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Diagonalization and conventions. The spin-dependent part is $\lambda (-k_y{\sigma }_x+k_x{\sigma }_y)$. For a Hermitian Hamiltonian $\lambda$ is real; the supplied lower-band spinor uses the convention $\lambda >0$, which will be adopted below. The squared spin matrix is ${\lambda }^2(k_x^2+k_y^2)I$. Hence the two ordered energies areIn the $k_z=0$ plane, let $k=\sqrt{k_x^2+k_y^2}$ and $q_R=2m\lambda /{\hslash }^2>0$. ThenIf $\lambda <0$, use $q_R=2m|\lambda |/{\hslash }^2$ to retain the energy ordering, and interchange the spin directions assigned below. This is an isotropic Rashba effect; $q_R$ has units of inverse length, whereas $\lambda$ has units of energy times length.

Band minima. Define $k_R=q_R/2$ and $E_R={\hslash }^2{q}_R^2/(8m)$. Completing the square givesAlong $k_y=k_z=0$, the radial coordinate is $|k_x|$, so the lower branch has two minima:In the full plane these are points on a circle of minima $k=k_R$. Both branches meet at $k=0$, $E=0$. Their ordered energy curves contain $|k_x|$: the lower curve descends from the crossing on either side, while the upper rises. Treating them as globally labeled smooth spin branches without switching labels across $k_x=0$ would obscure the lower-band minima.

Occupied regions and Fermi contours. At zero temperature each spin-split eigenstate with energy below $E_F$ is occupied. PutFor $-E_R<E_F<0$, $0<D<k_R$. The lower-band roots are $k_R-D$ and $k_R+D$, while the upper band is entirely empty. Thus the central disk $k<k_R-D$ is unoccupied, the annulus $k_R-D<k<k_R+D$ is singly occupied, and the exterior is unoccupied. Both Fermi circles belong to the lower band; there is no doubly occupied region.

At $E_F=0$, the lower band is occupied for $0<k<2k_R=q_R$, giving a singly occupied disk with outer Fermi circle at $q_R$. The two bands also meet the Fermi energy at the origin. That isolated degenerate point is not a finite doubly occupied disk, and occupation exactly at the Fermi energy is a boundary convention of zero area. Outside the outer circle the states are empty.

For $E_F>0$, $D>k_R$. The lower branch has outer radius $k_{-}=D+k_R$, while the upper branch has inner radius $k_{+}=D-k_R$. The disk $k<k_{+}$ is doubly occupied, with one state in each band at every momentum. The annulus $k_{+}<k<k_{-}$ is singly occupied, in the lower band only, and the exterior is unoccupied. These are the Fermi contours of an isotropic Rashba band; double occupancy here refers to two orthogonal spinor states at the same momentum, not two Electrons occupying one identical quantum state.

Inferring the splitting. Above the crossing, the two positive radii differ by $2k_R=q_R$:The minimum radius is therefore $k_R=0.05{\mathring{\mathrm{A}}}^{-1}$. It is the difference of the measured radii that is required for $E_F>0$; below the crossing the two lower-band radii instead add to $q_R$.

Spinors and spin direction. Write $a=(k_y+i{k}_x)/k$, which has $|a|=1$ for $k>0$. An orthogonal normalized partner to the supplied lower-band state isIndeed $u_{-}^{†}{u}_{+}=(-|a{|}^2+1)/2=0$. Direct multiplication by the spin matrix gives eigenvalues $-\lambda k$ and $+\lambda k$ for $u_{-}$ and $u_{+}$, respectively, so orthogonality has selected the other band rather than merely another arbitrary spin basis.

For a normalized spinor $(\alpha ,\beta {)}^T$, the Pauli expectation values are $(2\mathrm{Re}({\alpha }^{\ast }\beta ),2\mathrm{Im}({\alpha }^{\ast }\beta ),|\alpha {|}^2-|\beta {|}^2)$. Consequently the Rashba spin texture isThe spin lies in the plane and is perpendicular to momentum. On the outer circle $k_1$, which belongs to $E_{-}$, it points clockwise as viewed from $+z$; at $\mathbf{k}=(k_1,0)$ it points along $-y$. On the inner circle $k_2$, belonging to $E_{+}$, it points counterclockwise; at $\mathbf{k}=(k_2,0)$ it points along $+y$. Opposite momenta have opposite spins, consistent with time-reversal symmetry. At the origin the eigenstates are degenerate and no unique spin orientation is defined.

A spin texture describes how a spin expectation value or spin polarization varies with position or wavevector. It specifies a vector field, including its magnitude when that matters; a direction field alone is undefined wherever that magnitude vanishes. A Rashba spin texture is an example in which spin expectation values wind around circles in the plane of wavevectors.