| r − r $^{′}$|  → 2 R ; this is the manifestation of the usual break- down of the quasiclassical approximation in the vicinity of the classical turning point. (iii) To fix the relative phases of G ( j ) + and G ( j ) − we choose a given circular trajectory and consider the evolu- tion of G ( r , r $^{′}$; ϵ ) as r moves along the circle. Namely, as r passes the turning point at | r − r $^{′}$| = 2 R , G ( j ) + ( r , r $^{′}$; ϵ ) should transform into G ( j ) − ( r , r $^{′}$; ϵ ). In the vicinity of the turning point, | r − r $^{′}$| = 2 R (1 + z ), z ≪ 1, we have ϑ$_{±}$ ≈ π ∓ √ − 2 z . Thus, ϑ$_{±}$ can be viewed as two branches of the same analytical function, and ϑ$_{+}$ → ϑ$_{−}$ when z makes a circle around z = 0 in the complex plane. Since ϑ$_{±}$ + sin ϑ$_{±}$ ≈ π ∓ ( − 2 z ) 3 / $^{2}$/ 6, in order to get a decaying exponential in the classically forbidden region | r − r $^{′}$| > 2 R (the positive semiaxis of z ), the circle must be counterclockwise. Thus, which determines the poles corresponding to πϵR/ ℏ v be- ing an integer multiple of 2 π , i. e., Bohr-Sommerfeld quantization rule. It gives the exact expression for the Landau levels, | ϵ$_{n}$/v | = √ 2 n | e ℏ B/c | . Here we consider the matrix element M ( q ; ϕ$_{in}$, ϕ$_{out}$ ) for the transition from the initial state, corresponding to the incident photon with frequency ω$_{in}$ polarized at the angle ϕ$_{in}$ to the x axis, and no phonons, to the fi- nal state, corresponding to the scatterd photon with fre- quency ω$_{in}$ − 2 ω$_{q}$ polarized at the angle ϕ$_{out}$ to the x axis, and two phonons with momenta q , − q and frequencies ω$_{q}$ ( q is measured from the K or K ′ point, and for the phonons in the two valleys the frequencies ω ( K ) q = ω ( K $^{′}$) − q due to the time-reversal symmetry; we denote ω ( K ) q ≡ ω$_{q}$ and omit the valley index). Then, for unpolarized excita- tion and detection, the frequency-resolved intensity I (Ω) is given by (iv) To match the phases of G ( j ) − and G ( j +1 + ), we con- sider r → r ′ and require that G ( j ) − ( r , r $^{′}$; ϵ )+ G ( j +1) + ( r , r $^{′}$; ϵ ) satisfies the homogeneous Dirac equation. Indeed, the δ -function term is produced by G (0) + component, see re- mark (i). At r → r ′ the Green’s function satisfying the homogeneous Dirac equation can be can be constructed in the B → 0 limit. Let us introduce ¯ G$_{0}$ ( r − r $^{′}$; ϵ ), the Green’s function with analytical properties opposite to those of G$_{0}$ : precisely as in Eq. (A14). The matrix element is given by the loop of four Green’s function with electron-photon and electron-phonon ver- tices (here we write it in the coordinate representation) $^{29}$: We will assume γ$_{ω}$$_{in}$$_{/}$$_{2}$R/v ≫ 1 (non-quantizing field), then the probability of a half- or full rotation is exponen- tially small, and only the G (0) + contribution to the Green’s functions remains. The integral over ϵ is dominated by small | ϵ | ∼ γ$_{ω}$$_{in}$$_{/}$$_{2}$ , so the energy arguments of the Green’s functions can be assumed to have definite signs. Thus, the imaginary part of the exponent can be written as It also satisfies Eq. (A12), as seen from the antisymme- try of the Dirac operator with respect to the simultaneous change ϵ → − ϵ , i → − i , γ$_{ϵ}$ → − γ$_{ϵ}$ . Thus, G$_{0}$ ( r − r $^{′}$; ϵ ) − ¯ G$_{0}$ ( r − r $^{′}$; ϵ ) represents the sought solution, corresponding to the flux of particles focusing at the point r $^{′}$. Hence, the correspondence is G ( j ) − ( r → r $^{′}$; ϵ ) ∼ − ¯ G$_{0}$ ( r − r $^{′}$; ϵ ) (converging wave), G ( j +1) + ( r r $^{′}$; ϵ ) ∼ G$_{0}$ ( r − r $^{′}$; ϵ ) (out- going wave), which produces precisely the combination of signs as in Eq. (A14). Remarks (iii) and (iv) can be restated in very simple terms. Upon every half-rotation sin ϑ changes sign, so the square root produces a factor of e iπ/ $^{2}$. In addition, upon a full rotation ψ$_{n}$ acquires the Berry phase e $^{iπ}$. Hence, the sum over j in Eqs. (A13), (A14) reduces to a simple geometric progression, where the action S is that given by Eq. (A6), with the “+” subscript omitted and the energy argument explic- itly introduced. S ( r$_{2}$ , r$_{1}$ ; ω$_{in}$/ 2 − ω$_{q}$ + ϵ ) − qr$_{1}$ + S ( r$_{1}$ , r$_{0}$ ; ω$_{in}$/ 2 + ϵ ) + S ( r$_{0}$ , r ′ $_{1}$; ω$_{in}$/ 2 − ϵ ) + qr ′ 1 + S ( r ′ $_{1}$, r$_{2}$ ; ω$_{in}$/ 2 ω$_{q}$ + ϵ ) , (A28) ∞ ∑ j =0 e jπ ( i | ϵ | / ℏ − 2 γ ) R/v = 1 1 − e π ( i | ϵ | / ℏ − 2 γ ) $_{R/v}$, (A25) ¯ 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}$′}$| $^{/v}$. (A24) M ( q ; ϕ$_{in}$, ϕ$_{out}$ ) ∝ ∫ dϵ 2 π ∫ d $^{2}$r$_{2}$ d $^{2}$r$_{1}$ d $^{2}$r$_{0}$ d $^{2}$r ′ 1 × Tr { (Σ$_{x}$ cos ϕ$_{out}$ + Σ$_{y}$ sin ϕ$_{out}$ ) G ( r$_{2}$ , r$_{1}$ ; ϵ + ω$_{in}$/ 2 − ω$_{q}$ ) × Σ$_{z}$ e − i qr 1 G ( r$_{1}$ , r$_{0}$ ; ϵ + ω$_{in}$/ 2) (Σ$_{x}$ cos ϕ$_{in}$ + Σ$_{y}$ sin ϕ$_{in}$ ) × G ( r$_{0}$ , r ′ $_{1}$; ϵ − ω$_{in}$/ 2) Σ$_{z}$ e i $_{qr}$′ 1 G ( r ′ $_{1}$, r$_{2}$ ; ϵ ω$_{in}$/ 2 + ω$_{q}$ ) } . (A27) I (Ω) ∝ 2 π ∫ 0 dϕ$_{in}$ 2 π dϕ$_{out}$ 2 π ∫ d $^{2}$q (2 π ) 2 × |M ( q ; ϕ$_{in}$, ϕ$_{out}$ ) | $^{2}$δ (Ω − 2 ω$_{q}$ ) . (A26) e iπ/ 4 √ sin ϑ ( j ) + → e − iπ/ 4 √ | sin ϑ ( j ) − |$^{,}$ (A23) 3. Two-phonon Raman matrix element 8