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

The perturbative study of Ref. [6] suggested that the Hofstadter model would provide an exciting setting for studying the relationship between geometry and FQHE physics in a controlled fashion. Motivated by the geometric stability hypothesis [5], in Ref. [7], we further studied the geometry of the Hofstadter bands in the small flux per plaquette limit. We found that, as with the dispersion and Berry curvature, fluctuations in the quantum metric are exponentially suppressed as the magnetic flux decreases. In contrast, the deviation from saturation of the geometric inequalities falls off polynomially. This suggests that these inequalities are the dominant measure of closure of the density operator algebra near the continuum limit.

Continuing this line of reasoning, we also studied the many-body physics of interacting particles fractionally filling the bands, since we expect the density operator algebra to tell us about the stability of the FQH state to excitations. We found a correlation between the degree of saturation of the geometric inequalities and the size of the excitation gap above the many-body ground state. This provided further evidence in a controlled setting that geometry of single-particle bands can be used to predict the many-body physics of particles in these bands.

Cumulatively, along with our previous work, these results increase our confidence in the ability to predict the regions of parameter space that are most favorable for forming FQHE states. This has experimental implications in that one can calculate *a priori* parameter values that optimize geometric quantities and hence are more likely to lead to stable FQH phases.

### References

[5] T. S. Jackson, G. Moller, and R. Roy, Nat. Commun. **6**, 8629 (2015).

[6] F. Harper, S. H. Simon, and R. Roy, Phys. Rev. B **90**, 075104 (2014).

[7] D. Bauer, T. S. Jackson, and R. Roy, Phys. Rev. B **93**, 235133 (2016).