Functional RG in nonequilibrium

在第一个阶段，导出了两种新的非平衡RG方法。一种是基于Liouville空间中的动力学方程，在系统与储层之间的耦合中展开，即所谓的实时频率空间RG （RTRG-FS）。另一种方法是基于Keldysh形式和weterich RG格式，在局部系统的库仑相互作用中展开，即所谓的非平衡态泛函RG方法（FRG-NE）。在下一阶段，计划进一步发展这些方法，并解决Kondo模型，单杂质Anderson模型（SIAM），相互作用谐振能级模型（IRLM）和Luttinger液体中许多基本感兴趣的开放问题。一个重要的开放问题涉及到近藤模型（或等价的SIAM）的强耦合极限，其中所有的能量尺度都与近藤温度处于同一阶，因此自旋涨落不能再被扰动处理。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SIAMwe先前发现，FRG-NE方案适用于中等库仑相互作用的强耦合机制。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动的影响。另一个具有挑战性的课题是将RTRG-FS推广到显式时变场和谐波场应用于量子点的情况。我们将开发小振幅（线性响应），小频率（绝热量子泵浦）和大频率（Floquet理论，傅里叶分量的展开）的系统方案。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态响应。为了理解电荷和自旋涨落在非平衡状态下的基本相互作用，我们将在弱自旋涨落的良好控制下将RTRG-FS方案应用于SIAM。将考虑任意库仑相互作用、温度、磁场、栅极和偏置电压，并计算所有稳态特性（电平重整化、谐振线形）以及进入稳态的时间演变。关于量子线，一个尚未解决的基本问题是描述非平衡状态下由库仑相互作用引起的能量弛豫和消相。基于我们在量子点上的经验，我们计划通过考虑两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相关部分。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，并了解在存在两个以上障碍时相消和相平均之间的竞争。

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.