Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University Itinerant Spin Dynamics in Structures of ... - Jacobs University
Appendix E Summation over the Fermi Surface The Cooperon Hamiltonian in the 2D case is given by H c = τv 2 {〈cos 2 (ϕ)〉(Q+2m e a.S) 2 x +〈sin 2 (ϕ)〉(Q+2m e a.S) 2 y +4m 2 e γ Dv 2 〈cos 2 (ϕ)sin 2 (ϕ)〉(Q+2m e a.S) x .S x −4m 2 eγ D v 2 〈sin 2 (ϕ)cos 2 (ϕ)〉(Q+2m e a.S) y .S y +(2m 3 e γ Dv 2 ) 2 (〈cos 2 (ϕ)sin 4 (ϕ)〉Sx 2 +〈sin 2 (ϕ)cos 4 (ϕ)〉S 2 y)}, (E.1) with wave vector Q. We set m e ≡ 1, f 1 := 〈sin 2 (ϕ)〉, f 2 := 〈cos 2 (ϕ)〉, f 3 := 〈sin 2 (ϕ)cos 2 (ϕ)〉, f 4 := 〈sin 4 (ϕ)cos 2 (ϕ)〉, f 5 := 〈sin 2 (ϕ)cos 4 (ϕ)〉. (E.2) (E.3) (E.4) (E.5) (E.6) (E.7) Using the Matsubara trick we write ∫ 2π 0 dϕ 2π = 2 πN N∑ 1 √ 1− ( ) . (E.8) s 2 N s=1 148
Appendix E: Summation over the Fermi Surface 149 This gives us f 1 = 2 N−1 ∑ πN s=1 f 2 = 2 πN f 3 = 2 πN f 4 = 2 πN f 5 = 2 πN s 2 N √1− ( ) , (E.9) 2 s 2 N √ N∑ ( s ) 2, 1− (E.10) N s=1 N∑ ( s ) √ 2 ( s ) 2, 1− (E.11) N N s=1 N∑ ( s ) √ 4 ( s ) 2, 1− (E.12) N N s=1 N∑ s=1 Writing Eq.(E.1) in a compact way gives us Eq.(4.49). ( ( )3 s 2 ( s 2 2 1− . (E.13) N) N)
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Appendix E: Summation over the Fermi Surface 149<br />
This gives us<br />
f 1 = 2 N−1<br />
∑<br />
πN<br />
s=1<br />
f 2 = 2<br />
πN<br />
f 3 = 2<br />
πN<br />
f 4 = 2<br />
πN<br />
f 5 = 2<br />
πN<br />
s 2<br />
N<br />
√1− ( ) , (E.9)<br />
2 s 2<br />
N<br />
√ N∑ ( s<br />
) 2,<br />
1−<br />
(E.10)<br />
N<br />
s=1<br />
N∑ ( s<br />
)<br />
√<br />
2 ( s<br />
) 2,<br />
1−<br />
(E.11)<br />
N N<br />
s=1<br />
N∑ ( s<br />
)<br />
√<br />
4 ( s<br />
) 2,<br />
1−<br />
(E.12)<br />
N N<br />
s=1<br />
N∑<br />
s=1<br />
Writ<strong>in</strong>g Eq.(E.1) <strong>in</strong> a compact way gives us Eq.(4.49).<br />
( ( )3<br />
s 2 ( s 2 2<br />
1− . (E.13)<br />
N)<br />
N)
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