content_image/423251.jpg content_image/423267.jpg content_image/423268.jpg content_image/423252.jpg content_image/423263.jpg content_image/423249.jpg content_image/423250.jpg content_image/423257.jpg content_image/423262.jpg
broken spin rotational symmetry, we take the scattering matrix as an element of the circular symplectic ensem- ble of random matrices [25]. RMT does not depend on the exact form of SOI, but only requires that SOI totally breaks spin rotational symmetry. Our results are thus ex- pected to apply for Rashba and Dresselhaus SOI equally well, as long as one has a ballistic chaotic quantum dot with totally broken spin rotational symmetry. For N$_{P}$ = 1, we get from Ref. [19] 

<img src='content_image/423251.jpg'>

from which we conclude, together with Eq. (9), that 

RMT thus predicts the vanishing of the spin polarization on average. The latter is understood in the standard mesoscopic sense, as an average over changes in the dot shape, the overall chemical potential or the position of the leads. 

Chemical potentials nevertheless exhibit spatial fluctu- ations in a single sample, whose magnitude can be esti- mated by calculating var µ ( β ) P . Of further interest are also mesoscopic fluctuations of the sample-integrated chemi- cal potentials whose magnitude we analyze with var µ ( β ) $_{tot}$. To calculate the variance of µ ( β ) P in Eq. (9), we take the variance of the numerator and divide it by the squared average of the denominator, a procedure which neglects subdominant corrections O ( N − 1 $_{L,R}$). The variance of the spin-dependent transmission coefficients have been cal- culated in Ref. [16]. For i = L, R , we have 

<img src='content_image/423267.jpg'>

with N$_{T}$ = N$_{L}$ + N$_{R}$ + N$_{P}$ . For our case with N$_{P}$ = 1, we obtain 

<img src='content_image/423268.jpg'>

For N$_{L}$ = N$_{R}$ ≡ N ≫ 1, Eq. (14) goes into Eq.(1). 

With the further assumptions, discussed in the pre- vious section, that (i) spin relaxation processes are frozen in the probe and (ii) the correlator 〈 T ( αβ ) P L T ( αβ ) P ′ L 〉 − 〈 T ( αβ ) P L 〉〈 T ( αβ ) P ′ L 〉 sharply decays when the distance between the probe positions P and P ′ reaches the Fermi wave- length, we get the typical total spin-dependent chemical potentials 

As justified above, we identify these potentials with those on the dot. Assuming that they are generated by a spin 

<img src='content_image/423252.jpg'>

<img src='content_image/423263.jpg'>

polarization, we convert them into the variance of the spin polarization by dividing them with 2 δ . For N$_{L}$ = N$_{R}$ ≡ N , we obtain the variance of the total number of polarized spins on the dot as 

<img src='content_image/423249.jpg'>

which, for N → ∞ , gives Eq. (3). The validity of Eqs. (14–15) is numerically confirmed below in Fig. 3. 

The power of our scattering theory is perhaps even more evident when one realizes that it allows to calculate correlation functions between local spin polarizations in- side the sample and spin currents in the leads. The spin current in lead L is given by 

<img src='content_image/423250.jpg'>

and each of these three terms gives a separate contri- bution to the correlator 〈 I ( α ) L µ ( β ) P 〉 − 〈 I ( α ) L 〉〈 µ ( β ) P 〉 . For α = x, y, z , the contribution of the third term van- ishes because it contains the average of three spin- dependent transmission coefficients. Furthermore, RMT gives covar( T ( α 0) P i , T ( β 0) jk ) ∝ δ$_{ik}$δ$_{αβ}$δ$_{α}$$_{0}$ if j ̸ = P . Therefore, for α = x, y, z , the first two terms do not contribute ei- ther, unless the probe is placed in the immediate vicinity, i.e. within a single Fermi wavelength of the contact to the L lead. In this case, nonvanishing covariances appear in 〈〈 I ( α ) L µ ( β ) P 〉〉 , see Eq. (13b), and one obtains 

<img src='content_image/423257.jpg'>

where δ$_{P x}$$_{L}$ requires that the probe be placed within k − 1 F of the boundary between the SOI-coupled dot and the lead. Summing over all possible probe positions, one obtains a universal correlator 〈〈 I ( α ) L µ ( β ) tot 〉〉 = O ( N $^{0}$) for N$_{L,R}$ ∼ N ≫ 1, which is consistent with the universality of spin currents fluctuations [16–19]. RMT thus predicts that coherent polarizations correlate with the spin cur- rents only if they are right at the sample exit. 

We numerically illustrate the validity of our scattering theory of spin polarizations and confirm our main predic- tion, Eqs. (14-15). We use tight-binding models defined by the Hamiltonian 

where the operators c † is ( c$_{is}$ ) create (destroy) a spin-1 / 2 fermion with spin s = ↑ , ↓ and σ α s,s ′ indicate elements of Pauli matrices. To give a proof-of-principle for our ap- proach, we consider a diffusive model, while quantitative 

## IV. NUMERICS 

<img src='content_image/423262.jpg'>

