n n n n R r’ r FIG. 6: (Color online) Two classical trajectories connecting the points r , r $^{′}$. the polar angle ϕ$_{n}$ of the unit vector n (we use the basis where Σ$_{x}$ , Σ$_{y}$ are represented by Pauli matrices), Thus defined, ψ$_{n}$ is not a single-valued function: when n is rotated by 2 π , ψ$_{n}$ acquires a minus sign. We can fix the sign by requiring that at r → r $^{′}$, ψ$_{n}$$_{+}$ = − ψ$_{−}$$_{n}$$_{−}$ = ψ$_{r}$$_{−}$$_{r}$ ′ , ψ$_{n}$$_{−}$ = ψ$_{−}$$_{n}$$_{+}$ = ψ$_{r}$ $^{′}$− $_{r}$, and they evolve continu- ously during the motion along the circle. We make several further remarks concerning the above expression for G ( r , r $^{′}$), Eqs. (A13), (A14). (i) For r → r ′ G ( r , r $^{′}$; ϵ ) reduces to the expression for the quasiclassical Green’s function for B = 0, valid at | ϵ || r − r $^{′}$| / ( ℏ v ) ≫ 1. (ii) The difference between the exact Green’s function and G ( r , r $^{′}$; ϵ ) given by Eqs. (A13), (A14) is of higher order in ℏ . This can be checked explicitly by calculating the gradients: Now we discuss the Green’s function for the Dirac equation, Here Σ = (Σ$_{x}$ , Σ$_{y}$) are the pseudospin matrices in the sublattice space, 1 is the unit matrix, and γ$_{ϵ}$ > 0 is the half of the electron or hole inelastic scattering rate, introduced phenomenologically (see Sec. IVC of Ref. 29 for the discussion of its role in Raman scattering). The factor sgn ϵ corresponds to the chronologically ordered Green’s function. so that at r = r ′ It is proportional to the eigenvector corresponding to the energy − ϵ ; thus, it can be compensated by a correction to the pre-exponential factor which does not vanish when acted upon by ϵ + iv Σ · ( ∇ S − ie A /c ), and thus is of the next order in ℏ : The imaginary part of the argument of the exponential is just the classical action discussed in the previous subsec- tion. Replacement ϵ → | ϵ | is required by the analytical properties of the chronologically ordered Green’s function (otherwise it would correspond to the retarded Green’s function). The damping factor e − γ$_{ϵ}$ϑR/v is determined by the total length of the corresponding arc ϑR , includ- ing 2 πjR from j full rotations. The eigenvector ψ$_{n}$ of ( n · Σ ) ψ$_{n}$ = ψ$_{n}$ sgn ϵ , | n | = 1, can be written in terms of δG ( j ) + is smaller than G ( j ) + by the dimensionless param- eter ℏ v/ ( ϵR ). However, it has a stronger divergence at 2. Quasiclassical Green’s function [ ϵ + i ℏ γ$_{ϵ}$ sgn ϵ + v Σ · ( i ℏ ∇ + e c A )] G ( r , r $^{′}$) = 1 δ ( r − r $^{′}$) . (A12) G$_{0}$ ( r − r $^{′}$; ϵ ) = − e iπ/ 4 2 ( 1 sgn ϵ + r − r ′ | r r $^{′}$| · Σ ) × √ | ϵ | / ( ℏ v ) 3 2 π r − r $^{′}$| e ( i | ϵ | / ℏ − γ$_{ϵ}$ ) | r − $^{$_{r}$′}$| $^{/v}$, (A16) ψ$_{n}$ = 1 √$_{2}$ ( e − iϕ$_{n}$/ $^{2}$sgn ϵ e iϕ$_{n}$/ 2 ) . (A15) dϕ$_{n}$$_{±}$$_{(}$$_{r}$$_{,}$$_{r}$ $^{′}$) = − [ n$_{∓}$ × d r ]$_{z}$ R sin ϑ$_{±}$ = [ n$_{∓}$ × d r $^{′}$]$_{z}$ R sin ϑ$_{±}$ , (A17) ∇ sin ϑ$_{±}$ = 1 R cos ϑ$_{±}$ cos( ϑ$_{±}$/ 2) r − r ′ | r r $^{′}$| , (A18) ∇ ψ$_{n}$$_{+}$ = 1 2 i Σ$_{z}$ ψ$_{n}$$_{+}$ ∇ ϕ$_{n}$$_{+}$ = 1 2 i Σ$_{z}$ ψ$_{n}$$_{+}$ [ e$_{z}$ × n$_{−}$ ] R sin ϑ$_{+}$ , (A19) ( Σ · ∇ ) ψ$_{n}$$_{+}$ = − ( n$_{−}$ · Σ) ψ$_{n}$$_{−}$ 2 R sin ϑ$_{+}$ , (A20) [ ϵ + iγ$_{ϵ}$ sgn ϵ + i ℏ v Σ · ∇ ] G ( j ) + ( r , r $^{′}$; ϵ ) = = − i ℏ v R cos $^{2}$( ϑ$_{+}$/ 2) [ n$_{+}$ × Σ ]$_{z}$ G ( j ) + ( r , r $^{′}$; ϵ ) . (A21) G ( r , r $^{′}$) = ∞ ∑ j =0 [ G ( j ) + ( r , r $^{′}$) + G ( j ) − ( r , r $^{′}$) ] , (A13) The quasiclassical Green’s function is represented as a sum over all trajectories of the classical motion from r ′ to r , i. e., sum over the short and long arcs, “ ± ”, and over the number of rotations j = 0 , 1 , . . . : G ( j ) ± ( r , r $^{′}$) = e ± iπ/ $^{4}$sgn ϵ √ | ϵ | / ( ℏ v ) 3 2 πR | sin ϑ ( j ) ± ( r , r $^{′}$) | × × ψ$_{n}$$_{±}$$_{(}$$_{r}$$_{,}$$_{r}$ ′ $_{)}$ψ † − n $_{∓}$( r , $_{r}$′ $_{)}$e − i ( eB/ 2 ℏ c )[ r × $^{$_{r}$′}$]$_{z}$ × × e i [ ϑ ( j ) ± ( r , $^{$_{r}$′}$)+sin ϑ ( j ) ± ( r , $^{$_{r}$′}$)] | ϵ | R/ (2 ℏ v ) × e − ϑ ( j ) ± ( r , $^{$_{r}$′}$) γ$_{ϵ}$$^{R/v}$. (A14) δG ( j ) + ( r , r $^{′}$; ϵ ) = i ℏ v 2 ϵR cos $^{2}$( ϑ$_{+}$/ 2) [ n$_{+}$ × Σ ]$_{z}$ G ( j ) + ( r , r $^{′}$; ϵ ) . (A22) 7