Magnetic properties of the $\alpha -T_3$ model: Magneto-optical conductivity and the Hofstadter butterfly

The $\alpha -T_3$ model interpolates between the pseudospin $S=1/2$ honeycomb lattice of graphene and the pseudospin $S=1$ dice lattice via parameter $\alpha$. We present calculations of the magnetic properties of this hybrid pseudospin model, namely the absorptive magneto-optical conductivity and the Hofstadter butterfly spectra. In the magneto-optics curves, signatures of the hybrid system include a doublet structure present in the peaks, resulting from differing Landau level energies in the $K$ and $K^{\prime }$ valleys. In the Hofstadter spectra, we detail the evolution of the Hofstadter butterfly as it changes its periodicity by a factor of three as we vary between the two limiting cases of the $\alpha -T_3$ model.

Figure 1

(a) The $\alpha -T_3$ lattice, in a perpendicular magnetic field $B$. Hopping between sites $A$ and $B$ (which form a HCL) takes place with strength $t$. Sites labeled $C$, located at the centers of the hexagons, are coupled only to $B$ sites with variable hopping amplitude $\alpha t$. (b) Landau level energies in units of ${\gamma }_B$ as a function of the parameter $\alpha$ for the first four values of $n$. The $K$ and $K^{\prime }$ valleys are shown in solid and dashed blue lines, respectively. The Landau levels of the flat band are plotted in red. (c) Low-energy dispersion about one $K$ point for $B=0.$

Figure 2

(a) Cone-to-cone interband and (b) flat band-to-cone overlap functions as a function of Fock number $n$ for $\alpha =0.75$. Overlap functions $f_1$ and $f_2$ are shown in purple and green, respectively. The large $n$ limit of these functions is given by Eq. (14) and denoted by a dotted red line in (a) at $C/2$ and in (b) at $1/4-C$. Here $C=0.2304$. The $K$- and $K^{\prime }$-valley components of the overlap functions are denoted by solid and dashed lines, respectively which are simply a guide to the eye as only the values at integer $n$ apply. In the inset (c) we show $C\left(\alpha \right)$.

Figure 3

Absorptive, longitudinal component of the optical conductivity [${\sigma }_{xx}\left(\omega \right)$] for a chemical potential of (a) $\mu =0.01{\gamma }_B$, (b) $\mu =0.5{\gamma }_B$, and (c) $\mu =1.01{\gamma }_B$ for a range of $\alpha$ values as shown. The flat band-to-cone contributions are shaded red, while the cone-to-cone contributions are shaded blue. Their sum is shown with a thin black curve. Calculations are done using a scattering rate of $\mathrm{\Gamma }=0.025{\gamma }_B$.

Figure 4

Same as Fig. 3, but for the absorptive part of the transverse component of the optical conductivity [${\sigma }_{xy}\left(\omega \right)$].

Figure 5

Absorptive, longitudinal component of the optical conductivity under magnetic fields up to 16 T, for $\alpha =0,0.5,0.75,1$. We used a scattering rate of $\mathrm{\Gamma }=2.5$ meV and a chemical potential that falls below the first positive valued Landau level for all values of $\alpha$ considered.

Figure 6

Snowshoe diagram [23] for $\alpha =0.25$. (a) Relative positions of LLs in the $K$ and $K^{\prime }$ valleys are shown with open blue circles connected by dashed and dotted lines, respectively. Horizontal dashed lines show the four chemical potentials considered in Figs. 7 to 10. Arrows represent possible transitions between LLs in the $K$ valley, assuming a chemical potential of $\mu =0.1{\gamma }_B$. (b) LL in the $K^{\prime }$ valley with arrows depicting all possible transitions for $\mu =0.1{\gamma }_B$.

Figure 7

Absorptive, diagonal component of the optical conductivity for $\alpha =0.25$ showing $\mu /{\gamma }_B=0.1$, 0.5, 1.0, and 1.2 from top to bottom, respectively. The flat band-to-cone (cone-to-cone) contributions are shaded red (blue), and their sum is represented by a thin black line. Vertical dashed (dotted) lines show the energies associated with a number of transitions in the $K$ ($K^{\prime }$) valley. Red, blue, and green vertical lines mark the energy of flat band-to-cone, cone-to-cone interband, and cone-to-cone intraband transitions, respectively. In order from left to right, the following transitions are marked: $T_{0_0,1_{+}}, T_{2_{+},3_{+}}, T_{1_{+},2_{+}}^{\prime }, T_{1_{+},2_{+}}, T_{0_0,1_{+}}^{\prime }, T_{1_0,2_{+}}, T_{1_{-},2_{+}}, T_{1_{-},2_{+}}^{\prime }, T_{2_{-},3_{+}}$. Only a subset of these are labeled above the plot.

Figure 8

Same as Fig. 7 but for the absorptive, off-diagonal component of the optical conductivity.

Figure 9

Same as Fig. 7 but for the absorptive, optical conductivity for right-hand polarized light.

Figure 10

Same as Fig. 7 but for the absorptive, optical conductivity for left-hand polarized light. The labels on the top have been changed to emphasize left-directed transitions.

Figure 11

DC Hall conductivity $\text{Re}{\sigma }_{xy}\left(\mu \right)$ for (a) $\alpha =0.5$ and (b) $\alpha =1$. The cone-to-cone and flat band-to-cone contributions are shown in blue and red, respectively, and their sum is shown by a green curve. We show agreement with our previously published results of the Hall conductivity that were obtained using the Streda formula in dashed magenta [15].

Figure 12

(a) The $\alpha -T_3$ lattice with three atoms per unit cell at sites $A, B$, and $C$, represented by blue, red, and green circles, respectively. Primitive lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ and basis vector $\delta$ are depicted by arrows originating from lattice site $(n_1,n_2)$. (b) Schematic comparing the smallest plaquette for the $\alpha -T_3$ lattice for $\alpha \ne 1$ (dotted rhombus) versus $\alpha =1$ (the entire hexagon).

Figure 13

Hofstadter butterflies for six representative $\alpha$ values calculated for $q$ up to 50.

Figure 14

Low-field Hofstadter spectra for (a) $\alpha =0$ and (b) $\alpha =0.25$. The red arrow in (b) highlights the splitting of the LLs that is visible in the $\alpha =0.25$ butterfly. For very low field, bands are compared with the spectra for (c) $\alpha =0$ and (d) $\alpha =0.25$ with $K$ and $K^{\prime }$ bands shown by solid and dashed lines, respectively. The low-field bands are given by Eq. (22).