G^{R}, G^{A}, G^{<}

, where

G^{<}

is the “less” particle correlation function (Langreth‐Wilkins (LW) convention ). These Green's functions obey their Dyson equations, whose integro‐differential form is (with

ℏ = 1

, and

σ = ↑, ↓

; opposite spin denoted by

\bar{σ}

)

\begin{matrix} true \\ right normali \partial_{t} G_{σ}^{R} (t, t^{'}) = ε_{b} G_{σ}^{R} (t, t^{'}) + \int_{t^{'}}^{t} Σ_{σ}^{R} G_{σ}^{R}, \end{matrix}

for the propagators (only the equation for the retarded

G^{R}

is shown), while for

G^{<}

we have:

\begin{matrix} true \\ right normali \partial_{t} G_{σ}^{<} (t, t^{'}) = ε_{b} G_{σ}^{<} (t, t^{'}) + \int_{t_{0}}^{t} Σ_{σ}^{R} G_{σ}^{<} + \int_{t_{0}}^{t^{'}} Σ_{σ}^{<} G_{σ}^{A} . \end{matrix}

The propagators are insensitive to the initial conditions and obey the equal time boundary condition

...

…”

Section: Formalism: Non‐equilibrium Green's Functionsmentioning

confidence: 99%

“…First, the Ansatz itself consists in an approximate factorization of the particle correlation function, the GKBA decoupling ,

G_{σ}^{<} false(t, t^{'} false) = - G_{σ}^{R} false(t, t^{'} false) n_{σ} false(t^{'} false) + n_{σ} false(t false) G_{σ}^{A} false(t, t^{'} false)

The Ansatz simplifies to a few scalar relations here, to be compared with the general case where

G^{<} = - G^{R} ρ + ρ G^{A}

with ρ denoting the one‐particle density matrix. Second, the theory is assumed to be self‐consistent, i.e., the self‐energy matrix is assumed to be a functional of the Green's function,

Σ = Σ [G]

.…”

Section: Formalism: Generalized Master Equations (Gme)mentioning

confidence: 99%

“…In the simplest form, it is assumed that there exists a time

τ^{★}

, such that

\begin{matrix} true \\ right Σ^{♮} (t, t^{'}) & \approx & left 0 for | t - t^{'} | > τ^{★}, ♮ = R, A, < \end{matrix}

t = t^{'}

. It is in agreement with the Bogolyubov concept of the decay of correlations . We will use for convenience and simplicity and call

τ^{★}

by the conventional name of collision duration time.…”

Section: Formalism: Generalized Master Equations (Gme)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

G^{<}

and

G^{>}

. Still, the components of the NEGF retain some inner interconnection which may be termed FD structure out of equilibrium .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Non‐equilibrium dynamics of open systems and fluctuation‐dissipation theorems

Špička

Velický

Kalvová

2017

Fortschritte der Physik

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Ion Impact Induced Ultrafast Electron Dynamics in Finite Graphene‐Type Hubbard Clusters

Bönitz

Balzer

Schlünzen

et al. 2019

Physica Status Solidi (b)

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Strongly correlated systems of fermions have an interesting phase diagram arising from the Hubbard gap. Excitation across the gap leads to the formation of doubly occupied lattice sites (doublons) which offers interesting electronic and optical properties. Moreover, when the system is driven out of equilibrium interesting collective dynamics may arise that are related to the spatial propagation of doublons. Here, a novel mechanism that was recently proposed by the authors [Balzer et al., Phys. Rev. Lett. 121, 267602 (2018)] is verified by exact diagonalization and nonequilibrium Green functions (NEGF) simulationsfermionic doublon creation by the impact of energetic ions. The formation of a nonequilibrium steady state with homogeneous doublon distribution is reported. The effect should be particularly important for correlated finite systems, such as graphene nanoribbons, and directly observable with fermionic atoms in optical lattices. It is demonstrated that doublon formation and propagation in correlated lattice systems can be accurately simulated with NEGF. In addition to two-time results, single-time results within the generalized Kadanoff-Baym ansatz (GKBA) with Hartree-Fock propagators (HF-GKBA) is presented. Finally systematic improvements of the GKBA that use correlated propagators (correlated GKBA) and a correlated initial state are discussed.

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Non-equilibrium Green’s Functions for Coupled Fermion-Boson Systems

Karlsson

Leeuwen

2018

Handbook of Materials Modeling

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Electron systems out of equilibrium: Nonequilibrium Green's function approach

Cited by 24 publications

References 220 publications

Non‐equilibrium dynamics of open systems and fluctuation‐dissipation theorems

Non‐equilibrium dynamics of open systems and fluctuation‐dissipation theorems

Ion Impact Induced Ultrafast Electron Dynamics in Finite Graphene‐Type Hubbard Clusters

Non-equilibrium Green’s Functions for Coupled Fermion-Boson Systems

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