Fractional Quantum Hall Physics in Topological Flat Bands (1302.6606v1)

Abstract: We present a pedagogical review of the physics of fractional Chern insulators with a particular focus on the connection to the fractional quantum Hall effect. While the latter conventionally arises in semiconductor heterostructures at low temperatures and in high magnetic fields, interacting Chern insulators at fractional band filling may host phases with the same topological properties, but stabilized at the lattice scale, potentially leading to high-temperature topological order. We discuss the construction of topological flat band models, provide a survey of numerical results, and establish the connection between the Chern band and the continuum Landau problem. We then briefly summarize various aspects of Chern band physics that have no natural continuum analogs, before turning to a discussion of possible experimental realizations. We close with a survey of future directions and open problems, as well as a discussion of extensions of these ideas to higher dimensions and to other topological phases.

• The paper presents a comprehensive review demonstrating numerical and theoretical evidence for fractional quantum Hall states in flat Chern bands.
• It uses methods like exact diagonalization and mapped trial wave functions to draw parallels with traditional Landau level physics.
• It examines geometric and lattice effects, outlining criteria for achieving robust topological order in realistic condensed matter systems.

Fractional Quantum Hall Physics in Topological Flat Bands

The paper presents a detailed and pedagogical review of fractional quantum Hall (FQH) states realized within the framework of topological flat bands, specifically focusing on fractional Chern insulators (FCIs). The discussion draws significant parallels to the FQH effect in Landau levels, presenting both numerical and theoretical insights into these novel quantum states.

Overview

Traditional FQH states are realized in two-dimensional electron gases subjected to strong magnetic fields at low temperatures, resulting in quantized Hall conductance. This work extends the paper of such non-trivial topological phases to lattice systems with flat Chern bands. These bands, arising without external magnetic fields, offer a promising venue for exploring high-temperature FQH-like states stabilized by lattice effects.

The paper highlights several key aspects such as the construction of topological flat band models, the numerical evidence for FQH phases in these bands, and theoretical connections and differences with the Landau level problem. It examines the necessary conditions for lattice Chern bands to exhibit similar topological properties to Landau levels, including considerations of Berry curvature and flatness conditions.

Numerical and Theoretical Analysis

1. Numerical Evidence: The paper underscores the use of exact diagonalization techniques on small lattice systems to reveal topological order at fractional fillings. Key indicators include ground state degeneracies, spectral flow under twisted boundary conditions, and the computation of Hall conductance.
2. Trial Wave Functions and Pseudopotentials: By mapping Landau gauge orbitals onto Chern bands, the paper discusses constructing model Hamiltonians with fractionalized ground states. A critical tool is the mapping between Wannier orbitals and Landau levels, enabling a correspondence between known wave functions and states in Chern bands.
3. Projected Density Operators: The paper rigorously examines the underlying algebra of projected density operators within Chern bands, drawing analogies with Girvin-Macdonald-Platzman algebra in Landau levels. The Berry curvature's role as an effective magnetic field in determining fractionalization properties is critical in this analysis.
4. Geometrical Considerations: The work investigates the influence of the Fubini-Study metric and Berry curvature uniformity in tuning the resemblance to Landau level physics, suggesting criteria for favorable topological band structures.
5. Parton Constructions and Phases: The paper explores parton mean-field constructions for FCIs, illustrating transitions between compressible and incompressible phases within Chern bands—an exploration of Mott transitions and composite particle formations.

Lattice Effects and Extensions

The interplay between lattice symmetry and topological order introduces unique phenomena absent in conventional FQH systems. The paper of bands with higher Chern numbers, lattice defects' impact on topological phases, and symmetry-enriched topological phases are particularly noted for their theoretical richness and potential implications.

Practical Implications and Future Directions

The realization of FCIs could afford topological states' robustness at higher temperatures and less extreme conditions than traditional FQH systems. Experimental proposals in oxide interfaces and optical lattice setups are discussed as viable systems for observing these phases.

The review concludes by speculating on future research directions, such as the role of disorder, the impact of long-range interactions, experimental characterization techniques, and potential extensions to other topological systems, including time-reversal invariant systems.

Conclusion

Overall, the paper provides an exhaustive review of fractional Chern insulators within topological flat bands, offering a comprehensive bridge between lattice systems and traditional FQH physics. This work opens new avenues for exploring quantum Hall effects under lattice conditions, potentially leading to new quantum materials with exotic topological properties.