Path-integral Monte Carlo study of electronic states in quantum dots in an external magnetic field

We explore the correlated electron states in harmonically confined few-electron quantum dots in an external magnetic field by the path-integral Monte Carlo method for a wide range of the field and the Coulomb interaction strength. Using the phase structure of a preceding unrestricted Hartree-Fock calculation for phase fixing, we find a rich variety of correlated states, often completely different from the prediction of mean-field theory. These are finite temperature results, but sometimes the correlations saturate with decreasing temperature, providing insight into the ground-state properties.

Figure 1

The relative weight of the UHF ground state in the lowest Landau level, i.e., $〈{\mathrm{\Psi }}_0^{\text{HF}}\left|{\mathcal{P}}_{\text{LLL}}\right|{\mathrm{\Psi }}_0^{\text{HF}}〉/N$ for $N=8$ electrons (quantum dot oxygen), where ${\mathcal{P}}_{\text{LLL}}$ is the projection to the lowest Landau level. The contours start at 0.25 by the arc around the lower right corner; the increment is 0.05.

Figure 2

Examples of UHF ground-state structures in quantum dot lithium. (a) The particle density at $\gamma =0.6$ and $\lambda =3$ in state $(\frac{3}{2},C_{3v})$. (b) The spin density at $\gamma =0.6$ and $\lambda =1.8$ in state $(\frac{1}{2},C_s)$.

Figure 3

Sections of the UHF phase diagram of quantum dot Lithium $\left(N=3\right)$ as a function of the magnetic field parameter $\gamma$ and the coupling strength $\lambda$. Labels A and B indicate PIMC parameters in Figs. 4 and 5, respectively.

Figure 4

(a) The angular correlation function $g\left(\theta \right)$ for quantum dot lithium $\left(N=3\right)$ in the fully spin-polarized sector at $\gamma =0.2$ and $\lambda =1.8$. (b) The same at $\gamma =0.6$ and $\lambda =2.3$. The radial density is shown in the insets. Label A in Fig. 3.

Figure 5

[(a)–(c)] The angular correlation function $g\left(\theta \right)$ for partially spin-polarized $\left(S_z=\frac{1}{2}\right)$ quantum dot lithium $\left(N=3\right)$ for (a) $\gamma =0.6$ and $\lambda =0.9$, (b) $\gamma =0.8$ and $\lambda =1.4$, (c) $\gamma =1.4$ and $\lambda =0.4$. (d) Real-space spin density histogram for the parameters of (c), ${\beta }^{*}=3.2$. Label B in Fig. 3.

Figure 6

(Top) Sections of the UHF phase diagram of quantum dot beryllium $\left(N=4\right)$ as a function of the magnetic field parameter $\gamma$ and the coupling strength $\lambda$. The $\gamma =0$ line is exceptional; here, the sequence of transitions is $(1,C_{\infty })\stackrel{\lambda =1.3}{\to }(0,C_{2v})\stackrel{\lambda =2.8}{\to }(1,C_s)\stackrel{\lambda =3.2}{\to }(2,C_{4v})$. Labels A, B, C, and D indicate PIMC parameters in Figs. 7, 8, 9, and 10, respectively. (Bottom) Sketch of the particle and spin density peak configurations in the relevant symmetry-breaking states of quantum dot beryllium. Note that $(0,C_s)$ is relevant in PIMC only.

Figure 7

The angular correlation function $g\left(\theta \right)$ for fully spin-polarized quantum dot beryllium $\left(N=4\right)$ for $\gamma =0.2$ to 0.8, at the coupling strength values just following the value where UHF predicts full spin polarization. The radial density is shown for comparison in the inset for one case. ${\beta }^{*}=2.4$ for all PIMC results. Label A in Fig. 6.

Figure 8

[(a) and (b)] The angular correlation function $g\left(\theta \right)$ and the real-space particle density, respectively, in a PIMC simulation of fully polarized quantum dot beryllium $\left(N=4\right)$ at $\gamma =1.6$ and $\lambda =1$. [(c) and (d)] Corresponding results at $\lambda =2.5$. Label B in Fig. 6.

Figure 9

The angular correlation functions $g_{\text{same}}\left(\theta \right)$ and $g_{\uparrow \downarrow }\left(\theta \right)$ (a) at $\gamma =0.2,\lambda =0.6$, (b) at $\gamma =0.2,\lambda =1.2$, (c) at $\gamma =0.6,\lambda =1.2$ (here, ${\beta }^{*}=3.2)$, respectively, in the $S_z=0$ sector of quantum dot beryllium $\left(N=4\right)$. The radial densities are shown in the insets. Label C in Fig. 6.

Figure 10

PIMC results in the $S_z=1$ sector of quantum dot beryllium $\left(N=4\right)$. [(a) and (c)] The angular correlation function $g\left(\theta \right)$ and the real-space spin density histogram using ${\beta }^{*}=3.2$ at $\gamma =1.2$ and $\lambda =0.8$. [(b) and (d)] The same at $\gamma =1.6$ and $\lambda =0.4$. Label D in Fig. 6.

Figure 11

(Top) Sections of the UHF phase diagram of quantum dot boron $\left(N=5\right)$ at the magnetic field parameter $\gamma =\frac{{\omega }_c}{\omega }$ fixed at multiples of 0.2. We do not attempt to draw phase boundaries by connecting the transition points because of the limited resolution in the $\gamma$ direction. (Bottom) Sketch of the spin density peaks in the relevant symmetry-breaking states of quantum dot boron. Note that $(\frac{3}{2},C_s),(\frac{3}{2},C_{4v})$, and $(\frac{1}{2},C_s^{‴})$ are relevant in PIMC only. Labels A, B, and C indicate PIMC parameters in Figs. 12, 13, and 14, respectively.

Figure 12

The angular correlation function in the fully spin-polarized sector of quantum dot boron $\left(N=5\right)$. The insets show the radial density. (a) Typical pentagonal structure at $\gamma =1,\lambda =1.6$, as found everywhere $\gamma \ne 0.2$ (b) Square-coordinated structure at $\gamma =0.2,\lambda =4$; for $\gamma =0.2$ such correlations are present for all $\lambda$. Label A in Fig. 11.

Figure 13

The angular correlation function $g\left(\theta \right)$ in the $S_z=\frac{3}{2}$ sector of quantum dot boron $\left(N=5\right)$ (a) at $\gamma =0.4$ and $\lambda =1.8$, (b) at $\gamma =1$ and $\lambda =1.2$, (c) at $\gamma =1.2$ and $\lambda =0.9$, and (d) at $\gamma =1.6$ and $\lambda =0.4$. The inset shows the radial particle densities, distinct for each spin and in total, at ${\beta }^{*}=2.4$. Label B in Fig. 11.

Figure 14

Angular correlation functions and radial densities (insets) in the $S_z=\frac{1}{2}$ sector of quantum dot boron $\left(N=5\right)$ at (a) for $\gamma =0.2$ and $\lambda =1.2$, (b) for $\gamma =0.4$ and $\lambda =1.2$, (c) for $\gamma =0.6$ and $\lambda =1.8$, and (d) $\gamma =0.8$ and $\lambda =1.2$. ${\beta }^{*}=2.4$ in all panels. Label C in Fig. 11.

Figure 15

(Top) Sections of the UHF phase diagram of quantum dot carbon $\left(N=6\right)$ at the magnetic field parameter $\gamma =\frac{{\omega }_c}{\omega }$ fixed at multiples of 0.2. We do not attempt to draw phase boundaries by connecting the transition points because of the limited resolution in the $\gamma$ direction. Labels A, B, C, and D indicate PIMC parameters in Figs. 16, 17, 18, and 19, respectively. (Bottom) Sketch of the spin density peaks in the relevant symmetry-breaking states of quantum dot carbon. Note that $(2,C_s^{\prime })$ is never an UHF ground state, but a state with such correlations occurs in PIMC. For $S_z=0$, equivalent structures are obtained by interchanging spin up and spin down.

Figure 16

(a) The angular correlation function $g\left(\theta \right)$ for fully spin-polarized quantum dot carbon $\left(N=6\right)$ at $\gamma =0.8$ and $\lambda =3.5$. PIMC confirms the sixfold configuration in a ring. (b) As the coupling increases to $\lambda =4.5$, the angular correlation shows fivefold symmetry. At the same time, a particle density peak forms in the center. (c) The same at $\gamma =1.4$ and $\lambda =2$. (d) The same at $\gamma =1.4$ and $\lambda =4$, which is slightly beyond the upper end of the $S_z=3$ interval in UHF. Label A in Fig. 15.

Figure 17

The angular correlation functions and the radial density (insets) for quantum dot carbon $\left(N=6\right)$ of spin $S_z=0$. (a) At $\gamma =0.2$ and $\lambda =0.5$, PIMC shows $(0,C_{3v})$ correlations while the UHF ground state is $(0,C_{\infty })$. (b) At $\gamma =0.4$ and $\lambda =1.4$, UHF and PIMC agree on the $(0,C_{3v})$ state. (c) At $\gamma =0.8$ and $\lambda =0.8,(0,C_s)$ type correlations emerge on the UHF $(0,C_{\infty })\text{−}(0,C_{3v})$ phase boundary. (d) At $\gamma =0.8$ and $\lambda =1.7$, UHF predicts $(0,C_s)$ but PIMC indicates $(0,C_{3v})$ type correlations. ${\beta }^{*}=2.4$ in all panels. Label B in Fig. 15.

Figure 18

The angular correlation functions $g_{\uparrow \uparrow }\left(\theta \right),g_{\downarrow \downarrow }\left(\theta \right)$, and $g_{\uparrow \downarrow }\left(\theta \right)$ in the $S_z=1$ sector of quantum dot carbon $\left(N=6\right)$ at $\gamma =0.4$ and $\lambda =0.9$. (Inset) The total and the spin-resolved radial density at ${\beta }^{*}=1.6$; similar results are obtained for ${\beta }^{*}=0.8$ and 2.4. Label C in Fig. 15.

Figure 19

(a) The angular correlation function $g\left(\theta \right)$ for quantum dot carbon $\left(N=6\right)$ of spin $S_z=2$ at $\gamma =0.4$ and $\lambda =1.9$. (b) The radial density and $g\left(\theta \right)$ at $\gamma =0.6$ and $\lambda =2.7$. (c) The angular correlation function at $\gamma =0.8$ and $\lambda =1.8$. [(d) and (e)] The same at $\gamma =1$ and $\lambda =1.3$, and the spin density histogram for ${\beta }^{*}=2.4$. Label D in Fig. 15.

Figure 20

(Top) Sections of the UHF phase diagram of quantum dot nitrogen $\left(N=7\right)$ at the magnetic field parameter $\gamma =\frac{{\omega }_c}{\omega }$ fixed at multiples of 0.2. Labels A, B, C, and D indicate PIMC parameters in Figs. 21, 22, 23, and 24, respectively. (Bottom) Sketch of the spin density peaks in the relevant symmetry-breaking states of quantum dot nitrogen.

Figure 21

The angular correlation function $g\left(\theta \right)$ for fully spin-polarized quantum dot nitrogen $\left(N=7\right)$ for $\gamma =0.2$, 1, 1.2, and 1.4, at the coupling strength values just following the value where UHF predicts full spin polarization. The inverse temperature is ${\beta }^{*}=2.4$. The inset compares the radial density for $\gamma =0.2$ and $\lambda =2.3$ from PIMC and UHF. Label A in Fig. 20.

Figure 22

Angular correlation and radial density (insets) by PIMC in the $S_z=\frac{1}{2}$ sector of quantum dot nitrogen $\left(N=7\right)$. The parameters are (a) $\gamma =0.2$ and $\lambda =1.3$, (b) $\gamma =0.4$ and $\lambda =1.1$, (c) $\gamma =0.4$ and $\lambda =1.7$, (d) $\gamma =0.6$ and $\lambda =1.1$, and (e) $\gamma =0.8$ and $\lambda =1.2$. Label B in Fig. 20.

Figure 23

The angular correlation functions and the radial particle density (inset) in the $S_z=\frac{3}{2}$ sector of quantum dot nitrogen $\left(N=7\right)$ for $\gamma =1,\lambda =1.3$, and ${\beta }^{*}=3.2$. Label C in Fig. 20.

Figure 24

The angular correlation functions and the radial particle density (inset) in the $S_z=\frac{5}{2}$ sector of quantum dot nitrogen $\left(N=7\right)$. The parameters are (a) $\gamma =0.6$ and $\lambda =1.8$, (b) $\gamma =0.6$ and $\lambda =2.4$, (c) $\gamma =1.2$ and $\lambda =1.1$, and (d) $\gamma =1.4$ and $\lambda =0.8$. We use ${\beta }^{*}=2.4$, except for (d) where ${\beta }^{*}=3.2$. Label D in Fig. 20.

Figure 25

(Top) Sections of the UHF phase diagram of quantum dot oxygen $\left(N=8\right)$ at the magnetic field parameter $\gamma =\frac{{\omega }_c}{\omega }$ fixed at multiples of 0.2. Labels A to E indicate PIMC parameters in Figs. 26, 27, 28, 29, and 30, respectively. (Bottom) Sketch of the spin density peaks in the relevant symmetry-breaking states of quantum dot oxygen. Note that $(2,C_s)$ and $(2,C_s^{\prime })$ never occur as a UHF ground state, but such correlations do occur in PIMC. For $S_z=0$, equivalent structures are obtained by interchanging spin-up and spin-down.

Figure 26

The angular correlation function $g\left(\theta \right)$ for fully spin-polarized quantum dot oxygen $\left(N=8\right)$ for $\gamma =0.2$ at (a) $\lambda =2.8$ and (b) $\lambda =5.2$; The insets show the radial density. Label A in Fig. 25.

Figure 27

The angular correlation functions and the radial density (insets) for quantum dot oxygen $\left(N=8\right)$ of spin $S_z=0$ (a) for $\gamma =0.2$ and $\lambda =1.7$, (b) for $\gamma =0.4$ and $\lambda =2.4$, (c) for $\gamma =0.4$ and $\lambda =1.6$, and (d) for $\gamma =0.6$ and $\lambda =1.7$. ${\beta }^{*}=2.4$ in all panels. Label B in Fig. 25.

Figure 28

The angular correlation functions and the radial density (insets) for quantum dot oxygen $\left(N=8\right)$ of spin $S_z=1$ (a) for $\gamma =0.8$ and $\lambda =0.5$, (b) for $\gamma =0.8$ and $\lambda =1.7$, and (c) for $\gamma =1.4$ and $\lambda =0.7$. ${\beta }^{*}=2.4$ in all panels. Label C in Fig. 25.

Figure 29

The angular correlation functions and the radial density (insets) for quantum dot oxygen $\left(N=8\right)$ of spin $S_z=2$ (a) for $\gamma =0.6$ and $\lambda =1.8\left({\beta }^{*}=3.2\right)$, (b) for $\gamma =1$ and $\lambda =1.5\left({\beta }^{*}=2.4\right)$, and (c) for $\gamma =1.2$ and $\lambda =1.1\left({\beta }^{*}=3.2\right)$. Label D in Fig. 25.

Figure 30

First row: angular correlation functions and radial densities (insets) in the $S_z=3$ sector of quantum dot oxygen $\left(N=8\right)$ at $\gamma =0.8$ and (a) $\lambda =1.8$, (b) $\lambda =2.9$. (c) shows a spin density histogram for (b) at ${\beta }^{*}=3.2$. Second row: the same for $\gamma =1$ and (d) $\lambda =1.7$, (e) $\lambda =2.4$. (f) shows a spin density histogram for (e) at ${\beta }^{*}=2.4$. Label E in Fig. 25.

Figure 31

Angles defined for the integration of the cumulant action in Eq. (A10). $\mathrm{\Delta }{\mathbf{r}}^{\perp }$ is the clockwise $\pi /2$ rotation of $\mathrm{\Delta }\mathbf{r}$, i.e., $(y_1-y_0,x_0-x_1)$.