Luttinger-Liquid Behavior in Tunneling through a Quantum Dot at Zero Magnetic Field

Thermodynamic and transport properties of a two-dimensional circular quantum dot are studied theoretically at zero magnetic field. In the limit of a large confining potential, where the dot spectrum exhibits a shell structure, it is argued that both spectral and transport properties should exhibit Luttinger liquid behavior. These predictions are verified by direct numerical diagonalization. The experimental implications of such Luttinger liquid characteristics are discussed.

Figure 1

Comparison of the numerically evaluated gap $\Delta$ in the quantum dot spectrum to the prediction of Luttinger liquid theory [Eq. (6)]. There is a single fitting parameter, $V_{p=2}=0.055$ (see text). Inset: Plot of $({\Delta }^2-v_F^2)/v_F$ vs the interaction strength. Luttinger liquid theory predicts a straight line, of slope $2V_2/\pi$.

Figure 2

The numerically evaluated tunneling matrix elements $T_N$, which are proportional to the Coulomb blockade peak amplitudes, for several values of the interaction strength $\alpha$. In agreement with Luttinger liquid theory, a power-law dependence of $T_N$ on $N$ is observed, with an exponent that increases with the strength of the interactions.

Figure 3

The numerically evaluated tunneling matrix exponent $\beta$, derived from Fig. 2, as a function of the interaction strength $\alpha$. The dependence upon $\alpha$ exhibits a good agreement with the predictions of Luttinger liquid theory, Eqs. (8, 9).