Evolution of the Hofstadter butterfly in a tunable optical lattice

Recent advances in realizing artificial gauge fields on optical lattices promise experimental detection of topologically nontrivial energy spectra. Self-similar fractal energy structures generally known as Hofstadter butterflies depend sensitively on the geometry of the underlying lattice, as well as the applied magnetic field. The recent demonstration of an adjustable lattice geometry [L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, Nature (London) 483, 302 (2012)] presents a unique opportunity to study this dependence. In this paper, we calculate the Hofstadter butterflies that can be obtained in such an adjustable lattice and find three qualitatively different regimes. We show that the existence of Dirac points at zero magnetic field does not imply the topological equivalence of spectra at finite field. As the real-space structure evolves from the checkerboard lattice to the honeycomb lattice, two square-lattice Hofstadter butterflies merge to form a honeycomb lattice butterfly. This merging is topologically nontrivial, as it is accomplished by sequential closings of gaps. Ensuing Chern number transfer between the bands can be probed with the adjustable lattice experiments. We also calculate the Chern numbers of the gaps for qualitatively different spectra and discuss the evolution of topological properties with underlying lattice geometry.

Figure 1

The real-space potential and corresponding tight-binding lattice. The unit cell remains square for all experimental parameters [8]. The tunneling amplitudes $t_0,t_1$ are calculated by a fit to numerically obtained two lowest band energies. For $V_{\overline{x}}=14,V_x=0.79,V_y=6.45$ in units of $E_R$ and $\alpha =0.9$, we find $t_0=0.0391,t_1=0.0361$. Within a unit cell, two well-localized Wannier functions at potential minima ($A,B$) form the basis for tight-binding treatment. Modification of the potential depth parameters changes the distance $d$ between the sites $A$ and $B$, altering the tunneling rates between these sites.

Figure 2

Energy spectrum as a function of flux per unit cell. For $V_{\overline{x}}=4.5,V_x=1,V_y=5$, and $\alpha =0.95$, creating a lattice in the Dimer regime. All energies are in units of $E_R$; flux is measured in units of flux quantum. Tunneling amplitudes are $t_0=0.3488$ and $t_1=0.0843$ in units of recoil energy. The spectrum is formed by two square-lattice Hofstadter butterflies separated by a large gap controlled by $t_0$. The widths of both butterflies are proportional to $t_1$. The inset is used to display energy bands as a function of $k_x,k_y$ at zero magnetic field. The gap at zero magnetic field is mostly retained at nonzero field.

Figure 3

Hofstadter butterfly for $V_{\overline{x}}=12,V_x=0.79,V_y=6.45$, and $\alpha =0.9$, which is at the boundary of the honeycomb regime. Tunneling amplitudes are $t_0=0.0691$ and $t_1=0.0361$ in units of recoil energy. Comparing with Fig. 2, one can see that the two stacked Hofstadter butterflies get closer and touch only at the edges. As displayed in the inset, there are Dirac points at zero magnetic-field band structure. However, for all nonzero magnetic fields, there is a finite gap at zero energy.

Figure 4

Hofstadter butterfly for $V_{\overline{x}}=13.5,V_x=0.79,V_y=6.45$, and $\alpha =0.9$, deeper in the honeycomb regime. Tunneling amplitudes are $t_0=0.0441$ and $t_1=0.0361$ in units of recoil energy $E_R$. The central gap is closed at the edges ($\mathrm{\Phi }=0,1$) and the flux values $p/q=1/2,1/3,2/3$ but nowhere else. For $\frac{t_1}{t_0}=0.8186$, Eq. (19) shows that all fluxes with $q\ge 4$ have a gap at zero energy. The central gap is now split into four parts. Comparing the inset with the almost identical inset of Fig. 3 reveals the inability of zero magnetic-field band structure to determine the spectrum for nonzero field.

Figure 5

Hofstadter butterfly for $V_{\overline{x}}=14,V_x=0.79,V_y=6.45$, and $\alpha =0.9$, the parameters used to simulate graphene [8]. Tunneling amplitudes are $t_0=0.0391$ and $t_1=0.0361$ in units of recoil energy. For these values, the spectrum nearly matches the ideal honeycomb butterfly Fig. 6, except for minor gaps. The central gap is open for the flux values with denominator $q\ge 9$, producing narrow gaps near zero energy. As seen in the inset, the change in the zero-field band structure is only due to the movement of Dirac points.

Figure 6

Hofstadter butterfly of the ideal honeycomb lattice ($t_0=t_1$) as obtained in Ref. [16]. Energy is plotted in units of $t_0$. Notice that there is no gap at zero energy for any value of flux in contrast to Fig. 5. Still, the overall structure is remarkably similar.

Figure 7

The evolution of bands from the checkerboard to the honeycomb regime as a function of $\frac{t_1}{t_0}$ for flux $\mathrm{\Phi }=1/3$. Gaps are labeled with their Chern number. The gaps with nonzero Chern numbers are open for all $0<\frac{t_1}{t_0}\le 1$. The central gap closes at $\frac{t_1}{t_0}=\frac{1}{2^{1/3}}$. Near $\frac{t_1}{t_0}=0.1$, the gaps are the gaps of the two stacked square-lattice butterflies. In the other limit, they connect to the largest gaps of the honeycomb lattice without gap closings.

Figure 8

The evolution of bands from the checkerboard to the honeycomb at $\mathrm{\Phi }=1/5$ as a function of $\frac{t_1}{t_0}$. Gaps are labeled with their Chern number. The sequence of the Chern numbers in the spectrum near $\frac{t_1}{t_0}=0.1$ is $1,2,-2,-1,0,1,2,-2,-1$ as expected for two stacked square-lattice butterflies. In the other limit, the sequence is $1,2,3,-1,0,1,-3,-2,-1$, valid for the honeycomb lattice. The change in the sequence is possible due to the third (eighth) and the fourth (seventh) bands touching and exchanging Chern numbers at $\frac{t_1}{t_0}=0.5$. The central gap closes at $\frac{t_1}{t_0}=\frac{1}{2^{1/5}}$.

Figure 9

Evolution of the bands and gap Chern number for $\mathrm{\Phi }=1/13\text{and}5/13$. For large denominators, the evolution contains many band crossings and Chern number transfers. These transfers are necessary for the sequence of gap Chern numbers to connect the two limits. Notice that the largest gaps remain open throughout. Chern numbers for the smallest gaps are not displayed for visual clarity.