Composite Fermions on a Torus

Abstract

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 $\nu =\frac{n}{2pn+1}$, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form $\nu =\frac{{\nu }^{\ast }}{2p{\nu }^{\ast }+1}$, where $n$ and $p$ are integers and ${\nu }^{\ast }$ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors $\nu =1/3$, 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.

27 pages,7 figures

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Cited by in corpus (26)

• Wavefunctionology: The Special Structure of Certain Fractional Quantum Hall Wavefunctions
• Thirty Years of Composite Fermions and Beyond
• Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model
• Entangled Pauli Principles: the DNA of Quantum Hall Fluids
• Abelian topological order of $\nu =2/5$ and $3/7$ fractional quantum Hall states in lattice models
• Fractional quantum Hall effect with unconventional pairing in monolayer graphene
• Dirac Fermion Hierarchy of Composite Fermi Liquids
• Composite-fermion pairing at half and quarter filled lowest Landau level
• Trial wave functions for a Composite Fermi liquid on a torus
• Berry phase of the composite-fermion Fermi Sea: Effect of Landau-level mixing
• Bardeen-Cooper-Schrieffer pairing of composite fermions
• Origin of the $\nu =1/2$ fractional quantum Hall effect in wide quantum wells
• Hall Viscosity of Composite Fermions
• Valley Stoner Instability of the Composite Fermi Sea
• Composite anyons on a torus
• Hall Viscosity of the Composite-Fermion Fermi Seas for Fermions and Bosons
• Continuous phase transitions between fractional quantum Hall states and symmetry-protected topological states
• Berry phase of the composite Fermi-liquid
• An Exactly Solvable Model for Strongly Interacting Electrons in a Magnetic Field
• Self-similarity of spectral response functions for fractional quantum Hall states
• On the modular covariance properties of composite fermions on the torus
• Entanglement scaling and charge fluctuations in a Fermi liquid of composite fermions
• Superlattice induced electron percolation within a single Landau level
• Torus geometry eigenfunctions of an interacting multi-Landau level Hamiltonian
• Topological Order in Neural Wavefunctions
• Excitation and tunneling spectra of a fractional quantum Hall system in the thin cylinder limit