Generalized correlation functions for conductance fluctuations and the mesoscopic spin Hall effect

We study the spin Hall conductance fluctuations in ballistic mesoscopic systems. We obtain universal expressions for the spin and charge current fluctuations, cast in terms of current-current autocorrelation functions. We show that the latter are conveniently parametrized as deformed Lorentzian shape lines, functions of an external applied magnetic field and the Fermi energy. We find that the charge current fluctuations show quite unique statistical features at the symplectic-unitary crossover regime. Our findings are based on an evaluation of the generalized transmission coefficients correlation functions within the stub model and are amenable to experimental test.

Figure 1

Diagrams representing the leading contributions to the average transmission coefficient $\langle {\mathcal{T}}_{ij}^{\left(\alpha \right)}\rangle$ of Eq. (7). Vertical lines represent contractions of the $U$ matrix elements. The white and black dots stand for the channel indices of the matrix $U$ and the Greek symbols represent the Pauli matrix indices ${\sigma }_{\rho ,\sigma }$ in the spin space. The first diagram (top) is known as the diffuson and accounts for the semiclassical diffusive processes. The two others diagrams are known as cooperons and give the mean quantum correction, namely, the weak (anti)localization corrections.

Figure 2

Generalized average transmission coefficient $\langle {\mathcal{T}}_{ij}^{\left(\alpha \right)}(\epsilon ,x=0)\rangle$ versus energy $\epsilon =E/\Gamma$ for different spin and terminal indices. The analytical results (stub model) are represented by the solid and dotted lines, while the results of numerical simulations (RMT Hamiltonian model) are represented by the symbols. The figure inset describes the different ${\mathcal{T}}_{ij}^{\left(\alpha \right)}$.

Figure 3

Transmission coefficient covariance $\mathrm{cov}\left[{\mathcal{T}}_{ij}^{\left(\alpha \right)}(\epsilon ,x=0){\mathcal{T}}_{kl}^{\left(\beta \right)}({\epsilon }^{\prime },x=0)\right]$ as a function of the energy difference $\Delta \epsilon =(E-E^{\prime })/\Gamma$ for different spin and terminal indices. The analytical results (stub model) are represented by the solid and dotted lines, while the results of numerical simulations (RMT Hamiltonian model) are represented by the symbols. The figure inset describes the different $\mathrm{cov}\left[{\mathcal{T}}_{ij}^{\left(\alpha \right)}\left(\epsilon \right){\mathcal{T}}_{kl}^{\left(\beta \right)}\left({\epsilon }^{\prime }\right)\right]$ considered.

Figure 4

Contour plot of $h_x(n,x)$ as a function of the magnetic field, represented by $x$, and the channel factor $n$. The color code is explained in the strip on the right-hand panel.

Figure 5

Contour plot of $h_{\epsilon }(n,x)$ as a function of the magnetic field, represented by $x$, and the channel factor $n$. The color code is explained in the strip on the right-hand panel.