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c0
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1 2νF J c0
I
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5
e
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FJ
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−δΓ(ε)
0
1 ˆ (|ε| + νS ) C0 (ε) = 1 + δΓ(ε)C0 (ε) − (L 0 − Γsf (ε))C0 2 X 1 X 1 1 = 1 + νS χ⊥0 ωS2 2 C (ε) − C0 (ε − ω) − 0 2 ω2 ω2 ω6=0
|ω|>|ε|
= 1 + LC0 (ε)
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χ⊥0 =
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1 C0 (ε) ω2
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C0 (ε) =
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ρS =
ωS νS , xS = , 2πT 2πT
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x = ε/(2πT ) = m + 1/2 ∈ Z + 1/2 I
1 2
II
SI (x, ρS ) = 2
0
sf
X 1 1 4 ψ 0 (x + 1/2) = 2 |x| + ρS n x + ρS |n|>|x|
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SII (x, ρS ) = =
X 1 1 − 2 n |x − n| + ρ S n6=0
X
0<|n|<|x|
1 1 = 2 n |x| + ρS
1 [ψ(x + ρS ) − ψ(x + ρS + 1)] + (x + ρS )2 1 1 + [ψ(x + 1/2) − ψ(ρS + 1/2)] + ψ 0 (x + 1/2) + 2 (x + ρS ) x + ρS 1 1 + [ψ(x + 1/2) − ψ(ρS + 1/2)] − ψ 0 (x + 1/2) 2 (x − ρS ) x − ρS
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S
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X 1 Tc0 (0) (1) ln = 2πT − C0 (ε) − C0 (ε) T |ε| ε>0
X 1 1 (SI (x, ρS ) − SII (x, ρS )) = ψ(1/2 + ρS ) − ψ(1/2) − χ⊥0 ρS x2S 2 x + ρ S x>0
0;29¢P 1=3QN7¡52Fv5<14?:FÃ6G379@FG<94;N36;6M<94;<9=Ú:@PNZ¡a¡75P 9 T = T + ∆T (h) ¨7« P 9 T = T (ν ) 3QN737?29@ÑFG9<94;N376;NV29@;<9¢Ú!:@PN\-I ¡ /2:76« P9@N2¡¨<:;FND4< :N7«÷97N¨EN <∆T:1÷9P /
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S
S
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X 1 ∆T (h) 1 (SI (x, ρoS ) − SII (x, ρoS )) (1 − ψ 0 (1/2 + ρoS )ρoS ) = χ⊥0 ρoS x2S o TAG 2 x + ρ S x>0
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o S
o S
o S
o2 S
o S
X 1 ∆T (h) 1 (1 − ψ 0 (1/2 + ρoS )ρoS ) = χ⊥0 ρoS x2S (SI (x, ρoS ) − SII (x, ρoS )) − o TAG 2 x + ρ S x>0 02 h γ − ρoS ψ 0 (ρoS + 1/2). + ν˜ νS νS
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S
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1 |ε|C0 (ε) = 1 − νz C0 (ε) − (Lˆ0 + Γ (ε))C0 (ε) = 1 − νz C0 (ε) − LC0 (ε). 2 hSz i 2 ψ(|x| + 1/2 + i xS ) S(S + 1) π 2 hSz i Γ (ε) = νS coth πxS − ψ(|x| + 1/2 + i xS ) S(S + 1) π
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(1,2)
(ε) + C0
(ε),
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C0 (ε) =
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C0
.0FG949@F ¤ (0)
LC0 (ε) =
(0)
(0)
(ε) = C0 (ε)LC0 (ε).
1 1 νS hSz i (SI (x) + SII (x)) 2 (2π)2 T S(S + 1)
X
2xS 1 = 2 2 + x |x − n| n S n iπ 2 ψ(1/2 + x − ixS ) − ψ(1/2) + coth πxS = ix + xS 2 2 iπ + ψ(1/2 + x + ixS ) − ψ(1/2) − coth πxS −ix + xS 2
SI (x) =
SII (x) =
1 X 2xS = |x| n2 + x2S |n|<|x|
=
2 (i(ψ(1/2 + x + ixS ) − ψ(1/2 + x − ixS )) + π coth πxS ) x
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∆T = Tc0 − T
2:Q<6>
X (1,1) X νz 1 1 νS hSz i X 1 ∆T (1,2) = 2πT C0 (ε) + C0 (ε) = 2πT + (SI (x) + SII (x)) 2 2 T S(S + 1) Tc0 ε 2 (2π) x ε>0 ε>0 x>0
°7Ý
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T (h)
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1 1 νS 1 X 1 T∞ − T = (SI (x) + SII (x)) Tc0 2 (2π)2 T S + 1 x>0 x
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xS → ∞
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2 π 4 2 SI (x) = + (−ψ(1/2)) , ln xS + 2(−ψ(1/2)) ln xS + x xS 6
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±5P 5
X1 4 SII (x) = [ln xS + (−ψ(1/2)) + 1] . x xS x>0 T (h)
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