Bosonization approach to the edge reconstruction of two-dimensional electron systems in a quantum dot

We consider edge reconstruction of electrons in a two-dimensional harmonic trap under a strong magnetic field. In this system the edge reconstruction occurs as a result of competition between electron-electron interaction and confining potential. We develop a bosonization scheme for two-dimensional electron systems. With this method we obtain the excitation spectrum and demonstrate that the edge reconstruction occurs when magnetic field reaches a critical value. We also show that the edge reconstruction depends on the number of electrons. We calculate the third-order terms of bosons in Hamiltonian and examine the effect of those terms with a perturbation theory.

Figure 1

The graph of $F(\hslash {\omega }_0,B)$ with respect to $B$ when $\hslash {\omega }_0=1.5\mathrm{meV}$.

Figure 2

Excitation energy spectra for several values of $F$ (magnetic field) when $\hslash {\omega }_0=1.5\mathrm{meV}$ and $N=50$.

Figure 3

Excitation energy spectra for several values of the number of electrons. $\hslash {\omega }_0=1.5\mathrm{meV}$ and $B=8.5\mathrm{T}$.

Figure 4

Occupation coefficients of single particle orbitals $⟨c_n^{†}{c}_n⟩$ considering up to quadratic order terms of bosons in Eq. (9). This graph shows the occupancies when a boson with an angular momentum $k=11$ is excited. The range of a reconstructed part is from $n=N-k+1$ to $n=N+k$ and expressed as a simple step function.

Figure 5

Excitation energy spectra when $F=4.12$ ($\hslash {\omega }_0=1.5\mathrm{meV}$, $B=9.5\mathrm{T}$, and $N=50$). Spectrum A is unperturbed and B is perturbed.

Figure 6

Occupation coefficients $⟨c_n^{†}{c}_n⟩$, including the effect of third-order terms of bosons when $N=50$ and $L_{\mathrm{tot}}-L_{\min }=6$ $(k=6)$.

Figure 7

Occupation coefficients $⟨c_n^{†}{c}_n⟩$, including the effect of third-order terms of bosons when $N=50$ and $L_{\mathrm{tot}}-L_{\min }=11$ $(k=11)$.