Enhancement of tunneling density of states at a Y junction of spin-$\frac{1}{2}$ Tomonaga–Luttinger liquid wires

We calculate the tunneling density of states (TDOS) in a dissipationless three-wire junction of interacting spin-1/2 electrons, and find an anomalous enhancement of the TDOS in the zero-bias limit, even for repulsive interactions for several bosonic fixed points. This enhancement is physically related to the reflection of holes from the junction for incident electrons, and it occurs only in the vicinity of the junction ($x<v_{\min }/2\omega$, where $v_{\min }$ is the minimum of the velocity of charge or spin excitations and $\omega$ is the bias frequency), crossing over to the bulk value, which is always suppressed at larger distances. The TDOS exponent can be directly probed in an STM experiment by measuring the differential tunneling conductance as a function of either the bias voltage or temperature as done in C. Blumenstein et al., Nat. Phys. 7, 776 (2011).

Figure 1

Schematic of a Y junction of three spin-1/2 TLL wires and a STM tip for measuring the TDOS. The red and green lines indicate the charge and spin sectors, respectively, and the interwire tunneling region is represented by the circular region.

Figure 2

Contour plot of the TDOS exponents in the vicinity of the junction, ${\mathrm{\Delta }}_0, {\mathrm{\Delta }}_1, {\mathrm{\Delta }}_2$, and ${\mathrm{\Delta }}_3$ in the ${\theta }_c-g_c$ plane for the SU(2) symmetric line $g_s=1$, in (a), (b), (c), and (d), respectively. Note that for $g_s=1$, the TDOS exponents in Eqs. (18) and (19) are independent of ${\theta }_s$. The fixed points corresponding to the dome shaped regions lying to the left of the $g_c=1$ vertical line in all four panels show an anomalous enhancement of the TDOS, even for repulsive interactions. The boxed numbers specify the numerical value of the exponent for the corresponding contour lines.

Figure 3

Contour plot of the TDOS exponents in the $g_s-g_c$ plane for various fixed points. (a), (b), (c), (d), and (e) display ${\mathrm{\Delta }}_0$ for various fixed points of ${\mathbb{M}}_{1c}{\mathbb{M}}_{1s}$ type, i.e., NN (${\theta }_c={\theta }_s=0$), DD (${\theta }_c={\theta }_s=\pi$), DN (${\theta }_c=\pi ,{\theta }_s=0$), ND (${\theta }_c=0,{\theta }_s=\pi$), and ${\chi }^{+}{\chi }^{+}$ (${\theta }_c={\theta }_s=2\pi /3$), respectively. (f) denotes TDOS exponent ${\mathrm{\Delta }}_1$ for the ${\mathrm{D}}_A{\mathrm{D}}_A$ fixed point of the ${\mathbb{M}}_{2c}{\mathbb{M}}_{2s}$ type. The region bounded by ${\mathrm{\Delta }}_i<1$, in all the panels signifies TDOS enhancement in the zero bias limit and we clearly see enhancement in the TDOS even for repulsive e-e interactions ($g_c<1$) in several cases. The boxed numbers specify the numerical value of the exponent for the corresponding contour lines.