Chiral Y junction of Luttinger liquid wires at strong coupling: Fermionic representation

We calculate the conductances of a three-terminal junction setup of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength $g_3\ne g$ in the third wire. We employ the fermionic representation in the scattering state picture, allowing for a direct calculation of the linear response conductances, without the need of introducing contact resistances at the connection points to the outer ideal leads. The matrix of conductances is parametrized by three variables. For these we derive coupled renormalization group (RG) equations, by summing up infinite classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y junction) is analyzed in detail. For repulsive interaction ($g,g_3>0$) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction ($g<0$ and/or $g_3<0$) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line $g=g_3$, at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.

Figure 1

The RG phase portrait, showing regions with different stable fixed points. Diagonal red stripes correspond to stable $N$ point; vertical green stripes, to $A$ point; horizontal blue stripes, to ${\chi }^{\pm }$ points. Gray-shaded and checkerboard regions indicate stable $M$ point.

Figure 2

Allowed values of conductances, $a,b,c$, lie inside the depicted body, $B$. The location of the RG fixed points $N$, $A$, ${\chi }^{\pm }$, $M$ is also shown. The line of fixed points connecting FPs $A$ and ${\chi }^{+}$ is indicated in yellow.

Figure 3

Allowed values of conductance $a$, $c$ for the isotropic case. The $S$ matrices (B1) characterized by parameters $\Gamma$, $\varphi$ lead to two quantities $a$ and $c$, as described by (B9). We took 500 pairs $\Gamma$, $\varphi$ at random; these are blue points on the plot. As discussed in Appendix , the domain $\Delta$ is bounded by a deltoid curve.

Figure 4

RG flows for the fully symmetric chiral case, for different values of interaction. The four starting points are the same in all plots. The nonuniversal positions of the chiral $C^{\pm }$ points are shown by open circles.

Figure 5

The RG phase portrait, showing regions with different set and character of the FPs. The region of attractive interactions is shown in more detail in the second panel. The numbers in the plot refer to its description in Table . The three dots correspond to the special points $K=K_3=2$, $K=K_3=3$, and $K=2$, $K_3=4/3$, discussed in the text.

Figure 6

The RG phase portrait, showing regions with different character of stable FPs. The numbers in the plot refer to the description in Table . The three dots mark the points $K=K_3=2$, $K=K_3=3$, and $K=2$, $K_3=4/3$, discussed in the text. The three straight red lines correspond to $(2K+K_3)/3=2,3,4$, respectively. The dashed line indicates the condition $K=K_3$.

Figure 7

The position of the Dirichlet $D$ point is shown in relation to the body of allowed conductances in the asymmetric nonchiral case. The fixed points $N$, $A_i$, $M$ are also shown.

Figure 8

Feynman diagrams depicting the integral equations for the renormalized interaction, Eqs. (C2) and (C1). A simple wavy line stands for the initial interaction matrix $\mathbf{g}$, and the double wavy line defines the quantity $\mathbf{L}$, leading to the effective interaction strength in the RG equations for the conductances.