Magnetic Field Effects on Transport Through a Strongly Correlated Dot Coupled to Luttinger Leads

Abstract

The effects of the magnetic field and intralead electron interaction on the transport have been investigated in the Kondo regime by means of the nonequilibrium Green functions with the equation of motion method. The numerical results show that for weak intralead interaction, in addition to the Kondo dip zero bias, the differential conductance shows two peaks separated from the origin by the Zeeman energy. When the intralead interaction becomes moderately strong, the Zeeman splitting peaks turn into Zeeman splitting dips. With the further increase in the intralead interaction, all the dips disappear and the differential conductance is characterized by a power law scaling in bias voltage in the limit of the strong intralead interaction. Our results are valuable for analyzing transport in carbon nanotubes.

Appendix

In Ref. [17], a conclusion was drawn that ”at the particle–hole symmetric point the EOM Green’s function was temperature independent” [47,48,49]. Here, we show that this conclusion is not right.

At first, we review the relevant formulas in Ref. [17], the Hamiltonian was

$$\begin{aligned} H= & {} \sum \nolimits _\sigma {({\varepsilon _0} + \sigma h)} {n_{\mathrm{{d}}\sigma }} + (U/2)\sum \nolimits _\sigma {{n_{\mathrm{{d}}\sigma }}} {n_{d\bar{\sigma }}}\nonumber \\&+ \sum \nolimits _{n\sigma } {{\varepsilon _{n\sigma }}a_{n\sigma }^\dag {a_{n\sigma }}} - \sum \nolimits _{mn\sigma } {{J_{mn}}a_{m\sigma }^\dag {a_{n\sigma }}} - \sum \nolimits _{n\sigma } {({J_{n\sigma }}d_{n\sigma }^\dag {a_{n\sigma }}} + J_{n\sigma }^*a_{n\sigma }^\dag {d_\sigma }).\nonumber \\ \end{aligned}$$

(A1)

The retarded Green’s function of the quantum dot was defined as ${G_\sigma } = \langle \langle {d_\sigma };d_\sigma ^\dag \rangle \rangle$. Its equation of motion was that

$$\begin{aligned} (z - {\varepsilon _{\mathrm{{d}}\sigma }}){G_\sigma } = 1 + U\langle \langle {n_{\mathrm{{d}}\bar{\sigma }}}{d_\sigma };d_\sigma ^\dag \rangle \rangle - \sum \nolimits _n {{J_{n\sigma }}\langle \langle {a_{n\sigma }}} ;d_\sigma ^\dag \rangle \rangle . \end{aligned}$$

(A2)

The equation of motion of the Green’s function $\langle \langle {a_{n\sigma }};d_\sigma ^\dag \rangle \rangle \mathrm{{ }}$ was that

$$\begin{aligned} (z - {\varepsilon _{n\sigma }})\langle \langle {a_{n\sigma }};d_\sigma ^\dag \rangle \rangle = - \sum \nolimits _m {{J_{nm}}\langle \langle {a_{m\sigma }}} ;d_\sigma ^\dag \rangle \rangle - J_{n\sigma }^*{G_\sigma }. \end{aligned}$$

(A3)

After defining matrices ${{(H_{\text {net}}^{\sigma })}_{mn}}={{\varepsilon }_{n\sigma }}{{\delta }_{mn}}-{{J}_{mn}}$ and

$$\begin{aligned} {{M}_{mn\sigma }}(z)={{[{{(z-H_{\text {net}}^{\sigma })}^{-1}}]}_{mn}}. \end{aligned}$$

(A4)

The following equation was reached,

$$\begin{aligned} \langle \langle {a_{n\sigma }};d_\sigma ^\dag \rangle \rangle = - \sum \nolimits _m {{M_{nm\sigma }}(z)} J_{m\sigma }^*{G_\sigma }.\mathrm{{ }} \end{aligned}$$

(A5)

Then, a self-energy was defined:

$$\begin{aligned} {{\Sigma }_{\sigma }}(z)=\sum \nolimits _{nm}{{{J}_{n\sigma }}{{M}_{nm\sigma }}}(z)J_{m\sigma }^{*}. \end{aligned}$$

(A6)

Equations (A3), (A4), (A5) and (A6) here are just Eqs. (A3), (A4), (A5) and (A7) in Ref. [17]. After some other steps, the expression of ${G_\sigma }$ was derived as $G_{\mathrm{d}\sigma }^< (\omega ) = G_{\mathrm{d}\sigma }^r(\omega ){\Sigma ^ < }(\omega )G_{\mathrm{d}\sigma }^a(\omega )$ in [17]. Let us consider the cases of $2{{\varepsilon }_{0}}+U=0$ and $h=0$. At this point, the Anderson Hamiltonian becomes particle–hole symmetric. Then, the expression of ${G_\sigma }$ was simplified to be

$$\begin{aligned} {{[{{G}_{\sigma }}(z)]}^{-1}}=z-{{\Sigma }_{\sigma }}(z)-\frac{{{U}^{2}}}{4[z-{{\Sigma }_{\sigma }}(z)-2{{\Sigma }_{{\bar{\sigma }}}}(z)]}, \end{aligned}$$

(A7)

which was Eq. (14) in [17]. This expression was thought to be temperature independent, because ${M_{mn}}_\sigma \left( z \right)$ was so [17].

Now we present another expression of the ${M_{mn}}_\sigma \left( z \right)$. By the method given in Ref. [47], it is easy to give

$$\begin{aligned} \langle \langle {a_{n\sigma }};d_\sigma ^\dag \rangle \rangle = - \sum \nolimits _m {\langle \langle } {a_{n\sigma }};a_{m\lambda }^\dag \rangle \rangle J_{m\lambda }^*{G_\sigma }.\mathrm{{ }} \end{aligned}$$

(A8)

Comparison of Eqs. (A5) and (A8) demonstrates that

$$\begin{aligned} {M_{nm\sigma }} = \langle \langle {a_{n\sigma }};a_{m\sigma }^\dag \rangle \rangle . \end{aligned}$$

(A9)

Therefore, the ${{M}_{mn}}_{\sigma }\left( z \right)$’s are actually the Green’s functions of the conductors. The Green’s functions are certainly temperature dependent, so that the ${{M}_{mn}}_{\sigma }\left( z \right)$’s do, and the ${{\sum }_{\sigma }}\left( z \right)$ as well. The conclusion is that Eq. (A7) is temperature dependent.

Why was Eq. (A7) regarded to be temperature independent in [17]? This was because the ${{M}_{mn}}_{\sigma }\left( z \right)$’s were believed so. Actually, they were not. By the definition Eq. (A4), it is seen that ${{M}_{mn}}_{\sigma }\left( z \right)$’s are the matrix elements of ${{(z-H_{\text {net}}^{\sigma })}^{-1}}$. The matrix elements mean the transition of ${{M}_{mn\sigma }}(z)={{[{{(z-H_{\text {net}}^{\sigma })}^{-1}}]}_{mn}}$ between states. Particularly, the diagonal elements are the average values in the states, which are certainly temperature dependent.