Hofstadter butterfly evolution in the space of two-dimensional Bravais lattices

The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold-atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work [F. Yılmaz et al., Phys. Rev. A 91, 063628 (2015)], we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit-cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point-symmetry groups. We model the optical lattice with a sinusoidal real-space potential and obtain the tight-binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e., the triangular and the square lattices, displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transitions from the square to rectangular lattice and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.

Figure 1

Five Bravais lattices created by the potential in Eq. (3). From left to right, oblique, centered-rectangular (rhombic), triangular, rectangular, and square lattices. Lattice depths are $V_x=V_y=50$ (units of $E_R$), where $E_R={\hslash }^2/2m{\lambda }_1^2$ is the recoil energy. The unit cell and the primitive vectors are shown with thick lines. All lengths are scaled by ${\lambda }_1$ in Eq. (1).

Figure 2

The TB model with eight neighbors represented on the contour plot of the potential. The bold arrows are the primitive translation vectors. The TB model hopping parameters are denoted with four different symbols, $t_0,t_1,t_2$, and $t_3$.

Figure 3

The space of all possible 2D Bravais lattices, parametrized as a function of the angle between the primitive vectors and the ratio of their length. Lattices of high symmetry are shown by lines and points. The triangular lattice is shown as a yellow (T) point, and the square lattice is shown as a green (S) point. The rectangular lattice is shown by the light blue dashed curve, and the centered rectangular lattice is shown as two thick purple solid curves.

Figure 4

Evolution of the energy spectrum between the square lattice and the triangular lattice in four representative points for $\theta$. One of the major energy gaps disappears as the triangular lattice is approached. The closing is accompanied by infinitely many smaller gap closures. These results are in agreement with [9] but show that the transition in terms of the angle $\theta$ happens mainly near the triangular lattice limit.

Figure 5

The hopping amplitudes calculated along the transition between the square lattice and the triangular lattice as a function of $\theta$. The parameters are calculated for $V_X=V_Y=20$ (units of $E_R$). The NN hopping parameters $t_0,t_1$ have the same magnitude and decrease until they are equal to $t_2=0.43$ (units of $E_R$) in the triangular-lattice limit. The NNN hopping $t_3$ is an order of magnitude smaller, $t_3\approx 0.025$.

Figure 6

A representative energy spectrum of the oblique lattice. We display the spectrum from $\varphi =[-1,1]$ to emphasize the doubling of the $\varphi$ periodicity and the symmetry under $\varphi \to -\varphi$. All 2D Bravais lattices have this symmetry, which follows from inversion.

Figure 7

The energy gaps are calculated in the rectangular limit compared with perturbative treatment of 1D chains. In the first order, the spectrum has two robust energy gaps indicated by thick blue curves. The smaller gaps in the spectrum can be generated by higher orders in perturbative tunneling.

Figure 8

An overall picture of the evolution of the Hofstadter butterfly in the phase space of two-dimensional Bravais lattices.

Figure 9

Evolution of the energy spectrum from the square lattice to the rectangular lattice and from the triangular lattice to the centered rectangular lattice. The first transition is along the $\theta =\pi /2$ line, while the second transition is calculated along the centered rectangular lattice curve shown in Fig. 3. The greater the ratio of the primitive vectors is, the closer the system is to one-dimensional chains (see Fig. 7), where the subband splitting is weak.

Figure 10

The evolution of the energy bands for $\varphi =\frac{1}{3}$ and $\varphi =\frac{1}{5}$ on the path $\overline{ATSBA}$ shown in Fig. 3. Gaps are labeled with their Chern numbers. For $\varphi =1/3$, there are two topologically distinct regions, characterized by $1,-1$ and 1,2. For $\varphi =1/5$, there are four topologically distinct regions.

Figure 11

The topological regions of the two-dimensional Bravais lattice phase space for $\varphi =1/3$. All lattices can be classified into two regions: those with Chern numbers $-1,1$, including the square lattice and the rectangular lattice, and those with 2,1, including the triangular lattice and the centered rectangular lattice. The region which contains the triangular lattice roughly tracks the centered rectangular lattice shown by the purple dashed curve.