Two-dimensional quantum-dot helium in a magnetic field: Variational theory

A trial wave function for two-dimensional quantum-dot helium in an arbitrary perpendicular magnetic field (a system of two interacting electrons in a two-dimensional parabolic confinement potential) is introduced. A key ingredient of this trial wave function is a Jastrow pair correlation factor that has a displaced Gaussian form. The above choice of the pair correlation factor is instrumental in assuring the overall quality of the wave function at all values of the magnetic field. Exact numerical diagonalization results are used to gauge the quality of the proposed trial wave function. We find out that this trial wave function is an excellent representation of the true ground state at all values of the magnetic field including weak (or zero) and strong magnetic fields.

Figure 1

Variational mean square distance between two electrons, ${\alpha }^2⟨{\mid {\vec{\rho }}_1-{\vec{\rho }}_2\mid }^2⟩$, for 2D quantum-dot helium system in a perpendicular magnetic field as a function of dimensionless Coulomb coupling parameter $\lambda =e^2\alpha ∕\left(4\pi {ϵ}_0{ϵ}_r\hslash {\omega }_0\right)$ for two values of magnetic field corresponding to $\gamma ={\omega }_c∕{\omega }_0=0$ and 1. The line joining the data points at zero magnetic field serves as a guide to the eye.

Figure 2

Contour plots of the CPD function $P(\vec{\rho },{\vec{\rho }}_0)$ corresponding to the displaced Gaussian ground-state wave function for 2D quantum-dot helium at zero magnetic field $(\gamma =0)$ and $\lambda =10$. The black dot denotes the location of the fixed electron sittuated at position $\alpha {\vec{\rho }}_0=\left(\alpha x_0\ne 0,0\right)$. Distances are given in dimensionless units where $\alpha$ is the inverse oscillator length $\sqrt{m{\omega }_0∕\hslash }$.

Figure 3

Variational ground-state energy of 2D quantum-dot helium in a perpendicular magnetic field $ϵ=E∕\left(\hslash {\omega }_0\right)$ as a function of dimensionless Coulomb coupling parameter $\lambda =e^2\alpha ∕\left(4\pi {ϵ}_0{ϵ}_r\hslash {\omega }_0\right)$ for values of magnetic field corresponding to $\gamma ={\omega }_c∕{\omega }_0=0$, 2, 4, and 6. The solid lines represents the “adjusted” classical energy ${ϵ}_c+{ϵ}_0$ as a function of $\lambda$ calculated at the given $\gamma$ values.

Figure 4

The ground-state energy curves $ϵ=E∕\left(\hslash {\omega }_0\right)$ as a function of the magnetic field parameter $\gamma$ for the case of displaced Gaussian variational wave function (Var) and Laughlin’s wave function (Laughlin) for two values of the Coulomb correlation strength, $\lambda =2$ and 6. The solid lines joining the data points serve as a guide to the eye.