11 Appendix C: The Wigner function of a spin the next paragraph. For ε = 0 the four fixed points are: The Stratonovich-Wigner-Weyl correspondence (SWWC) associates with any Hermitian operator of a spin ( j $^{)}$, a real sphere(2 j ) function A$_{W}$ (Ω). Both spaces have the same dimension (2 j +1) $^{2}$, and the association is one-to-one. This appendix provides the reasoning and the practical formulas following [43, 44]. The inner product of two operators is defined as trace[ ˆ A † ˆ B ]. An orthonormal-like set of projector-like op- erators, that are knows and the Stratonovich-Weyl oper- ators, can be defined, such that with corresponding energies E$_{−}$ , E$_{x}$ , and E$_{+}$ that are given by Eqs.(10-12). We can also determine the borders of the separatrix by solving the equation f ( θ$_{1}$ $^{′}$, 2 ′ ) = f ( θ$_{x}$ ). Thus we get the following expressions for the outer borders of the islands The value of the critical bias Eq. (9) is obtained by solving the equation f $^{′}$( θ ) = 0 together with f $^{′′}$( θ ) = 0. An equivalent set of questions is where Ω = ( θ, ϕ ), and the “delta” function is δ$_{j}$ (Ω − Ω $^{′}$) = 2 j ∑ ℓ =0 ℓ ∑ m = − ℓ Y $^{ℓm}$(Ω) Y ℓm $^{∗}$(Ω$^{′}$) (C3) θ$_{1}$ $^{′}$, 2 ′ = arcsin(1 − (2 /u )) . (B9) C$_{lm}$ = ∑ m $^{′}$,m′′ ( − 1) j − m ′ √ 4 π 2 l +1 2 j +1 ( j l j − m ′ m m ′′ ) ψ ∗ m ′ ψ$_{m}$ ′′ . ρ$_{W}$ (Ω) = ∑ l,m C$_{lm}$Y $^{lm}$(Ω) , (C6) ˆ T lm = ∑ m $^{′}$,m′′ ( − 1) j − m ′ √$_{2}$$_{l}$ + 1 ( j l j − m ′ m m ′′ ) | m $^{′}$〉〈 m $^{′′}$| ˆ P Ω = √ 4 π 2 j + 1 2 j ∑ l =0 l ∑ m = − l Y $^{lm}$(Ω) ˆ T $^{lm}$, (C5) s$_{1}$ ′ = 4 s 3 x − 3 s$_{x}$. (B17) A$_{c}$ ≈ 4 π (1 − u − 2 / $^{3}$)3 / $^{2}$. (B18) s$_{x}$ = cos( θ$_{x}$ ) = (1 − u − 2 / $^{3}$)1 / $^{2}$. (B16) u 2 4 s 4 − [ ε$_{c}$u ] s 3 + [1 − f$_{c}$u + ε 2 $_{c}$] s 2 + [2 ε$_{c}$f$_{c}$ ] s + [ f 2 c − 1] = 0 . f$_{c}$ ≡ f ( θ$_{x}$ ; ε$_{c}$ ) = − 1 2 u + 3 2 u 1 / $^{3}$. (B15) f ( θ, ϕ ) = 1 2 u (cos θ ) 2 − ε cos θ − sin θ cos ϕ. (B14) A$_{W}$ (Ω) = trace[ ˆ P Ω ˆ ρ ] . (C4) leading to the solution Accordingly any operator ˆ A can be represented by the phase-space function For ε$_{c}$ = ε , one island has zero area while the other has a critical area A$_{c}$ . For completeness we derive an estimate for A$_{c}$ , that we have used in [25]. The island is defined by the equation f ( θ, ϕ ) = const., where From the above definition it follows that the inner prod- uct of two Hermitian operators is given by a phase-space integral over the product of the corresponding Wigner functions as in Eq.(32). The actual construction of the ˆ P Ω is not a simple task. One procedure is to start with the non-orthonormal set of coherent state projectors | Ω 〉〈 Ω | and to perform an “or- thogonalization”, very similar to that employed in con- dense matter physics for the purpose of defining a Wan- nier basis. The final result is conveniently expressed as which is an orthogonal transformation over the non- Hermitian mulitpole operators ˆ T lm defined as The outer turning point θ$_{1}$ ′ is identified as the second root of the equation f ( θ ) = f$_{c}$ . Defining s ≡ cos θ the equation takes the form The critical island is defined by the equation f ( θ, ϕ ) = f$_{c}$ where according to the previous paragraph One root with double degeneracy is obviously We use here the Wigner 3 j symbols. Conse- quently we can represent any operator ˆ A either by A$_{lm}$ = trace[( ˆ T $^{lm}$) † ˆ A ] or by A$_{W}$ (Ω) = trace[ ˆ P Ω ˆ A ]. In particular it follows that ρ = | ψ 〉〈 ψ | is represented by Then we can get the third root by solving a quadratic equation, leading to The area of the critical island can be found numerically, and a very good approximation is s 3 $_{x}$, namely where θ$_{−}$ = + π/ 2 , (B6) sin( θ ) f $^{′′}$( θ ) − cos( θ ) f $^{′}$( θ ) = 1 + u sin $^{3}$( θ ) = 0 , (B10) cos( θ ) f $^{′′}$( θ ) + sin( θ ) f $^{′}$( θ ) = ε − u cos θ = 0 . (B11)