Interplay of charge and spin in quantum dots: The Ising case

The physics of quantum dots is depicted succinctly by the universal Hamiltonian, where only zero-mode interactions are included. In the case in which the latter involve charging and isotropic spin-exchange terms, this would lead to a non-Abelian action. Here we address an Ising spin-exchange interaction, which leads to an Abelian action. The analysis of this simplified yet nontrivial model shed light on a more general case of charge and spin entanglement. We present a calculation of the tunneling density of states and dynamic magnetic susceptibility. We explain how the latter can be used for an experimental determination of the exchange interaction strength.

Figure 1

TDoS (in units of ${\nu }_0$) as a function of $\epsilon \equiv \varepsilon /E_c$ for $T=0.2E_c$ and $\Delta /T=0.1$ in (a) a CB valley (${\tilde{N}}_0=100$), (b) an intermediate region (${\tilde{N}}_0=100.35$), and (c) a CB peak (${\tilde{N}}_0=100.5$).

Figure 2

Dependence of the TDoS (in units of ${\nu }_0$) on the temperature (measured in $E_c$) at the bottom of a CB valley (${\tilde{N}}_0=70$) for $\Delta /E_c=0.02$.

Figure 3

$\frac{1}{\beta }\text{Im}{\chi }^{+-}$ as a function of frequency $\omega$ (in units of $\Delta$) for (a) $\frac{\Delta }{T}=0.1$ and $\frac{J}{\Delta }=0.1,0.2$, and 0.3 for the left, center, and rightmost curves, respectively, and (b) $\frac{J}{\Delta }=0.1$ and $\frac{\Delta }{T}=0.05,0.1$, and 0.2 for the top, center, and bottom curves, respectively.

Figure 4

Fit of numerically acquired data for FWHM to function $W=\alpha {\omega }_0$, yielding $\alpha =1.59$. $R^2$ for this fit is 0.999.