Abstract
In this paper we study the time evolution of an observable in the interacting fermion systems driven out of equilibrium. We present a method for solving the Heisenberg equations of motion by constructing excitation operators which are defined as the operators \documentclass[12pt]{minimal}\begin{document}$\hat{A}$\end{document} satisfying \documentclass[12pt]{minimal}\begin{document}$[\hat{H},\hat{A}]=\lambda \hat{A}$\end{document}[Ĥ,Â]=λÂ. It is demonstrated how an excitation operator and its excitation energy λ can be calculated. By an appropriate supposition of the form of \documentclass[12pt]{minimal}\begin{document}$\hat{A}$\end{document} we turn the problem into the one of diagonalizing a series of matrices whose dimension depends linearly on the size of the system. We perform this method to calculate the evolution of the creation operator in a toy model Hamiltonian which is inspired by the Hubbard model and the nonequilibrium current through the single impurity Anderson model. This method is beyond the traditional perturbation theory in Keldysh-Green's function formalism, because the excitation energy λ is modified by the interaction and it will appear in the exponent in the function of time.
Highlights
The interacting fermion systems have a central position in modern condensed matter physics.[1]
How to understand the real time evolution of an interacting system driven out of equilibrium remains a great challenge in spite of intense efforts in recent years
III we demonstrate how to construct the excitation operator and use it to calculate the time evolution of the creation operator in a toy model inspired by the Hubbard model
Summary
The interacting fermion systems have a central position in modern condensed matter physics.[1]. Numerical methods are developed like the real time Quantum Monte Carlo,[16,17,18,19] the time-dependent numerical renormalization group,[20,21,22] the scattering state numerical renormalization group,[23] the adaptive time-dependent density matrix renormalization group[24,25,26] and the non-equilibrium dynamical mean field theory.[27] The numerical method has the advantage that the result can be obtained in very high precision and the parameters of the model can be chosen arbitrarily They suffer the inconvenience that the large computer resources are required and the result does not incorporate the physics of the problem directly.
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