# The $SU(4)-SU(2)$ crossover and spin filter properties of a double quantum dot nanosystem

## TL;DR

This paper investigates the $SU(4)-SU(2)$ crossover in a double quantum dot system under magnetic field, revealing how spin polarization and Kondo states evolve, with implications for spin filtering and quantum state control.

## Contribution

It provides a detailed analysis of the $SU(4)-SU(2)$ crossover driven by magnetic field and gate potential, highlighting the transition between different Kondo ground states in a double quantum dot system.

## Key findings

- Identification of the $SU(4)$ to $SU(2)$ crossover driven by magnetic field and gate potential.
- Observation of a sudden change in spin polarization and the establishment of an orbital $SU(2)$ ground state.
- Demonstration of the competition between Zeeman energy and Kondo temperature in the crossover process.

## Abstract

The $SU(4)-SU(2)$ crossover, driven by an external magnetic field $h$, is analyzed in a capacitively-coupled double-quantum-dot device connected to independent leads. As one continuously charges the dots from empty to quarter-filled, by varying the gate potential $V_g$, the crossover starts when the magnitude of the spin polarization of the double quantum dot, as measured by $\langle n_{\uparrow}\rangle -\langle n_{\downarrow}\rangle$, becomes finite. Although the external magnetic field breaks the $SU(4)$ symmetry of the Hamiltonian, the ground state preserves it in a region of $V_g$, where $\langle n_{\uparrow}\rangle -\langle n_{\downarrow}\rangle =0$. Once the spin polarization becomes finite, it initially increases slowly until a sudden change occurs, in which $\langle n_{\downarrow}\rangle$ (polarization direction opposite to the magnetic field) reaches a maximum and then decreases to negligible values abruptly, at which point an orbital $SU(2)$ ground state is fully established. This crossover from one Kondo state, with emergent $SU(4)$ symmetry, where spin and orbital degrees of freedom all play a role, to another, with $SU(2)$ symmetry, where only orbital degrees of freedom participate, is triggered by a competition between $g\mu_Bh$, the energy gain by the Zeeman-split polarized state and the Kondo temperature $T_K^{SU(4)}$, the gain provided by the $SU(4)$ unpolarized Kondo-singlet state.