Landau Levels on a Torus

Jurgen Michael Honig, Józef Spałek

Ansar Fayyazuddin, Dingping Li

The lowest Landau level on a torus is studied. The dimension of the many-body Hilbert space is obtained and is found to be different from the formula given by Haldane. Our result can be tested in numerical investigations of the low-energy spectrum of fractional quantum Hall states on a torus.

Abhishek Anand, Songyang Pu, G. J. Sreejith

A short-ranged, rotationally symmetric multi-Landau-level model Hamiltonian for strongly interacting electrons in a magnetic field was proposed [A. Anand et al., Phys. Rev. Lett. 126, 136601 (2021)] with the key feature that it allows exact many-body eigenfunctions on the disk not just for quasiholes but for all charged and neutral excitations of the entire Jain sequence filling fractions. We extend this to geometries without full rotational symmetry, namely, the torus and cylinder geometries, and present their spectra. Exact diagonalization of the interaction on the torus produces the low-energy spectra at filling fraction $\ensuremath{\nu}=n/(2pn+1)$ that is identical, up to a topological ($2pn+1$)-fold multiplicity, to that of the integer quantum Hall spectra at $\ensuremath{\nu}=n$, for the incompressible state as well as all excitations. While the ansatz eigenfunctions in the disk geometry cannot be generalized to closed geometries such as torus or sphere, we show how to extend them to cylinder geometry. Meanwhile, we show eigenfunctions for charged excitations at filling fractions between $\frac{1}{3}$ and $\frac{2}{5}$ can be written on the torus and the spherical geometries.

L.R. Ribeiro, Claudio Furtado, Jose Roberto Nascimento

Songyang Pu, Ying-Hai Wu, Jainendra K. Jain

We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the ``unprojected'' wave functions satisfying the correct (quasi)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called ``proper states,'' which includes the incompressible ground states at electron filling factors $\ensuremath{\nu}=\frac{n}{2pn+1}$, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form $\ensuremath{\nu}=\frac{{\ensuremath{\nu}}^{*}}{2p{\ensuremath{\nu}}^{*}+1}$, where $n$ and $p$ are integers and ${\ensuremath{\nu}}^{*}$ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors $\ensuremath{\nu}=1/3$, 2/5, and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.