Features of spin-charge separation in the equilibrium conductance through finite rings

We calculate the conductance through rings with few sites $L$ described by the $t\text{−}J$ model, threaded by a magnetic flux $\Phi$ and weakly coupled to conducting leads at two arbitrary sites. The model can describe a circular array of quantum dots with large charging energy $U$ in comparison with the nearest-neighbor hopping $t$. We determine analytically the particular values of $\Phi$ for which a depression of the transmittance is expected as a consequence of spin-charge separation. We show numerically that the equilibrium conductance at zero temperature is depressed at those particular values of $\Phi$ for most systems, in particular at half filling, which might be easier to realize experimentally.

Figure 1

Scheme of the systems studied numerically. (a) $L=6$ and (b) 7. In both cases $M=3$.

Figure 2

(Color online) Energy levels for a system of four sites and three electrons for $J=0$. Solid lines correspond to the spin quantum number $n_s=0$, dashed lines to $n_s=-2$, and dotted lines to $n_s=-1$. The different colors indicate different charge configurations.

Figure 3

(Color online) Total integral of the transmittance for a system of $L=4$ sites and $N+1=4$ electrons for $J=0.01$, $t^{\prime }=0.3t$, and two positions of the leads at $M$.

Figure 4

Low-energy part of the transmittance as a function of gate voltage for a half-filled ring of $L=6$ sites, $J=0.03t$, $t^{\prime }=0.3t$, and $M=3$ near a level crossing of the intermediate states with $N=5$ particles. The crossing is at ${\varphi }_d\simeq 0.07743\pi$ and the different curves from left to right correspond to $\left(\varphi -{\varphi }_d\right)/\pi =-0.04$, $-0.02$, $-0.01$, 0, 0.01, 0.02, and 0.04.

Figure 5

(Color online) Intensity of the first peak in the transmittance as the gate voltage is lowered, $I_{\text{fp}}$, as a function of applied magnetic flux for a ring of six sites and six electrons, $M=3$ [Fig. 1a], $t^{\prime }=0.3t$, and several values of $J$. The curves with $J>0.001t$ are displaced vertically for the sake of clarity.

Figure 6

(Color online) Same as Fig. 5 for five electrons.

Figure 7

(Color online) Same as Fig. 5 for a ring of seven sites, seven electrons, and $M=3$ [Fig. 1b].