Seven Etudes on dynamical Keldysh Model

TL;DR

The paper presents seven pedagogical Etudes on dynamical Keldysh models with multicomponent Gaussian non-Markovian noise, deriving exact Green's functions, self-energies, and vertex structures via a closed-form Dyson solution constrained by Ward identities. A unifying framework of first-order ODEs for the Green's functions is developed across arbitrary component number $d$, with centrifugal terms inducing level repulsion for $d>1$ and distinct analytic structures for even versus odd $d$. The authors then build a comprehensive recurrence-based apparatus to enumerate skeleton Feynman diagrams for the self-energy, vertex, and T-matrix, first in the single-component case and then generalized to multi-component models, including large-$D$ expansions. They connect these formal results to physical realizations in complex quantum-dot devices and discuss potential applications to quantum transport and related disordered systems, while providing the mathematical machinery (Hilbert transforms, Dawson function, etc.) needed to handle the analytic structure of the solutions. The work thus offers exact solvable models for dynamical Gaussian disorder with rich diagrammatics, enabling precise spectral and transport analyses of multi-component quantum-dot and related systems.

Abstract

We present a comprehensive pedagogical discussion of a family of models describing the propagation of a single particle in a multicomponent non-Markovian Gaussian random field. We report some exact results for single-particle Green's functions, self-energy, vertex part and T-matrix. These results are based on a closed form solution of the Dyson equation combined with the Ward identity. Analytical properties of the solution are discussed. Further we describe the combinatorics of the Feynman diagrams for the Green's function and the skeleton diagrams for the self-energy and vertex, using recurrence relations between the Taylor expansion coefficients of the self-energy. Asymptotically exact equations for the number of skeleton diagrams in the limit of large N are derived. Finally, we consider possible realizations of a multicomponent Gaussian random potential in quantum transport via complex quantum dot experiments.

Figures (14)

• Figure 1: Diagrammatic expansion for the Green's function. The double line corresponds to bold Green's function. The single lines denote bare Green's functions. The wavy lines stand for the correlator of the Gaussian classical random field.
• Figure 2: Experimental realization of the Keldysh model in quantum dots: a) single QD; b) serial double QD; c) linear (serial) triple QD; d) triangular triple QD.
• Figure 3: Left: single quantum dot with the noisy back gate. Right: quantum well with fluctuating energy level.
• Figure 4: Slow noise produced by the back gate in double QD. Top: "diagonal" noise describing fluctuations of the level positions; Bottom: $U(1)$ fluctuations of the barrier. Three $2\times 2$ Pauli matrices constitute the basis for the fundamental representation of the $SU(2)$ group.
• Figure 5: Slow noise produced by the back gate in the serial triple QD. First and second rows: fluctuations of the level positions in each of three QD. Third row: symmetric fluctuations of the barriers. Fourth row: anti-symmetric fluctuations of the barriers. Bottom row: fluctuations of the co-tunneling. The same processes will naturally appear in the triangular QD Fig. \ref{['qdex1']}d and describe direct tunneling between the first and the third dot. Eight $3\times 3$ Gell-Mann matrices constitute the basis for the fundamental representation of the $SU(3)$ group.