Composite fermions on a torus

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 $\nu =\frac{{\nu }^{*}}{2p{\nu }^{*}+1}$, where $n$ and $p$ are integers and ${\nu }^{*}$ 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.

Figure 1

Composite fermions on a torus.

Figure 2

(a) An example of a “proper state.” No composite fermion in any $\mathrm{\Lambda }$ level has a hole directly underneath it in any $\mathrm{\Lambda }$ level. (A composite fermion is depicted as an electron bound to two flux quanta represented by vertical arrows.) (b) An example of a state that is not proper.

Figure 3

Examples of certain important types of proper states: (a) the 2/5 ground state, (b) a CF quasihole at $\nu =2/5$, (c) a CF quasiparticle at $\nu =2/5$, and (d) a neutral CF exciton at $\nu =2/5$.

Figure 4

The Coulomb energy per particle for the ground states at (a) $\nu =1/3$, (b) $\nu =2/5$, and (c) $\nu =3/7$. The circles are the CF energies and the squares are the energies from exact diagonalizations. The energy is quoted in units of $e^2/\epsilon l$ and includes self-interaction.

Figure 5

The pair correlation functions for an $N=30$ particle system at $\nu =2/5$.

Figure 6

The excitation gap for (a) $\nu =1/3$ and (b) $\nu =2/5$ in units of $e^2/\epsilon l$.

Figure 7

The ground state energy for $N=10,N_{\varphi }=25$ as a function of cutoff. The cutoff is the largest value of $\left|m\right|$ and $\left|n\right|$, where $m$ and $n$ are defined through $\mathbf{q}=(\frac{2\pi m}{L_1},\frac{2\pi n}{L_2})$.