Equivalence of Chern bands and Landau levels from projected Interactions (2503.19017v1)

Abstract: We introduce a universal formulation of the generalized real space interactions with translational invariance, when projected into a single Landau level, can be equivalent to density-density interaction projected into any Chern bands (e.g. Landau levels, continuous moire Chern bands and discrete lattice Chern bands). By constructing a complete basis of generalized pseudopotentials, we define the projected interaction range that can be used to characterize important features of different Chern bands relevant to the robustness of interacting topological phases. The analytical construction allows us to study general Chern bands beyond the torus geometry, for example on the disk or spherical geometry where the edge dynamics and curvature effects can be explored. For moire systems (and lattice Chern bands with large incompressibility gap) one can see transparently the topological Hall viscosity for the fractional states are identical to those in the Landau levels. The physical unprojected interaction also provides a natural embedding for lattice systems, shedding light on the concerns of relying on quantum geometric tensor (varying with lattice embedding in real space) to understand the properties of interacting phases of matters.

Equivalence of Chern Bands and Landau Levels from Projected Interactions

The paper of two-dimensional topological bands with a non-zero Chern number is a key area in understanding emergent quantum phases of matter and engineering novel low-dimensional materials. The paper by Bo Yang offers a robust framework to analyze these bands, notably establishing a conceptual bridge between Chern bands and Landau levels (LLs) through the lens of projected interactions. This essay provides a detailed exploration of the paper's contributions, highlighting significant theoretical developments and implications for future research in quantum Hall systems and Chern bands.

The fundamental assertion of the paper is the equivalence between generalized real-space interactions, when projected into a single Landau level, and density-density interactions projected into any Chern bands. These include Landau levels, continuous Moiré Chern bands, and discrete lattice Chern bands. This correspondence is derived by constructing a complete basis of generalized pseudopotentials, enabling the definition of a projected interaction range, which characterizes significant features of various Chern bands pertinent to the stability of interacting topological phases.

Main Contributions and Results

1. Universal Interaction Framework: The paper introduces a universal formulation for real-space interactions characterized by translational invariance. This approach allows interactions in any Chern band to be represented as generalized interactions in Landau levels. This novel framework facilitates a unified treatment of Chern bands across diverse systems, such as moiré superlattices and lattice Chern insulators.
2. Projected Interaction Range: The authors employ a generalized pseudopotential basis to define the projected interaction range, $\bm{r}_c$, for Chern bands. This range determines the number of Landau levels that a Chern band can be effectively reduced to under certain conditions. Consequently, this method provides a tangible measure to evaluate the feasibility and robustness of fractional quantum Hall (FQH) states in different Chern bands.
3. Ideal Chern Bands: The paper identifies a subset of Chern bands, termed as "ideal Chern bands", where the projected interaction range is finite. These bands can emulate the dynamical properties of LLs very closely. Notably, for ideal Chern bands, key topological properties, such as guiding center Hall viscosity, are consistent with those in LLs, suggesting an emergent full rotational symmetry within specific null spaces of the interacting system.
4. Beyond Ideal Chern Bands: For non-ideal Chern bands, $\bm{r}_c$ is generally infinite, implying an absence of simple real-space interactions that fully replicate the behavior seen in ideal conditions. The implication here is profound, indicating that perturbative corrections or tuning away from ideal conditions might significantly alter the phase diagram of the system, possibly leading to new quantum states.
5. Real Space Interaction Mapping: The projection of interactions, originally defined as density-density interactions, into band structures is extended to both LLs and lattice Chern bands. This methodology underscores the intrinsic similarity between these systems while emphasizing the role of single-particle geometric properties like Berry curvature fluctuations and the quantum geometric tensor.
6. Geometry and Metric Consideration: It is notable that the paper addresses the influence of guiding center metrics by leveraging an orthonormal set of pseudopotentials. This provides a geometric interpretation, particularly in analyzing the edge dynamics and Hall viscosity in topological phases.

Implications and Future Prospects

Practically, the equivalence established between Chern bands and Landau levels could have wide implications for designing and realizing robust FQH states in lattice systems, especially in engineered moiré materials and twisted bilayer graphenes. The insight offered by the projected interaction range metric $\bm{r}_c$ can guide experimentalists in tuning systems to achieve desired topological properties. The insights into generalized pseudopotentials can further influence the paper of non-abelian excitations and many-body interactions, expanding the theoretical framework for topological quantum computation.

Theoretically, the research underscores the importance of band geometry in determining the stability and nature of topological phases, challenging some of the traditional perspectives on the universality of quantum geometric effects. Future extensions might also consider higher Chern number systems ($C>1$) or interactions beyond two-body terms, explaining non-trivial entanglement patterns or topological invariants in these complex systems.

Overall, by linking Chern bands and Landau levels through their interaction properties, the paper provides a foundational step towards a more cohesive understanding of topological phases beyond the typical confines of quantum Hall effects and offers numerous pathways for both experimental and theoretical exploration.