Renormalization group study of the conductances of interacting quantum wire systems with different geometries

We examine the effect of interactions between the electrons on the Landauer-Büttiker conductances of some systems of quantum wires with different geometries. The systems include a long wire with a stub in the middle, a long wire containing a ring which can enclose a magnetic flux, and a system of four long wires which are connected in the middle through a fifth wire. Each of the wires is taken to be a weakly interacting Tomonaga-Luttinger liquid, and scattering matrices are introduced at all the junctions present in the systems. Using a renormalization group method developed recently for studying the flow of scattering matrices for interacting systems in one dimension, we compute the conductances of these systems as functions of the temperature and the wire lengths. We present results for all three regimes of interest, namely, high, intermediate, and low temperature. These correspond, respectively, to the thermal coherence length being smaller than, comparable to, and larger than the smallest wire length in the different systems, i.e., the lengths of the stub or each arm of the ring or the fifth wire. The renormalization group procedure and the formulas used to compute the conductances are different in the three regimes. In particular, the dimensionality of the scattering matrix effectively changes when the thermal length becomes larger than the smallest wire length. We also present a phenomenologically motivated formalism for studying the conductances in the intermediate regime where there is only partial coherence. At low temperatures, we study the line shapes of the conductances versus the energy of the electrons near some of the resonances; the widths of the resonances are found to go to zero with decreasing temperature. Our results show that the Landauer-Büttiker conductances of various systems of experimental interest depend on the temperature and lengths in a non-trivial way when interactions are taken into account.

Figure 1

The stub system, showing two long wires labeled 1 and 3, and a stub labeled 2. The lower end of the stub where three wires meet and the upper end of the stub are denoted by $A$ and $B$, respectively.

Figure 2

The ring system, showing two long wires labeled 1 and 3, the two arms of the ring labeled 2 and 4, and two three-wire junctions labeled $A$ and $B$.

Figure 3

The four-wire system, showing four long wires labeled 1, 2, 3, and 4, a connecting wire in the middle labeled 5, and two three-wire junctions labeled $A$ and $B$.

Figure 4

Effective descriptions of the various systems at low temperature, $L_T>L_0$. The stub and ring systems effectively reduce to a two-wire system with a junction as in (a), while the four-wire system reduces to a four-wire system with a junction as in (b).

Figure 5

A wire with two voltage probes $P_1$ and $P_2$ at a point labeled $A$. The two probes cause phase randomization of right and left moving waves, respectively.

Figure 6

$T_{13}$ for the stub system as a function of $L_T∕L_S$ for $\alpha =0.2$, $L_S∕d=10$, $\eta =e^{i\pi ∕2}$, and different values of $r^{\prime }$. The four sets of curves are for $r^{\prime }=-0.10,-0.33,-0.54$, and $-0.80$ from top to bottom.

Figure 7

$T_{13}$ as a function of $L_T∕L_S$ for $\alpha =0.2$, $L_S∕d=10$, $r^{\prime }=-0.54$, and different values of $\eta$. The four sets of curves are for $\eta =-1,e^{i0.8\pi },e^{i0.6\pi }$, and 1 from top to bottom.

Figure 8

$T_{13}$ for the ring system as a function of $L_T∕L_S$ for $\alpha =0.2$, $L_S∕d=10$, $r^{\prime }=-0.18$, $\Phi =0$, and different values of $\eta$. The four coherent curves are for $\eta =1,e^{i0.08\pi },e^{i0.16\pi }$, and $e^{i0.4\pi }$ from top to bottom. The incoherent curve is the same for all $\eta$.

Figure 9

$T_{13}$ as a function of $2k_F{L}_S\mathrm{mod}2\pi$ for $\alpha =0.2$, $L_S∕d=10$, $r^{\prime }=-0.19$, $\Phi =0$, and different values of $L_T∕L_S$. The three sets of curves are for $L_T∕L_S=1,50$, and ${10}^5$ from top to bottom.

Figure 10

$T_{13}$ as a function of $2k_F{L}_S\mathrm{mod}2\pi$ for $\alpha =0.2$, $L_S∕d=10$, $\Phi =0.5\pi$, and different values of $L_T∕L_S$. The four sets of curves are for $L_T∕L_S=1,50,3000,$ and ${10}^6$ from top to bottom. Note that there is a transmission zero at $2k_F{L}_S=0$ mod $2\pi$.

Figure 11

$T_{13}$ as a function of $2k_F{L}_S\mathrm{mod}2\pi$ for $\alpha =0.2$, $L_S∕d=10$, $\Phi =0$, and different values of $L_T∕L_S$. The four sets of curves are for $L_T∕L_S=1,150,\mathrm{20\hspace{0.17em}000}$, and ${10}^6$ from top to bottom. Note that there is a transmission zero at $2k_F{L}_S=0$ mod $2\pi$.

Figure 12

$T_{13}$ as a function of the scaled variable $[2k_F{L}_S$ mod $2\pi ]{(L_T∕L_S)}^{\alpha }$ for $\alpha =0.2$, $L_S∕d=10$, $r^{\prime }=-0.32$, $\Phi =0$, and different values of $L_T∕L_S$. The four sets of curves are for $L_T∕L_S=1,2.7,7.4$, and $20$ from top to bottom. The inset shows the same plots without scaling, i.e., $T_{13}$ as a function of $2k_F{L}_S$.

Figure 13

$T_{13}$ for the four-wire system as a function of $L_T∕L_S$ for $\alpha =0.2$, $L_S∕d=10$, and $\eta =1$. The four sets of curves (incoherent and interpolating) are for $r^{\prime }=-0.43,-0.28,-0.17$, and $-0.07$ from top to bottom. The coherent curve stays at $1∕4$ for all values of $r^{\prime }$ since $\eta =1$.

Figure 14

$T_{13}$ as a function of $L_T∕L_S$ for $\alpha =0.2$, $L_S∕d=10$, and $r^{\prime }=-0.28$. The four sets of curves (interpolating and coherent) are for $\eta =1,e^{i0.3\pi },e^{i0.6\pi }$, and $e^{i0.9\pi }$ from top to bottom. The incoherent curve is independent of $\eta$.

Figure 15

$T_{13}$ as a function of $L_T∕L_S$ for $\alpha =0.2$, $L_S∕d=10$, $r^{\prime }=-0.33$, and $\eta =e^{i0.52\pi }$. $T_{13}$ first increases and then decreases at very low temperatures.