4 Performing integration over momentum space for the spherical Fermi surfaces in the limit γ$_{0}$k$_{F}$ ≪ ε$_{F}$ , we ob- tain The corresponding gyrotropy conductivity is To find the gyrotropy conductivity in the supercon- ducting phase with the cubic symmetry, one needs to perform summation and integration in the eqn. (20) us- ing G and F in the superconducting phase. The integral in eqn. (20) consists of two terms × { [ G R $_{+}$( ω $^{′}$, k$_{+}$ ) − G A $_{+}$( ω $^{′}$, k$_{+}$ )] G A $_{−}$( ω ′ − ω, k$_{−}$ ) +[ G R $_{−}$( ω $^{′}$, k$_{−}$ ) − G A $_{−}$( ω $^{′}$, k$_{−}$ )] G R $_{+}$( ω ′ + ω, k$_{+}$ ) − [ F R $_{+}$( ω $^{′}$, k$_{+}$ ) − F A $_{+}$( ω $^{′}$, k$_{+}$ )] F A $_{−}$( ω ′ − ω, k$_{−}$ ) − [ F R $_{−}$( ω $^{′}$, k$_{−}$ ) − F A $_{−}$( ω $^{′}$, k$_{−}$ )] F R $_{+}$( ω ′ + ω, k$_{+}$ ) } . (32) j g i ( ω, q ) = e$_{ijl}$ e $^{2}$ω 12 π $^{2}$γ$_{0}$ k$_{F}$ q$_{l}$E$_{j}$ ( ω, q ) . (27) λ = i e $^{2}$ω 12 π $^{2}$γ$_{0}$ k$_{F}$ . (28) G R,A λ ( ω, k ) = u 2 $_{λ}$( k ) ¯ hω − ϵ$_{λ}$ ( k ) ± iδ + v 2 $_{λ}$( k ) ¯ hω + ϵ$_{λ}$ ( k ) ± iδ , (33) F R,A λ ( ω, k ) = t$_{λ}$ ( k )Δ 2 ϵ$_{λ}$ ( k ) [ 1 ¯ hω + ϵ$_{λ}$ ( k ) ± iδ − 1 ¯ hω − ϵ$_{λ}$ ( k ) ± iδ ] , (34) I$_{l}$ = ∫ d $^{3}$k (2 π ¯ h ) $_{3}$ˆ k$_{l}$ [ J$_{+}$$_{−}$ ( k , ω ) − J$_{−}$$_{+}$ ( k , ω )] , (29) Here, the Green functions are and where where IV. GYROTROPY CONDUCTIVITY IN THE SUPERCONDUCTING STATE and k$_{±}$ = k ± q / 2. Taking into account and Transforming the summation into an equivalent con- tour integration $^{16}$, eqn. (30) can be written as after integration with respect to ω $^{′}$, we can rewrite eqn. (32) as: u 2 $_{λ}$( k ) v 2 $_{λ}$( k ) } = 1 2 ( 1 ± ξ$_{λ}$ ( k ) ϵ$_{λ}$ ( k ) ) , (35) 2 2 ϵ $_{λ}$( k ) = ξ $_{λ}$( k ) + Δ $^{2}$, (36) J$_{+}$$_{−}$ ( k , ω ) = T ∞ ∑ m = −∞ [ G$_{+}$ ( K$_{+}$ ) G$_{−}$ ( K$_{−}$ ) − F$_{+}$ ( K$_{+}$ ) F † $_{−}$( K$_{−}$ )] , (30) J$_{−}$$_{+}$ ( k , ω ) = T ∞ ∑ m = −∞ [ G$_{−}$ ( K$_{+}$ ) G$_{+}$ ( K$_{−}$ ) − F$_{−}$ ( K$_{+}$ ) F † $_{+}$( K$_{−}$ )] . (31) G R λ − G A λ = − 2 πi [ u 2 $_{λ}$δ (¯ hω − ϵ$_{λ}$ ) + v 2 $_{λ}$δ (¯ hω + ϵ$_{λ}$ )] , (37) F R λ − F A λ = πit$_{λ}$ Δ ϵ$_{λ}$ [ δ (¯ hω − ϵ$_{λ}$ ) − δ (¯ hω + ϵ$_{λ}$ )] , (38) J$_{+}$$_{−}$ ( k , ω ) = ¯ h 4 πi ∮ dω $^{′}$tanh ¯ hω ′ 2 T J$_{+}$$_{−}$ ( k , ω ) = − 1 2 [( tanh ϵ$_{+}$ ( k$_{+}$ ) 2 T − tanh ϵ$_{−}$ ( k$_{−}$ ) 2 T )( u 2 $_{+}$( k$_{+}$ ) u 2 $_{−}$( k$_{−}$ ) ϵ$_{+}$ ( k$_{+}$ ) − ϵ$_{−}$ ( k$_{−}$ ) − ¯ hω + v 2 $_{+}$( k$_{+}$ ) v 2 $_{−}$( k$_{−}$ ) ϵ$_{+}$ ( k$_{+}$ ) − ϵ$_{−}$ ( k$_{−}$ ) + ¯ hω ) + ( tanh ϵ$_{+}$ ( k$_{+}$ ) 2 T + tanh ϵ$_{−}$ ( k$_{−}$ ) 2 T u 2 $_{+}$( k$_{+}$ ) v 2 $_{−}$( k$_{−}$ ) ϵ$_{+}$ ( k$_{+}$ ) + ϵ$_{−}$ ( k$_{−}$ ) − ¯ hω + v 2 $_{+}$( k$_{+}$ ) u 2 $_{−}$( k$_{−}$ ) ϵ$_{+}$ ( k$_{+}$ ) + ϵ$_{−}$ ( k$_{−}$ ) + ¯ hω )] − 1 2 Δ 2 4 ϵ$_{+}$ ( k$_{+}$ ) ϵ$_{−}$ ( k$_{−}$ ) [ − tanh ϵ$_{+}$ ( k$_{+}$ ) 2 T − tanh ϵ$_{−}$ ( k$_{−}$ ) 2 T )( 1 ϵ$_{+}$ ( k$_{+}$ ) − ϵ$_{−}$ ( k$_{−}$ ) − ¯ hω + 1 ϵ$_{+}$ ( k$_{+}$ ) − ϵ$_{−}$ ( k$_{−}$ ) + ¯ hω ) + ( tanh ϵ$_{+}$ ( k$_{+}$ ) 2 T + tanh ϵ$_{−}$ ( k$_{−}$ ) 2 T 1 ϵ$_{+}$ ( k$_{+}$ ) + ϵ$_{−}$ ( k$_{−}$ ) − ¯ hω + 1 ϵ$_{+}$ ( k$_{+}$ ) + ϵ$_{−}$ ( k$_{−}$ ) + ¯ hω )] . (39) For the second term under integral in the eqn. (29) we Here we have ignored the shifts in the arguments of the phase factors: t$_{λ}$ ( k ± q / 2 ) ≈ t$_{λ}$ ( k ) leading to the small corrections of the order of γ$_{0}$k$_{F}$ /ε$_{F}$ to the main terms.