SU(3) fermions in a three-band graphene-like model

Two-dimensional graphene is fascinating because of its unique electronic properties. From a fundamental perspective, one among them is the geometric phase structure near the Dirac points in the Brillouin zone, owing to the SU(2) nature of the Dirac cone wave functions. We ask if there are geometric phase structures in two dimensions which go beyond that of a Dirac cone. Here we write down a family of three-band continuum models of noninteracting fermions which have more intricate geometric phase structures. This is connected to the SU(3) nature of the wave functions near threefold degeneracies. We also give a tight-binding free fermion model on a two-dimensional graphene-like lattice where the threefold degeneracies are realized at fine-tuned points. Away from them, we obtain new “three-band” Dirac cone structures with associated nonstandard Landau level quantization, whose organization is strongly affected by the non-SU(2) or beyond-Dirac geometric phase structure of the fine-tuned points.

Figure 1

The dispersion for $H_K^{\text{3A}}$ [Eq. (3)].

Figure 2

The dispersion for $H_K^{\text{3A}}$ [Eq. (3)] at a constant magnitude of momentum ($p=1$) about $\mathbf{p}=(0,0)$ [see Eq. (4)]. The arrows are a guide to move across the bands in a smooth way, where the rule is to follow the same colored arrows while crossing the line degeneracies.

Figure 3

Case 1: $l_1^{+}=0,n^{-}=0,f^{-}=l_1^{-}$ In the above, $g^{-}\ne f^{-}$ and $m_2^{-}\ne l_1^{-}$. When all of these quantities are equal to each other, we obtain the special case as in Fig. 1 corresponding to our starting point $H_K^{\text{3A}}$.

Figure 4

Case 2: Representative picture of the effect of $f^{-}\ne l_1^{-}$ and/or $n^{-}\ne 0$ for $l_1^{+}=0$.

Figure 5

Case 3a: $l_1^{+}\ne 0n^{-}=0,f^{-}=l_1^{-}$.

Figure 6

Case 3b: $l_1^{+}\ne 0,n^{-}=0,f^{-}\ne l_1^{-}$. The three pictures are in increasing order in the difference between $f^{-}$ and $l_1^{-}$.

Figure 7

Case 4 ($l_1^{+}\ne 0,n^{-}\ne 0$): First picture is for $n^{-}<n_{\text{cric}}^{-}$. Second picture is for $n^{-}=n_{\text{cric}}^{-}$. The third plot is when $n^{-}>n_{\text{cric}}^{-}$ and bottom two band gap out.

Figure 8

Case 5 ($n^{+}\ne 0$). This figure represents the case where $n^{+}<0$. If we change to $n^{+}>0$ then the bottom band becomes standalone.

Figure 9

The graphene-like lattice on which fermions live is shown above. Red points are $a$ sublattice sites, green points are $b$ sublattice points, blue points are $c$ sublattice sites. The underlying Bravais lattice is still triangular the same as graphene.

Figure 10

(Left) Dispersion of the graphene-like lattice Hamiltonian for $\delta t_0=0$ and $\delta t_1=0$ over the full Brillouin zone. Near the valley points $\vec{K},{\vec{K}}^{\prime }$, they reproduce the continuum band structure of $H_{\mu }^{\text{3A}}$ as in Fig. 1. (Right) A cut along $p_y=0$ is shown to better display the noncontractible loop on which the twofold line degeneracy lives.

Figure 11

For $\delta t_0=0.5$ and $\delta t_1=-0.25$ the dispersion has this behavior where the bottom band and the middle band has one Dirac point and the middle band and the top band has two Dirac points.

Figure 12

Three figures are for $\delta t_0=0.5$ and $\delta t_1=0.35.0.25,0.22$, respectively. As we see here the extra Dirac point travels and gets finally gapped out.

Figure 13

For $\delta t_0=0.5$ and $\delta t_1=-0.25$ the dispersion has this behavior along $k_y=0$ line.

Figure 14

Here we present the spectrum for $\delta t_0=0.5,\delta t_1=-0.25$ where $p=1,q=1000$. In the inset it shows that for the bottom band the spectrum is $\sim \sqrt{n}$ and they are doubly degenerate.

Figure 15

Orbit of the electron in the top band near the two same energy band degeneracy for the top band. The red points signify the location of the Dirac points.

Figure 16

Hofstadter butterfly for our model lattice with $\delta t_0=0.1$ and $\delta t_1=-0.1$. Here $x$ axis represents the flux in units of quantum flux enclosed by the unit cell and the $y$ axis represents the energy.