Geometric stability of topological lattice phases

Abstract

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.

Phase transition and intrinsic metric of dipolar fermions in the quantum Hall regime

“…The notion of geometry has also inspired the construction of a more general class of FQH states with non-Euclidean metric 17 , which were used to characterize intrinsic non-trivial metrics emergent from many-body interactions of various experimental systems. More recently, an exciting possibility of the co-existence of topological order with broken symmetry 18,19 , leading to the "nematic" FQH effect, has also been proposed [20][21][22][23][24] . In this case, the nematic order arises due to spontaneous symmetry breaking, supported by recent numerical calculation 25 and experiments using hydrostatic pressure 26 .…”

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

“…7 That reference gave qualitative arguments linking properties of this algebra with the stability of FQHlike states which were numerically investigated in Ref. 8. The results obtained there suffered from the limitation that underlying band geometry could only be varied indirectly; moreover, one would like to be able to unify these numerical examples with a theoretical picture capable of making analytic statements.…”

Momentum-space cigar geometry in topological phases

“…It represents the natural metric of an effective curved momentum-space in condensed matter systems. Recent works point out that this kind of geometry has deep physical implications in topological phases [25][26][27][28][29][30][31][32][33]. Similar ideas about curved momentumspace geometry have been also developed in high-energy physics literature [34][35][36][37][38][39][40].…”

“…The quantum metric plays an important role in many-body systems and generally carries different information with respect to the Berry phase. The Bures metric has been connected to physical properties and observables of two-dimensional systems, such as density operators [25][26][27], quantum phase transitions [56][57][58], superfluid weight [59], orbital susceptibility [60,61]. Here, we focus on the geometric properties of gapped boundary of three-dimensional insulators in class AII, where a gap is induced by an external Zeeman field or by an ferromagnet on the surface [22].…”

“…This implies that the integral in eq. (27) has to be considered in the limit L A → 0. We remind also that this horizon is zero-dimensional, being the boundary of the one-dimensional space-like region, which is the radial line in our case.…”