Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators

We develop a first quantization description of fractional Chern insulators that is the dual of the conventional fractional quantum Hall (FQH) problem, with the roles of position and momentum interchanged. In this picture, FQH states are described by anisotropic FQH liquids forming in momentum-space Landau levels in a fluctuating magnetic field. The fundamental quantum geometry of the problem emerges from the interplay of single-body and interaction metrics, both of which act as momentum-space duals of the geometrical picture of the anisotropic FQH effect. We then present a novel broad class of ideal Chern insulator lattice models that act as duals of the isotropic FQH effect. The interacting problem is well-captured by Haldane pseudopotentials and affords a detailed microscopic understanding of the interplay of interactions and nontrivial quantum geometry.

Figure 1

(a)–(f) Real-space lattice guiding-center wave functions for the ideal isotropic $\mathcal{C}=3$ model, evaluated from Eq. (8). Insets depict analogous conventional LLL wave functions. (g) Berry curvature ${\mathrm{\Omega }}_{\mathbf{k}}$ (inset) and associated MLL spectra for the FCI models discussed in the main text. $m$ is the guiding center index; dotted lines indicate ideal flat-$\mathrm{\Omega }$ spectrum ${\epsilon }_m=(\lambda \mathcal{C}m/2\pi )$. The $\mathcal{C}$-fold degeneracy of MLLs reflects the $\mathcal{C}$-component basis for $\mathcal{C}>1$ models. The guiding-center structure remains robust in the presence of fluctuations of ${\mathrm{\Omega }}_{\mathbf{k}}$.

Figure 2

(a)–(c) Haldane pseudopotentials for the $\mathcal{C}=1$ three-orbital and $\mathcal{C}>1$ ideal droplet models (on-site and NN interactions). Only odd (even) pseudopotentials determine interactions between same-species fermions (bosons). Error bars quantify the residual center-of-mass $M$ variation, strongly suppressed for the ideal $\mathcal{C}=2,3$ models due to emergent conservation of guiding center. (d) Log plot of guiding-center and $SU(\mathcal{C})$ symmetry breaking in the interacting problem with matrix elements $V_{m_1\dots m_4}^{n_1\dots n_4}$, quantified as the ratio of the norm of $\mathrm{\Delta }M=m_1+m_2-m_3-m_4$ ($\mathrm{\Delta }N=n_1+n_2-n_3-n_4$ mod $\mathcal{C}$ component) symmetry-breaking and -preserving two-body matrix elements. The guiding-center index identifies with ${\mathrm{C}}_4$ symmetry. (e) Role of broken rotational symmetry on the lattice, depicted via leading-order $V_{m,m+4}^{\prime }$ of generalized pseudopotentials $V_{m{m}^{\prime }}^{\prime }=⟨m,M|H|m^{\prime },M⟩$.