# Geometric stability of topological lattice phases

The theoretical work in Refs [3–4] led us to propose a “geometric stability hypothesis,” which states that an approximate version of the single-mode approximation remains valid in FCI models, and that curvature and metric fluctuations along with the trace inequality are the relevant parameters measuring the deviation of a single-particle lattice Hamiltonian from Landau level physics. In Ref. [5] we conducted extensive numerical simulations on a scale beyond anything carried out in previous literature, comprehensively sampling the parameter space at thousands of values for three of the best-studied FCI models. Using a two-step sampling procedure, we were able to distinguish the effects caused by fluctuations of the Berry curvature and Hilbert space metric; the manuscript is the first work to demonstrate the importance of the latter factor.

Our work identified the quantum geometry of bands as the dominant factor determining the stability of FCIs in practice, which provides the first steps toward a unified theoretical description of FQH and FCI physics. Remarkably, we found that easily computed, single-particle quantities could be used to accurately estimate the parameters maximizing the gap of the interacting many-body system—it’s rare that one can find any computationally tractable behavior in a strongly correlated system, by its very nature. This could provide a significant shortcut to the design of laboratory realizations of high-temperature FCI states. Note that the notion that the gap would have any dependence on single-particle behavior is impossible to explain within the context of the continuum FQHE, in which the gap is entirely due to Landau level-projected interactions.

*Figure 2: Summary of band geometry results obtained in [5] for the Ruby lattice FCI model proposed in [24]; qualitatively similar results were found for other models. (a) Ruby lattice and convention for chiral hoppings. (b) Band geometry for parameters minimizing the many-body gap for N = 8 bosons at half-filling. Eigenvectors of the quantum metric are plotted as ellipses, with coloring determined by the fluctuations of Berry curvature from its mean value. (c) Berry curvature fluctuations σ _{B} versus gap for 2500 parameter values in the FCI phase. Insets: Fluctuations in quantum metric (σ_{g}) and averages of determinant and trace inequalities for Hamiltonians constrained to have a fixed value of σ_{B}. A clear linear trend present at low σ_{B} is washed out when curvature fluctuations are higher.*

### References

[3] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Phys. Rev. B **85**, 241308 (2012).

[4] R. Roy, Phys. Rev. B **90**, 165139 (2014).

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