Fermi-edge singularity in chiral one-dimensional systems far from equilibrium

We study the effects of strong coupling of a localized state charge to one-dimensional electronic channels out of equilibrium. While the state of this charge and the coupling strengths determine the scattering phase shifts in the channels, the nonequilibrium partitioning noise induces the tunneling transitions to the localized state. The strong coupling leads to a nonperturbative backaction effect which is manifested in the orthogonality catastrophe and the Fermi-edge singularity in the transition rates. We predict an unusually pronounced manifestation of the non-Gaussian component of noise that breaks the charge symmetry, resulting in a nontrivial shape, and a shift of the position of the tunneling resonance.

Figure 1

Scheme of the experimental setup: a Mach-Zender interferometer and a quantum dot are formed by the 1D electronic channels (solid black lines with arrows) on the edges of the 2D electron gas (shaded areas) in the quantum Hall regime at filling factor $\nu =2$. The quantum dot with the single relevant level at ${\varepsilon }_0$ close to the Fermi energy is Coulomb interacting (green lines) with the surrounding channels $a=U,D,L,R$, with coupling strengths $U_a$, and has a weak tunnel coupling with amplitudes ${\tau }_{\mathrm{L},\mathrm{R}}$ with the inner channels (red dotted lines). Voltage bias $\Delta \mu$ is applied to one of the outer channels, which creates nonequilibrium excitations after the left partitioning quantum point contact (labeled ${\text{QPC}}_{\mathrm{L}}$) with transparency $T$. The visibility of the interference pattern with respect to the Aharonov-Bohm phase ${\varphi }_{\mathrm{AB}}$ is observed after the right QPC.

Figure 2

The transition rates ${\Gamma }_{+}$ and ${\Gamma }_{-}$ (in arbitrary units) as functions of the level energy parameter ${\varepsilon }_0$ for symmetrical screening ${\eta }_{\mathrm{D}}=0.5$ and (left) exponent $\alpha =0.5$ and (right) $\alpha =-0.5$ at different transparencies $T=0.15$ (blue) and $T=0.85$ (red). Analytic results in the Markovian limit are shown by the dashed lines, and the numerical data are shown by the solid lines. As expected for screening ${\eta }_{\mathrm{D}}=0.5$, the positions of the corresponding peaks for $T\to 0$ and $T\to 1$ are located at ${\varepsilon }_0=0$ and ${\varepsilon }_0=-\Delta \mu /2$, respectively. While the analytic curves maintain consistent power-law tails on both sides, the nonequilibrium effects are represented by the characteristic suppression of the tails of the numerical curves; see, for example, $T=0.85$ at $\varepsilon =\Delta \mu /2$ (red line, top left panel and inset) and at $\varepsilon =-\Delta \mu$ for $T=0.15$ (blue line, bottom left panel and inset). Note the symmetry (15) between rates ${\Gamma }_{+}$ and ${\Gamma }_{-}$ for small and large transparency.

Figure 3

The visibility profiles found from the analytics (dashed lines) and numerics (solid lines) for transparencies $T=0.15$ (blue) and $T=0.85$ (red) for $\alpha =-0.5$ and screening ${\eta }_{\mathrm{D}}=0.5$ as in the experiment [5], corresponding to the phase shift of $\pi$ on the upper arm; hence the complete loss of visibility is achieved. Each analytic curve has a mirror symmetry and is also identical to its equivalent for $\alpha =0.5$ due to the symmetry incidental of the Markovian approximation. The numerical results manifest a well-pronounced asymmetry of the dip due to the non-Gaussian effects at strong coupling.

Figure 4

The tunneling conductance (in arbitrary units) for $T=0.15$ (blue) and $T=0.85$ (red) for the Markovian limit (dashed lines) and the numerics (solid lines) for (top) $\alpha =0.5$ and (bottom) $\alpha =-0.5$. Notice how the numeric curves hit the horizontal axis at ${\varepsilon }_0=\Delta \mu$ and ${\varepsilon }_0=-\Delta \mu /2$. For $\alpha =-0.5$ the Markovian limit breaks down considerably sooner, which explains the noticeable discrepancy between the numerics and analytic plots.

Figure 5

The position ${\varepsilon }_0^{☆}/\Delta \mu$ of the visibility dip center as a function of the QPC transparency. Solid lines are the numerical results for $\alpha =0.5$ (blue) and $\alpha =-0.5$ (red). The dashed line is for the Markovian limit, although it is applicable only asymptotically for $T\to 0$ and $T\to 1$. The dotted line presents the Gaussian case with suppression of the higher cumulants, reflecting simply the average charge density on the channel as the partitioning QPC transparency changes.