Figure 2: Numerical proof-of-principle for the local measurement of spin accumulations on a single, diffusive, Rashba-coupled lattice of size 20 × 50 traversed by an electric current in the positive x -direction (from left to right). The on-site disorder and Rashba spin-orbit interaction correspond to a mean free path ℓ = 6 a and a spin-orbit length l$_{so}$ = 9 a . The system has hard-wall boundary conditions at its top and bottom boundaries. From left to right, spin accumulations µ ( β ) with β = x , y and z . Only µ ( y ) displays a finite average magnetoelectric effect (central panel), as predicted in Ref. [2]. Fluctuations exist for all spin orientations. The color scale is in units of the voltage bias eV . Note the different color scales in different panels. efficients of Eq. (7) are straightforwardly obtained from S . For the diffusive model, this method is certainly not the most efficient numerically. We still rely on it, because for the RMT model, the infinite-range hopping renders the implementation of recursive Green’s function algo- rithms problematic. Note that it is necessary to first generate random Hamiltonians from which random scat- tering matrices are constructed in order to consistently define probe positions and investigate whether spin-spin correlations decay sharply beyond a Fermi wavelength. To investigate the decay of these correlations, we gener- ate a random H and take confirmations of our theory are solely based on the RMT model. For the diffusive model, i = ( i$_{x}$, i$_{y}$ ) labels a site on a two-dimensional lattice of size L$_{x}$ × L$_{y}$ = ( N$_{x}$a ) × ( N$_{y}$a ) with spacing a , giving the size M = N$_{x}$ × N$_{y}$ of the Hamiltonian matrix. We further take h ( α ) ij = it$_{so}$δ$_{α,x}$δ$_{i}$$_{x}$$_{,j}$$_{x}$ ( δ$_{i}$$_{y}$$_{+}$$_{a,j}$$_{y}$ + δ$_{i}$$_{y}$$_{−}$$_{a,j}$$_{y}$ ) − it$_{so}$δ$_{α,y}$δ$_{i}$$_{y}$$_{,j}$$_{y}$ ( δ$_{i}$$_{x}$$_{+}$$_{a,j}$$_{x}$ + δ$_{i}$$_{x}$$_{−}$$_{a,j}$$_{x}$ ) − tδ$_{α,}$$_{0}$ [ δ$_{i}$$_{x}$$_{,j}$$_{x}$ ( δ$_{i}$$_{y}$$_{+}$$_{a,j}$$_{y}$ + δ$_{i}$$_{y}$$_{−}$$_{a,j}$$_{y}$ )+ δ$_{i}$$_{y}$$_{,j}$$_{y}$ ( δ$_{i}$$_{x}$$_{+}$$_{a,j}$$_{x}$ + δ$_{i}$$_{x}$$_{−}$$_{a,j}$$_{x}$ )] + w$_{i}$δ$_{α,}$$_{0}$δ$_{i,j}$ , corresponding to an on-site disorder potential with randomly distributed w$_{i}$ ∈ [ − w/ 2 , w/ 2], a nearest-neighbor hopping t and a Rashba spin-orbit interaction t$_{so}$ . These parameters give a spin- orbit length ℓ$_{so}$ ≈ πat/t$_{so}$ and an elastic mean free path ℓ ≈ 48 at $^{2}$/w2 close to half filling. For the RMT model, we take ∑ α h ( α ) ij σ α to be randomly distributed quaternions [24, 25], with time- reversal symmetry requiring h (0) ij = h (0) ji , and h ( α ) ij = − h ( α ) ji for α = x, y, z . Generally, the indices i, j have no particular physical meaning in RMT and the size M of the Hamiltonian matrix is a free parameter. Here we attach the system to external leads, and also consider i and j to label sites on a two-dimensional lattice whose Hamiltonian has an infinite-range random hopping. Once the sharp decay of spin-spin correlations is confirmed, M = k 2 $_{F}$A/ 4 π gives the total number of uncorrelated probe positions. For both models, all our simulations fix the Fermi energy slightly below half-filling. To construct the scattering matrix, we follow the method of Ref. [33]. We introduce a M × N$_{T}$ projection matrix W which connects the system to external leads. This matrix has only one nonzero element per row, at a spot determined by the coupling between the system and the leads. For an ideal contact, all nonzero matrix elements of W are unity. The N$_{T}$ × N$_{T}$ scattering matrix S associated with H is given by S = 1 − 2 πiW $^{†}$( E$_{F}$ − H + iπW W $^{†}$)− $^{1}$W , (20) We next confirm our conjecture of a sharp decay of correlations, and to that end, we calculate the spin-spin We first show numerical proof-of-principle for the spin- probe mode. In Fig. 2, we present data for the local spin polarizations µ ( β ) P , β = x, y, z . The three panels clearly show that, for transport in the x -direction and a Rashba SOI, there is a finite average magnetoelectri- cally induced polarization along the y -axis only, as pre- dicted in Ref. [2]. Furthermore, there are local (as well as sample-to-sample) fluctuations of the spin polarization in all directions, as predicted in Ref. [23]. All accumu- lations vanish on average in the RMT model, moreover spatial fluctuations occur on the smallest possible scale of a single site, as expected, and we therefore do not show them here. where for each choice of W , only a single component of the column vector ⃗ v (whose index indicates its length) is nonzero and equal to one. The position of that nonzero component determines the position of the probe on the two-dimensional lattice. Our Hamiltonians are defined on rectangular two-dimensional lattices, and we contact the transport modes at two of its edges, which is repre- sented by the two identity matrices in Eq. (21). W =   I$_{N}$$_{L}$ 0 0 0 ⃗ v$_{M}$$_{−}$$_{N}$$_{L}$$_{−}$$_{N}$$_{R}$ 0 0 0 I$_{N}$$_{R}$   , (21) with the overall Fermi energy E$_{F}$ . The transmission co- 6 4 8 12 16 20 10 20 30 40 50 X Y −0.03 −0.02 −0.01 0 0.01 0.02 4 8 12 16 20 10 20 30 40 50 X Y −0.04 −0.03 −0.02 −0.01 4 8 12 16 20 10 20 30 40 50 X Y −0.03 −0.02 −0.01 0 0.01 0.02 0.03