Kondo effect in systems with dynamical symmetries

TL;DR

This paper explores the Kondo effect in quantum dots with multiple nearly degenerate spin states, revealing how dynamical symmetries like SO(n) and SU(n) emerge and can be controlled experimentally through tunneling and gate voltages.

Contribution

It introduces the concept of dynamical symmetry groups in quantum dots with complex spin multiplets, highlighting the role of hidden operators and their experimental controllability.

Findings

Dynamical symmetry groups SO(n) and SU(n) can be realized in quantum dot systems.

Gate voltages allow control over the symmetry group parameter n.

External magnetic fields can isolate Runge-Lenz operators, leading to SU(n) symmetry.

Abstract

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 ${\bf R}_{n}$ are needed (in order to fully determine its quantum states), which have non-zero matrix elements between states of different spin multiplets $<S_{i}M_{i} |{\bf R}_{n}|S_{j}M_{j} > \ne 0$. 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 ${\bf s}$ with the operators of the dot then contains new exchange terms, $J_{n} {\bf s} \cdot {\bf R}_{n}$ beside the ubiquitous ones $J_{i} {\bf s}\cdot {\bf S}_{i}$. The operators ${\bf S}_{i}$ and ${\bf R}_{n}$ generate a dynamical group (usually SO(n)). 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 then, under favorable circumstances, the exchange interaction involves solely the Runge-Lenz operators ${\bf R}_{n}$ and the corresponding dynamical symmetry group is SU(n). For example, the celebrated group SU(3) is realized in triple quantum dot with four electrons.