Tunneling into a Finite Luttinger Liquid Coupled to Noisy Capacitive Leads

Tunneling spectroscopy of one-dimensional interacting wires can be profoundly sensitive to the boundary conditions of the wire. Here, we analyze the tunneling spectroscopy of a wire coupled to capacitive metallic leads. Strikingly, with increasing many-body interactions in the wire, the impact of the boundary noise becomes more prominent. This interplay allows for a smooth crossover from standard 1D tunneling signatures into a regime where the tunneling is dominated by the fluctuations at the leads. This regime is characterized by an elevated zero-bias tunneling alongside a universal power-law decay at high energies. Furthermore, local tunneling measurements in this regime show a unique spatial dependence that marks the formation of plasmonic standing waves in the wire. Our result offers a tunable method by which to control the boundary effects and measure the interaction strength (Luttinger parameter) within the wire.

Figure 1

(a) A 1D metallic quantum wire of length $L$ is connected to metallic leads, depicted as an outer circuit that is characterized by an Ohmic resistance $R$ and the capacitance to the ground $C$. The leads act as electron reservoirs with well-defined Fermi-Dirac distributions. The tunneling density of states [Eq. (1)] at position $x$ along the wire is probed by a nearby scanning tunneling microscope (STM). (b) The zero temperature power spectral-density $S(\omega )$ of the RC-circuit’s noise [Eq. (3)] (blue solid line) and two asymptotic limits: (i) $\omega {\tau }_{\mathrm{RC}}\ll 1$ (orange dot-dashed line) corresponding to the behavior of ideal Ohmic leads [73] and (ii) $\omega {\tau }_{\mathrm{RC}}\gg 1$ (red dashed line) where high-energy fluctuations are damped by the circuit’s capacitance. (c) (Top) An electron from the STM induces 1D plasmonic excitations, for which the finite wire acts as an effective Fabry-Pérot interferometer with reflection (transmission) coefficients $r_{A,B}$ ($t_{A,B}$). (Bottom) A schematic view of the wire as left- and right-propagating modes connected to two identical leads, that impart current fluctuations $\delta j_{L/R}(t)$ onto the wire [Eq. (3)].

Figure 2

The Green’s function of the wire in the large-capacitance limit. (a) The imaginary (thin green line) and real (thick red line) part of a lesser Green’s function $G^{<}(L/2,t)$ [Eq. (7)]. The dashed lines show the analytically obtained asymptotic limits for long ($t\gg {\tau }_{\mathrm{RC}}$) and short ($t\ll {\tau }_{\mathrm{RC}}$) times. The shaded region (light blue) marks the time interval $\tau <t<{\tau }_{\mathrm{RC}}$ where the interaction-induced correlations in the wire compete with the RC noise. (b) The real part of $\log [\tilde{g}(x,t)]$ [Eq. (4)] exhibiting the nontrivial power-law behavior of the Green’s function depending on the position of the STM tip (solid lines). Furthermore, our analytical asymptotic result (dashed lines) [Eq. (11)] agrees with the numerical result (solid lines). In all plots, we use an experimentally realizable interaction parameter $U/v_F=15$, see, e.g., Refs. [33, 34], and large capacitance, ${\tau }_{\mathrm{RC}}/\tau =100$.

Figure 3

(a) The normalized TDOS $\nu /{\nu }_0$ in the large-capacitance regime (${\tau }_{\mathrm{RC}}/\tau =100$) calculated for five different STM positions in the wire. At high energies, $\omega \gg 2\pi /{\tau }_{\mathrm{RC}}$, the tunneling is suppressed and the TDOS exhibits a power-law decay. Interaction-induced Fabry-Pérot oscillations with a period of $2\pi /\tau$ are present at high energies. For low energies, the TDOS is constant that depends on the value of ${\tau }_{\mathrm{RC}}$. (b) An enlargement on (a) where the TDOS is rescaled by a factor ${\omega }^3$ such that the difference between different measuring positions inside the wire can be seen more clearly. (c) The TDOS of a finite (blue solid line) and infinite (black dashed line) TLL when the capacitance in the leads is set to zero. In a finite-length TLL, the zero-bias TDOS does not vanish but saturates at a finite value [57]. Note that the normalization of the TDOS is with respect to the value of noninteracting TDOS with vanishing capacitance, ${\nu }_0$. The interaction strength used in all plots is $U/v_F=15$.