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 (FQHE) 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 have a dominant effect on 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.

Geometric stability of topological lattice phases

The fractional quantum Hall (FQH) effect illustrates the range of novel phenomena which can arise in a topologically ordered state in the presence of strong interactions. The possibility of realizing FQH-like phases in models with strong lattice effects has attracted intense interest as a more experimentally accessible venue for FQH phenomena which calls for more theoretical attention. Here we investigate the physical relevance of previously derived geometric conditions which quantify deviations from the Landau level physics of the FQHE. We conduct extensive numerical many-body simulations on several lattice models, obtaining new theoretical results in the process, and find remarkable correlation between these conditions and the many-body gap. These results indicate which physical factors are most relevant for the stability of FQH-like phases, a paradigm we refer to as the geometric stability hypothesis, and provide easily implementable guidelines for obtaining robust FQH-like phases in numerical or real-world experiments.

Theory of minimum spanning trees II: exact graphical methods and perturbation expansion at the percolation threshold

Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph, and use it to develop a continuum perturbation expansion for the MST on critical percolation clusters in space dimension d. The perturbation expansion is proved to be renormalizable in d=6 dimensions. We consider the fractal dimension D_p of paths on the latter MST; our previous results lead us to predict that D_p=2 for d>d_c=6. Using a renormalization-group approach, we confirm the result for d>6, and calculate D_p to first order in epsilon=6-d for dleq 6 using the connection with critical percolation, with the result D_p = 2 - epsilon/7 + O(epsilon^2).

Theory of minimum spanning trees I: Mean-field theory and strongly disordered spin-glass model

The minimum spanning tree (MST) is a combinatorial optimization problem: given a connected graph with a real weight (cost) on each edge, find the spanning tree that minimizes the sum of the total cost of the occupied edges. We consider the random MST, in which the edge costs are (quenched) independent random variables. There is a strongly-disordered spin-glass model due to Newman and Stein [Phys. Rev. Lett. 72, 2286 (1994)], which maps precisely onto the random MST. We study scaling properties of random MSTs using a relation between Kruskals greedy algorithm for finding the MST, and bond percolation. We solve the random MST problem on the Bethe lattice (BL) with appropriate wired boundary conditions and calculate the fractal dimension D=6 of the connected components. Viewed as a mean-field theory, the result implies that on a lattice in Euclidean space of dimension d, there are of order W^{d-D} large connected components of the random MST inside a window of size W, and that d = d_c = D = 6 is a critical dimension. This differs from the value 8 suggested by Newman and Stein. We also critique the original argument for 8, and provide an improved scaling argument that again yields d_c=6. The result implies that the strongly-disordered spin-glass model has many ground states for d>6, and only of order one below six. The results for MSTs also apply on the Poisson-weighted infinite tree, which is a mean-field approach to the continuum model of MSTs in Euclidean space, and is a limit of the BL. In a companion paper we develop an epsilon=6-d expansion for the random MST on critical percolation clusters.