Kondo effect in systems with dynamical symmetries

This paper is devoted to a systematic exposure of the Kondo physics in quantum dots for which the low-energy spin excitations consist of a few different spin multiplets $|S_i{M}_i〉.$ Under certain conditions (to be explained below), some of the lowest energy levels $E_{S_i}$ are nearly degenerate. The dot in its ground state cannot then be regarded as a simple quantum top, in the sense that beside its spin operator other dot (vector) operators ${\mathbf{R}}_n$ are needed (in order to fully determine its quantum states), which have nonzero matrix elements between states of different spin multiplets $〈S_i{M}_i|{\mathbf{R}}_n|S_j{M}_j〉\ne 0.$ These Runge-Lenz operators do not appear in the isolated dot Hamiltonian (so in some sense they are “hidden”). Yet, they are exposed when tunneling between dot and leads is switched on. The effective spin Hamiltonian which couples the metallic electron spin $\mathbf{s}$ with the operators of the dot then contains exchange terms $J_n\mathbf{s}\cdot {\mathbf{R}}_n$ besides the ubiquitous ones $J_i\mathbf{s}\cdot {\mathbf{S}}_i.$ The operators ${\mathbf{S}}_i$ and ${\mathbf{R}}_n$ generate a dynamical group [usually $\mathrm{SO}\mathrm{}\left(n\right)\mathrm{}].$ Remarkably, the value of n can be controlled by gate voltages, indicating that abstract concepts such as dynamical symmetry groups are experimentally realizable. Moreover, when an external magnetic field is applied, under favorable circumstances the exchange interaction involves solely the Runge-Lenz operators ${\mathbf{R}}_n$ and the corresponding dynamical symmetry group is $\mathrm{SU}\mathrm{}\left(n\right).$ For example, the celebrated group $\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}$ is realized in a triple quantum dot with four electrons.