Statistics of spin fluctuations in quantum dots with Ising exchange

We explore the effect of single-particle level fluctuations on the Stoner instability in a quantum dot with a strong spin-orbit coupling in the framework of the universal Hamiltonian with the Ising exchange interaction. We reduce the problem to studying the statistics of extrema of a certain Gaussian process and demonstrate that, in spite of the randomness of the single-particle levels, the longitudinal spin susceptibility and all its moments diverge simultaneously at the point of the Stoner instability which is determined by the standard criterion involving the mean level spacing only.

Figure 1

The dependence of $\mathcal{P}\left(W\right)$ on $W/z^2$ at $T=3\delta$ obtained numerically for $J_z/\delta =0.94$ ($z\approx 0.5$) (upper solid curve) and $J_z/\delta =0.99994$ ($z\approx 16.8$) (lower solid curve). The black dotted curve is the CCDF for the normal distribution with mean and variance as one can find from the lowest order perturbation theory in $V$ for $T=3\delta$ and $J_z/\delta =0.94$ [28]. The red dashed curve is the CCDF of the degenerate process $\tilde{v}\left(h\right)$ for $T=3\delta$ and $J_z/\delta =0.99994$ [28]. Inset (a): Several realizations of the process $v\left(h\right)$; dashed lines $\pm 2h\sqrt{ln2}$ are guides for the eye. Inset (b): Comparison of the tail of $\mathcal{P}\left(W\right)$ obtained numerically for $J_z/\delta =0.99994$ ($z\approx 16.8$) and asymptotic result (15).