Spin-orbit interaction in quantum dots in the presence of exchange correlations: An approach based on a good-spin basis of the universal Hamiltonian

We discuss the problem of spin-orbit interaction in a two-dimensional chaotic or diffusive quantum dot in the presence of exchange correlations. Spin-orbit scattering breaks spin rotation invariance, and in the crossover regime between different symmetries of the spin-orbit coupling, the problem has no closed solution. A conventional choice of a many-particle basis in a numerical diagonalization is the set of Slater determinants built from the single-particle eigenstates of the one-body Hamiltonian (including the spin-orbit terms). We develop a different approach based on the use of a good-spin many-particle basis that is composed of the eigenstates of the universal Hamiltonian in the absence of spin-orbit scattering. We introduce a complete labeling of this good-spin basis and use angular momentum algebra to calculate in closed form the matrix elements of the spin-orbit interaction in this basis. Spin properties, such as the ground-state spin distribution and the spin excitation function, can be directly calculated in this basis. Our approach is not limited to the spin-orbit coupling problem but can be applied to any perturbation that is added to the universal Hamiltonian.

Figure 1

An example of a tree graph representing the normal-coupling scheme ${\lambda }_1\otimes {\lambda }_2\otimes {\lambda }_3\otimes {\lambda }_4\otimes {\lambda }_5$ for ${\lambda }_5>\cdots {\lambda }_2>{\lambda }_1$ singly occupied levels. The total spin $S$ of each state is indicated on the vertical axis. The $d_q\left(S\right)$ states with the same total spin $S$ are distinguished by the different paths $\gamma$ of intermediate spins. The internal points are shown by solid circles, whereas the terminal points are shown by open circles.

Figure 2

An example of a $q=7$ tree, showing the action of $A_{\mu \nu }^{\Sigma =1}$ in the canonical basis. The thick solid line represents the initial state $\overline{\gamma }$ with $S=3∕2$. Under the action of $A_{\mu \nu }^{\Sigma =1}$, this state is connected to states with $S^{\prime }=S$ (dotted-dashed path) and $S^{\prime }=S+1$ (dashed path). In the case shown, the intermediate spin in the transition is $S^{″}=S+1∕2$ (determined by the initial path $\overline{\gamma }$).

Figure 3

An example of spin-tree operations involved in the calculation of the reduced matrix element $\left({\gamma }^{\prime }{S}^{\prime }\parallel A_{\mu \nu }^{\Sigma =1}\parallel \gamma S\right)$ for case (i) and $S^{\prime }=S-1$. We consider $q=7$ singly occupied levels (the multiplicity of the states at the internal nodes $k$ is not shown explicitly). In panel (a) the thick solid line represents the initial path $\gamma$. The paths $\overline{\gamma }$ which result from the push-out operation of state $\nu$ at position 2 in the normal-ordered tree (denoted by a dotted vertical line) are obtained by taking any combination of dashed lines that leads to spin $S$ in the terminal node, starting from node 1. Note that there are nine such paths. In panel (b), we show by the thick solid line one of the possible paths $\overline{\gamma }$ from panel (a) and consider the action of the operator $A_{\mu \nu }^{\Sigma =1}$. The action of the annihilation operator in $A_{\mu \nu }^{\Sigma =1}$ can only result in a backward motion on path $\overline{\gamma }$ (labeled by 1), and leads to intermediate spin $S^{″}=S-\frac{1}{2}$. The action of the creation operator in $A_{\mu \nu }^{\Sigma =1}$ can lead to both $S^{\prime }=S-1$ and $S^{\prime }=S$. We choose the path (labeled by 2) that leads to the desired final state $S^{\prime }=S-1$. Finally, in panel (c) we consider the push-in operation to move $\mu$ back to its position 3 in the normal-ordered representation (denoted by the second dotted vertical line in (c)). The paths generated through this procedure can be obtained by any combination of the dashed lines that starts from the terminal node at $S$ and connects to the thick solid line at position 2. There are five such paths.