Interaction-induced conductance from zero modes in a clean magnetic graphene waveguide

We consider a waveguide formed in a clean graphene monolayer by a spatially inhomogeneous magnetic field. The single-particle dispersion relation for this waveguide exhibits a zero-energy Landau-like flat band, while finite-energy bands have dispersion and correspond, in particular, to snake orbits. For zero-mode states, all matrix elements of the current operator vanish, and a finite conductance can only be caused by virtual transitions to finite-energy bands. We show that Coulomb interactions generate such processes. In stark contrast to finite-energy bands, the conductance is not quantized and shows a characteristic dependence on the zero-mode filling. Transport experiments thereby offer a novel and highly sensitive probe of electron-electron interactions in clean graphene samples. We argue that this interaction-driven zero-mode conductor may also appear in other physical settings and is not captured by the conventional Tomonaga-Luttinger liquid description.

Figure 1

Schematic sketch of the MGW setup viewed from above. A magnetic field $+B{\widehat{\mathit{e}}}_z$ is applied everywhere in the graphene plane except for the waveguide region $\left|x\right|<d/2$, where the field is reversed. Near the field switching lines, quantum modes corresponding to classical snake orbits are realized, while only Landau-quantized cyclotron orbits are present far away from these lines. Several classical orbits corresponding to $n=1$ are schematically illustrated. The system dimensions are $L_x$ and $L_y$, respectively.

Figure 2

Dispersion relation of the MGW for $d=2l_B$. Only the marked three bands ($n=-1,0,1$) will be kept later on in Sec. 4c, assuming that the zero mode is partially filled. We note that the $n=1$ and $n=2$ bands are separated by an avoided crossing not apparent on the shown scale. Energy (momentum) is given in units of $E_B$ ($l_B^{-1}$), see Eq. (4).

Figure 3

Main panel: Single-particle gap $E_g$ vs waveguide width $d$ in units as in Fig. 2. Inset: Long-wavelength part of the dispersion relation for $n=1$ and several values of $d$.

Figure 4

Self-consistent zero-mode HF energies ${\varepsilon }_k$ (in units of $E_B$) for various filling factors $\nu$. The red dashed horizontal lines denote the chemical potential at the respective filling, with all lower states occupied. We consider $\alpha =0.5, d=2l_B, R=d, L_x=16l_B$, and $N_s=400$ basis states, see Eq. (24). This parameter set corresponds to $L_y=209.4l_B$, where $\left|k\right|=\pi N_s/L_y=6l_B^{-1}$ represents the zone boundary.

Figure 5

Interaction-induced Fermi momentum $k_F$ vs filling factor $\nu$ for the parameters in Fig. 4. The dotted curve is a guide to the eye only. Note the approximately linear $\nu$ dependence. The inset shows the effective Fermi velocity $v$ vs $\nu$, where $v$ is given in units of graphene's Fermi velocity $v_F$.

Figure 6

Diagrammatic expansion of the conductance (see main text). (a) The zeroth-order ($m=0$) diagram, while (b) and (c) refer to the two first-order ($m=1$) diagrams.

Figure 7

Same as Fig. 6 but for the second-order ($m=2$) diagrams.

Figure 8

Zero-mode waveguide conductance $G$ (in units of $G_0=e^2/h$) vs filling $\nu$, see Eq. (52), for two values of the fine structure constant $\alpha$. Dotted curves are guides to the eye only. All other parameters are as in Fig. 4. The inset shows the $\alpha$ dependence at two selected fillings.