Composite fermion theory of correlated electrons in semiconductor quantum dots in high magnetic fields

Interacting electrons in a semiconductor quantum dot at strong magnetic fields exhibit a rich set of states, including correlated quantum fluids and crystallites of various symmetries. We develop in this paper a perturbative scheme based on the correlated basis functions of the composite-fermion theory, that allows a systematic improvement of the wave functions and the energies for low-lying eigenstates. For a test of the method, we study systems for which exact results are known, and find that practically exact answers are obtained for the ground state wave function, ground state energy, excitation gap, and the pair correlation function. We show how the perturbative scheme helps resolve the subtle physics of competing orders in certain anomalous cases.

Figure 1

Schematic depiction of Slater determinant basis states for $N=6$ electrons at $L=95$, which maps into $L*=5$ with $2p=6$. The single electron orbitals at $L*=5$ are labeled by two quantum numbers, the LL index $n=0,1,\dots$, and the angular momentum $l=-n,-n+1,\dots$. The $x$-axis labels $n+l$ and the $y$-axis $n$. The dots show the occupied orbitals forming the Slater determinants ${\Phi }_{\alpha }^{L*=5}$ relevant up to the first order. The state shown at the top left has the lowest kinetic energy (if the kinetic energy is measured relative to the lowest Landau level, then, in units of the cyclotron energy, the total kinetic energy of this state is two). The other nine states have one higher unit of kinetic energy. The basis states ${\Psi }_{\alpha }^{L=95}$ are obtained according to Eq. (2), through multiplication by ${\prod }_{j<k}{(z_j-z_k)}^6$, which converts electrons into composite fermions carrying six vortices. That is shown schematically by six arrows on each dot. The single state at the top is relevant at the zeroth order, and all ten basis states are employed at the first order. (In fact, there are a total of 12 linearly independent states ${\Phi }_{\alpha }^{L*}$ at the first order for $L*=5$, but they produce only ten linearly independent states ${\Psi }_{\alpha }^L$ at $L=95$.)

Figure 2

(Color online) Pair correlation function for $N=6$ electrons at $L=99$. The position of one particle is fixed on the outer ring, coincident with the position of the missing peak. The ground state wave function used in the calculation is obtained from (a) exact diagonalization; (b) the zeroth-order CF theory; (c) the first-order CF theory; (d) the second-order CF theory. The “noise” in (a) and (d) results from the relatively large statistical uncertainty in Monte Carlo because of the more complicated wave function.

Figure 3

Comparison of the exact excitation gaps (◻) for $N=6$ with the gaps obtained in the first-order CF theory (●).