Functional RG in nonequilibrium

在第一阶段，两个新的非平衡RG方法已经得到。一种是基于Liouville空间的动力学方程，并扩展到系统与水库的耦合，即频率空间的实时RG（RTRG-FS）。另一种是基于Keldysh形式和Wetterich RG格式，并在库仑相互作用下对局域系统进行扩展，即所谓的非平衡泛函RG方法（FRG-NE）。在下一个阶段，计划进一步发展这些方法，并解决Kondo模型，单杂质安德森模型（SIAM），相互作用共振能级模型（IRLM）和Luttinger液体中的许多基本问题。一个重要的悬而未决的问题是关于近藤模型（或等价的SIAM）的强耦合极限，其中所有的能量标度都与近藤温度处于相同的量级，因此自旋涨落不再能被微扰处理。基于RTRG-FS中的一些初步和有希望的考虑，我们将尝试包括Z因子和2环项，并且我们将使用完全自洽的衰变率确定。对于SIAM，我们以前发现，FRG-NE计划的工作在强耦合制度的中度库仑相互作用。为了处理强库仑相互作用，我们将尝试包括三粒子顶点的某些静态部分，并研究它们对两粒子顶点流的影响。另一个具有挑战性的课题是将RTRG-FS推广到显式含时和谐波场应用于量子点的情况。我们将为小振幅（线性响应），小频率（绝热量子泵浦）和大频率（Floquet理论，傅立叶分量展开）开发系统方案。该方法将用于分析具有一个或两个能级的相互作用共振能级模型的动力学响应，为了理解非平衡态中电荷和自旋涨落的基本相互作用，我们将在弱自旋涨落的良好控制区域中将RTRG-FS方案应用于SIAM。任意的库仑相互作用，温度，磁场，栅极和偏置电压将被考虑，所有的定态性质（能级重整化，共振线形状）连同时间演化到station.Concerning量子线，一个基本的未解决的问题是描述的库仑相互作用引起的非平衡状态下的能量弛豫和退相。基于我们在量子点中的经验，我们计划通过考虑2粒子顶点的频率依赖性来研究这个问题。我们将包括Z因子的流动和自能虚部的频率依赖部分。目的是计算非平衡单粒子分布函数和零偏异常分裂的线形，并了解在两个以上障碍的存在下，退相和相位平均之间的竞争。

During the first period two new nonequilibrium RG methods have been derived. One is based on kinetic equations in Liouville space and expands in the coupling between system and reservoir, the so-called real-time RG in frequency space (RTRG-FS). The other is based on the Keldysh formalism and the Wetterich RG scheme and expands in the Coulomb interaction on the local system, the so-called functional RGmethod in nonequilibrium (FRG-NE). In the next period it is planned to further develop these methods and to adress many open questions of fundamental interest in the Kondo model, the single-impurity Anderson model (SIAM), the interacting resonant level model (IRLM), and Luttinger liquids. An important open question concerns the strong coupling limit of the Kondo model (or, equivalently, the SIAM), where all energy scales are of the same order as the Kondo temperature, so that spin fluctuations can no longer be treated perturbatively. Based on some preliminary and promising considerations within RTRG-FS, we will try to include the Z-factor and 2-loop terms, and we will use a fully selfconsistent determination of the decay rates. For the SIAMwe found previously that the FRG-NE scheme works in the strong coupling regime for moderate Coulomb interactions. To treat strong Coulomb interactions we will try to include certain static parts of the 3-particle vertex and study their influence on the flow of the two-particle vertex.Another challenging topic will be the generalization of RTRG-FS to the case when explicitly time-dependent and harmonic fields are applied to quantum dots. We will develop systematic schemes for small amplitudes (linear response), small frequencies (adiabatic quantum pumping), and large frequencies (Floquet theory, expansion in Fourier components). The method will be used to analyse the dynamic response of the interacting resonant level model with one or two levels.To understand the fundamental interplay of charge and spin fluctuations in nonequilibrium, we will apply the RTRG-FS scheme to the SIAM in the well-controlled regime of weak spin fluctuations. Arbitrary Coulomb interaction, temperature, magnetic field, gate and bias voltage will be considered, and all stationary properties (level renormalization, resonant line shapes) together with the time evolution into the stationary state will be calculated.Concerning quantum wires, a fundamental unsolved issue is the description of energy relaxation and dephasing induced by the Coulomb interaction in a nonequilibrium situation. Based on our experience in quantum dots, we plan to study this problem by considering the frequency-dependence of the 2-particle vertex. We will include the flow of the Z-factor and the frequency-dependent part of the imaginary part of the self-energy. The aim is to calculate the nonequilibrium one-particle distribution function and the line shape of the splitting of the zero-bias anomaly, and to understand the competition between dephasing and phase-averaging in the presence of more than two barriers.