Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model

We study how the stability of the fractional quantum Hall effect is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria play a dominant role in predicting the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.

Figure 1

Magnitude of Brillouin-zone fluctuations of band-geometric quantities, as a function of $N$. Red points correspond to the coefficient of the $cos\left(N{k}_x\right)+cos\left(N{k}_y\right)$ term in the Fourier series expansions of $B$, $g$, while black points correspond to the rms fluctuation of each quantity found from numerical integration over the BZ. Blue lines show the approximation given in Eqs. (25, 26, 27, 28). Our analysis only predicts the form of the leading $N$ dependence for ${\tilde{g}}_{xy}$ and $\tilde{D}$, which is shown with a thin line (the overall proportionality constant being undetermined).

Figure 2

Plot of the BZ-averaged value of the trace (5) and determinant inequalities (4). Points are obtained from numerically integrated values of $\langle T\rangle$ and $\langle D\rangle$. The blue lines are given by Eqs. (40) and (41). The thin black lines show the fluctuation amplitudes $\tilde{T}$ and $\tilde{D}$ plotted in the last row of Fig. 1 for comparison.

Figure 3

Plot of the residuals in the implicit relationship between (40) and (41). Points are obtained from values of $\langle T\rangle$ and $\langle D\rangle$ numerically integrated over the BZ. The blue line is a plot of the predicted $N$ dependence, ${\left|A\right|}^4{N}^4{e}^{-2\tilde{\sigma }N}$, as a guide to the eye.

Figure 4

Scaled many-body gaps as a function of BZ-averaged trace inequality (5) for FQH-like states at various values of $N$. Insets show behavior of points at larger values of $N$. Error bars reflect the finite accuracy of our diagonalization code. (a) Laughlin state of $N_p=8$ bosons at $\nu =1/2$. (b) Laughlin state of $N_p=8$ fermions at $\nu =1/3$. (c) Moore-Read state of $N_p=8$ bosons at $\nu =1$.

Figure 5

Gap in particle entanglement spectrum from tracing out four particles in ground-state density matrix. (a) Laughlin state of $N_p=8$ bosons at $\nu =1/2$. (b) Laughlin state of $N_p=8$ fermions at $\nu =1/3$.

Figure 6

Plots of the metric components and Berry curvature over a minimal periodic cell of the BZ (of size $2\pi /N\times 2\pi /N$) for the Hofstadter model at $N=3$.