Perturbative approach to flat Chern bands in the Hofstadter model

We present a perturbative approach to the study of the Hofstadter model for when the amount of flux per plaquette is close to a rational fraction. Within this approximation, certain eigenstates of the system are shown to be multicomponent wave functions that connect smoothly to the Landau levels of the continuum. The perturbative corrections to these are higher Landau level contributions that break rotational invariance and allow the perturbed states to adopt the symmetry of the lattice. In the presence of interactions, this approach allows for the calculation of generalized Haldane pseudopotentials, and in turn, the many-body properties of the system. The method is sufficiently general that it can apply to a wide variety of lattices, interactions, and magnetic field strengths.

Figure 1

Hofstadter's butterfly shows the fractal spectrum of the Hofstadter model (with energy $\epsilon$) as a function of magnetic flux per plaquette, $\varphi$. The entire pattern repeats for flux filling outside of the range 0 to 1. Landau levellike lines can be observed near to the points $(0,-4)$ and $(1/2,-2\sqrt{2})$, for example.

Figure 2

Harper's equation may be thought of as describing a particle moving in a cosine potential. Semiclassically, we envisage a ground-state wave function that consists of localized Gaussians in the troughs of the potential, coupled together by tunneling. For the lowest energy states, the turning point is a quadratic turning point, since the first derivative of the potential is vanishingly small.

Figure 3

Energy bands (top) and integrated density of states per unit area (bottom) for $\varphi =1/3$, $\varphi =1/3+1/30\equiv 11/30$, and $\varphi =1/3+1/33\equiv 4/11$. (Top) The energy spectra are plotted against $k_x$ for fifty equally spaced values of $k_y$ for each band. The spectra for the latter two field values show a clear energy gap above the lowest subband (which is just resolvable in each case). (Bottom) The density of states for the three complete bands is shown for $\varphi =1/3$, whilst the individual minibands are plotted for $\varphi \approx 1/3$ with bandwidth given by the $x$-error bars. The band structure of the “pure” $\varphi =1/3$ case carries over to the $\varphi \approx 1/3$ cases: the minibands arrange themselves into three bands and several subbands.

Figure 4

(Top) Rapidly oscillating cosine potential for $\varphi =1/3+1/30=11/30$. The eleven lattice sites in the troughs form the eleven minibands of the lowest band. (Bottom) Three effective potentials for $\varphi =1/3+1/30=11/30$. The three troughs of the effective potentials give rise to the three minibands which form the lowest subband.

Figure 5

First-order perturbative wave function plotted near to the origin for $\varphi =31/60$. The grey dashed lines indicate the continuum functions $A_j{\psi }^j\left(x\right)$.

Figure 6

Energy spectrum for three particles in the Hofstadter model with interaction strength $V/\left(4\pi l_B^2\right)={10}^{-3}$, harmonic trap strength $V_h={10}^{-5}$ and field strength given by $\delta =1/N={10}^{-6}$ (left) and $\delta =1/N=1/5$ (right). The single-particle basis states have a cutoff of $m_{\max }=10$, and only the lowest energy states are shown. The lowest energy linear part of spectrum starting at $m=2$ shows the Laughlin $\nu =1/2$ state and its edge excitations. The higher energy bulk excitations are not shown.

Figure 7

Dislocation corresponding to a Burgers vector of $\mathbf{b}=-\widehat{x}$.

Figure 8

The function $\tilde{\sigma }\left(N\right)$ (continuous line) and its limiting value (dashed line).

Figure 9

Comparison of analytic bandwidth and Berry curvature with numerics ($\alpha =2.3$ here).

Figure 10

Band structure for a repeating unit of the Brillouin zone for $\varphi =1/3$ (the full Brillouin zone has two additional copies of this section in the $k_y$ direction). The bands alternate in orientation from top to bottom and the troughs and peaks are located at $(k_x,k_y)=(0,0)$ and $(k_x,k_y)=(\pi /3,\pi /3)$. We may expand about any of these extreme points.

Figure 11

Cosine potential $-cos(2\pi \varphi n-k_y)$ for $\varphi =3/7$ and $k_y=\{0,2\pi /7,4\pi /7,6\pi /7\}$ for (a), (b), (c), and (d), respectively (we have actually chosen $k_y$ to be slightly greater than these values so that the site energies are not degenerate). The solid line shows the background cosine potential, while the points give the potential energy on each lattice site. The weight of the lowest band is centered on the lattice point with the lowest potential energy and is indicated by the large red points. This hops to the left by two sites whenever $k_y$ is changed by $2\pi /7$. When $k_y$ has changed by $2\pi$, the weight of the lowest band will have traversed 2 complete unit cells to left, corresponding to a Chern number of $-$2.

Figure 12

Effective potentials and one perturbative wave function for $\varphi =1/3+1/30$ and (left) $k=0$, (right) $k=\pi /3$. As we pump $k_y$, the wave-function weight moves monotonically to the right. When $k_y=2\pi$ the wave-function weight will have traversed one complete unit cell, corresponding to a Chern number of 1.

Figure 13

Anisotropic square lattice.

Figure 14

Section of square lattice showing both types of hopping.

Figure 15

Triangular lattice with Wigner-Seitz cell shaded in blue and a single plaquette shaded in red. We set the lattice spacing $a=1$.