Tunneling into quantum wires: Regularization of the tunneling Hamiltonian and consistency between free and bosonized fermions

Tunneling between a point contact and a one-dimensional wire is usually described with the help of a tunneling Hamiltonian that contains a $\delta$ function in position space. Whereas the leading-order contribution to the tunneling current is independent of the way this $\delta$ function is regularized, higher-order corrections with respect to the tunneling amplitude are known to depend on the regularization. Instead of regularizing the $\delta$ function in the tunneling Hamiltonian, one may also obtain a finite tunneling current by invoking the ultraviolet cutoffs in a field-theoretic description of the electrons in the one-dimensional conductor, a procedure that is often used in the literature. For the latter case, we show that standard ultraviolet cutoffs lead to different results for the tunneling current in fermionic and bosonized formulations of the theory, when going beyond leading order in the tunneling amplitude. We show how to recover the standard fermionic result using the formalism of functional bosonization and revisit the tunneling current to leading order in the interacting case.

Figure 1

Tunneling contact between a metallic contact and an interacting one-dimensional wire. In most of our discussion we take the wire to have a single chiral (i.e., unidirectional) mode. The inset shows a schematic representation of the unfolding procedure that allows the electrons in the metallic contact to be described as a one-dimensional chiral mode.

Figure 2

Sketch of the regularization schemes Eqs. (6) and (7) for the tunneling coupling Eq. (1). Choice II is sensitive to the respective chirality of electrons in the wire and in the contact and we make a distinction between choices IIa and IIb.

Figure 3

Conductances in Eq. (9) as a function of the tunneling parameter $t=\gamma /2\hslash \sqrt{uv}$. Different choices of the tunneling regularization are responsible for different qualitative behaviors in the limit $t\to \infty$, even if they all have the same behavior for $t\to 0$. The vertical dashed line signals the change of scale from $t$ to $1/t$ on the horizontal axis.

Figure 4

Numerical calculation of $\mathcal{I}$ in Eq. (34) as a function of $\omega =\alpha \mathrm{\Omega }/v$ for fixed $va/u\alpha =1$. Diamonds are calculated starting from Eq. (35), using the standard bosonization approach. Circles are calculated using the fermionic approach, using Wick's theorem for the four-point correlators in Eq. (17). The estimated errors of the numerical results are less than the symbols used to represent the data points. The limit $\alpha \to 0$ clearly differs for the two approaches.

Figure 5

Numerical calculation of $\mathcal{I}$ as a function of $va/u\alpha$. Diamonds are for the standard bosonization approach.

Figure 6

Spinless fermions on two semi-infinite lattices hopping on nearest neighbors with amplitude $\lambda$. The two lattices are coupled at their ends with hopping amplitude ${\lambda }^{\prime }\ne \lambda$. If ${\lambda }^{\prime }=0$, electrons in the two lattices are backscattered at the junction and cover the dashed path.