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Кузбасский государственный технический университет

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Физика

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Теории / Садовский М.В. Диаграмматика (2005)

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P										,,
% ! ] &, L					&, O & . ( ! W
	G(εnp) =				iεn + ξp
	G(εnp) =			(iεn)2 − ξp2 − \|∆\|2						&Q,
	F (εnp) =				∆
	F (εnp) =			(iεn)2 − ξp2 − \|∆\|2						&Q
# $ &,,, &,, & S #
&,,	&Q &
L " %
' 1	. =					−∆
	Gˆ−1(εnp) = iεn + ξp					−∆				(@ B +
(F − ≡ F +					−∆	iεn − ξp
(F − ≡ F +		G		F −		G	++	G	+−
	Gˆ(εnp) =	G		F −	=	G		G		(@ B$+
	Gˆ(εnp) =	F	+	˜	=					(@ B$+
		F		G		G−+		G−−

± -1 0 " ( + < (±pF ) " 8

% } ± (-"0 -"0+ % "

> (@ @,+ (@ B*+ !			=
(iεn + ξp)G˜(εnp) − ∆ F (εnp) = 1				(@ B?+
(iεn − ξp)F	+		˜	(@ B@+
		(εn) − ∆G(εnp) = 0

H &Q, iεn → ε + iδ *

&QL # # ( * N !

! &QO

TK W

ε < ∆| |

&QQ

N (EF ) =

−| |

N (ε)

√

|ε|

∆

ε2 ∆2

| |

U . & &O& 4 !

* # TK *

! ) # ( # |∆| ! 8 6

4 /2 *&

G ! 1 2 ^ * # * !

∆ 1 * ! 2& G # * ∆ ! # ! ! &JL ! # ! *

* bQ b+Q * # 8 *& ' ! #) ! # &J ! ! * !W

∂

< b

(τ ) >=

< a+

F (pτ =

0) &QL

−∂τ −

±Q

−

p Q

−

	∂	+ ω	< b+	(τ ) >= g		< a+	a	>= g	F ±(pτ = 0) &QO
−								− Q		−
−	∂τ	Q	±Q		Q	p±Q p		− Q	p	−
						p			p

S ) * 1% 2 ! # &JL

# 1( !2 ! # ! ! < a+± ap > & ( !

p Q

# ! ! ! 8 7 5 4 5 !&

& Pgu J < a+p±Qap > / ' 3 (Q = 2pF +=

< ρq >= ρ0δ(q) + ρ1δ(q ± Q), ρ1 < ap+±Qap >	(@ H*+
p
< ρ(x) >= ρ0 + ρ1 cos(2pF x + φ)

, . P - >JK<== = E J<NU0 < J0 0 G/M< K V<1

. & &OW G # ( TK $ ! (

G %* # $ ! ! ! * !* 1 ! 2 !

&QL &QO W

(iωm − ωQ) < b±Q >ωm = −gQ

F (pεn)

(iωm + ωQ) < b+

>ωm = gQ

F ±(pεn)

±Q

S ) ! !W

< b

+ b+

−

2ω

F −(ε

ωm2 +

ωQ2

−Q

ωm

p n

6 ωm = 0 $ ] ) ! !W

&L,

&LI

< bQ + b−Q	>ωm=0= − gQ			dξpT	n	F −(εnp)	&LJ
+		λωQ

*!! p ) ξp *

& , & 5 &JL &LJ * !W

∆ = gQ < bQ + b−+Q	>ωm=0= −ωQ		−EF dξpT	n	F −(εnp)	&L
		λ	EF

( ±EF

* !* ) !& G ) &Q

! * ! * ∆ TK W

1 = λ	0	dξ	ξ2 + ∆2	(T ) th	2T		&L
	EF		1		ξ2 + ∆2	(T )

! *) & &,IQ

. & &, W M 8 # ( * ! !

! * ! *

T = 0 ) ! ! W

Tp0 =	2γ	EF e− λ1	,	∆0 =	π	Tp0	&LQ

	π				γ

! * Tp0 & J

* ! * 1 ! # 2 % &

G * # ! # *) % * (

* ! ! ! & 6 * ! ! !

! !* * 8 # * ! !

* pF l 1 *) % . & &, & . &

&L *) ! W

G(εnp) = G0(εnp) + G0(εnp)Σ(εnpp)G(εnp) + G0(εnp)∆F (εnp) +
+G0(εnp)Σ(εnpp − Q)F (εnp)	&LL
F (εnp) = G0(εnp − Q)Σ(εnp − Qp − Q)F (εnp) + G0(εnp − Q)∆ G(εnp) +
+G0(εnp − Q)Σ(εnp − Qp)G(εnp)	&LO

# ! & . ( ! * * ! * 1 2 ξp−Q = −ξp Q = 2pF

&,IL &,IO &,J W

G(εnp) = [iε˜n + ξp]Det−1

F εnp ˜ Det−1

( ) = ∆

iε˜n = iεn − Σ(εnpp) ≡ iεn − Σn(εn)

˜ n = ∆ + Σ(εnp − Qp) ≡ ∆ + Σa(εn)

∆

Det iεn 2 − ξ2 − |˜ n|2

= ( ˜ ) p ∆

&O,

&OI

&OJ

! "

,J . P - >JK<== = E J<NU0 < J0 0 G/M< K V<1

! !* * # !

# ! !W

Σn(εn) = ρv2νF

−∞ dξpG(εnξp) = − 2

ε˜2

∆˜ n 2

∞

iε˜

)

∞

∆˜

Σa(εn) = ρv2νF −∞ dξpF (εnξp) = −

ε˜2

∆˜ n

Γ = 1

= 2πρv2νF & 5 &O $ &O ! !W

) n

iε˜n

= iεn +

iε˜n

)ε˜n2

+ |∆˜ n|2

∆n

= ∆n −

)ε˜n2

+ |∆˜ n|2

&OQ

&OL

S ! &OQ &OL ( * #

&,J &,JI &,J ! &, &, I !

! & # ! !

&Q, &Q &,,, &,, &

					ε˜n
			un =				&OO
					˜
					∆n
&OQ &OL !W
	εn	= un		Γ 1
			1 −				&,
	∆			∆
						un2 + 1

* un %* ) εn/∆ Γ/∆& / un !

ε˜n = εn +

&, ,

un2 + 1

∆n = ∆ +

+ 1

: &L ! # ∆n # * !W

∆ = −λT

dξpF +(εnξp)

&, I

1 = −λT

∆n

dξp

∆

&, J

(iεn)2

−

ξp2

∆˜ n

− |

G ! *! ] ! #

∆ → 0 * ! & 5

* !W

−∞

(iεn)2

ξp

1 + λT

∞

dξp

−

(iεn)2 ξp2

∆˜ n 2 − (iεn)2

ξp2

−∞

∞

−

∆n

− | |

−

λT

dξp

∆

G * * ! # ! * ! &

( ! * !W

= 1 − λ ln

2γEF

= λ ln

Tp0

πT

− λT

εn

∆n

ε˜n + ∆˜ n

−

εn

= −λT

∆

)

*!! n |εn| EF (

&, ! *!! ! n

& * # # * &OO * * !W

	Tp0			1			∆−1
ln	T = πT		\|		\|	−
ln	T = πT		\|	εn	\|	−		1 + u2
		n						n
		n						n

un * ! &, & G ∆ → 0 &, ! !W

un∆ → εn + Γ εn

|εn|


	un = ∆−1	(εn + Γsignεn)
5			1

1 + un2 → \|un\| =				[\|εn\| + Γ]

			∆

&, Q

&, L

&, O

&,,

&, Q * ! *) * ! * W

ln Tp		= 2πTp n 0	εn	− εn + Γ	&,,,
	Tp0		1	1
		≥

# ! &,O # * ! !* ! * ) &, Q

&J W

= ψ

+ 2πTp

− ψ

&,,

ln Tp

Tp0

G ! Γ ) &, ! !W

Tp ≈ Tp0 −

Γ = Tp0

−

&,,I

4τ

6 Γc Tp * #

* !W

Γc =	1	=	π	Tp0 =	∆00
			2γ		2	&,,J
	τc

, . P - >JK<== = E J<NU0 < J0 0 G/M< K V<1

∆00 # # T = 0 * ! &LQ &

. ! ! # * T = 0& ! W

∆0 = ∆(T = 0; Γ),

∆00 = ∆(T = 0; Γ = 0)

&,,

2π

...& &, J ( ! W

! *

T → 0

* ! *W

T n ... →

dε

1 = −λ

2π

ε|2

&,,

dξp ε˜ε2 − ξp2 − |∆˜

dε

∆ε

∆

ε˜ε ∆ε ) ! ! &OQ $ &, Q W

ε˜ε = ε +

√

&,,Q

1 − u2

∆ε

= ∆0 −

√

&,,L

1 − u2

∆0

= u 1 − ∆

√1 − u2

&,,O

&,,

∆ε → ∆0 ε˜ → ε * ! * !

1 + λ 0

ξp2

+ ∆02

= 1 + λ ln

∆0

= −λ ln

∆0

∞

dξp

2EF

∆00

−

−∞

)

∆˜

− ∆˜ ε2 − ε2

− ξp2 − ∆02

2π

−∞ ε˜2 − ξp2

∞ dε

∞

∆0

dε

∆0

−∞

∆ε

ε˜2

− ∆0

∆˜ 2

∞

−

)

−

T = 0 ! !

*) * W

dε

√

−

√

∆00

∆2

ε2

1 u2(x)

1 x2

∆0

∞

∆−1

∞

−

− &, ,

! &, ,

! x )

u ! !* * ! &,,O

2 * ! x = ∞ ↔ u∞ = ∞ x =

)

( !

∆02 − 1

∆0 > 1

0 ↔ u0

= 0

∆0 ≤

1 x = 0

↔ u0

∆0

−

1 arctg

Γ 2

−

+ ∆0

Γ 2

Γ 1

4 ∆0

∆0

≤

∆00

−2

−

≥

∆0

1/2

S ) ! * ∆ = 0 Γ > Γc = ∆00/2 = πTp0/2γ & &,,J &

P									,Q
G # ( W
	N (EF )	= −π	−∞ dξpIm ε˜ε2 − ξp2 − ∆˜ ε2 =
	N (ε)	1	∞			ε˜ε + ξp
		= Im			ε˜ε	= Im	√	u	&, I
				)
				)	∆˜ ε2 − ε˜ε2			1 − u2

u = u	ε	,	Γ	&,,O & &, I
	∆0		∆0

* ) |u| > 1& #) ε

&, I

* )& 5 &, I

* # *

M axε =

εg ! &,,O

! #

εg

∆0 & G

Γ = 0

εg = ∆00

! |u|

1& S

∆0

! !

& 3 ! *

= u ∆0 ,

∆0

*) # &,,O ! ug ∆0

εg

M ax u 1 − ∆0

√1 − u2

≡ M axF(u) = M ax ∆0

= ∆0

&, J

εg

H * F (ug) = 0 ! !W

ug2

= 0

− ∆0

− (1 − ug )3/2 ∆0

)1 − ug2

(1 − ug2)3/2 =

ug2 < 1

∆0

* * W

ug =

1 − ∆0

&, Q

2/3

G &, Q &,,O * ! # W

εg = ∆0

1 − ∆0

2/3

3/2

&, L

5 ! ! ! ! εg = 0

≥ 1 #

∆0

1 2 ! ! ∆0 = 0& ' *)

# T = 0

∆0 ≤ Γ ≤

∆00

2 exp

Γc

0.91Γc

&, O

∆00 *) Γ =

G Γ = ∆0

* ∆0

= exp

−π4

≈

−

* # %

!! &

> % " 8

" " " " ! <

% " ( ; . ),@*+

,L . P - >JK<== = E J<NU0 < J0 0 G/M< K V<1

, - * / #

6 ! * T > Tp0 !* (

! ( ! )

* * ! ! W

∆Q = gQ < bQ + bQ+ >, Q 2pF

&,I

' * * ! # ! ωm = 0&

4 #) * ! (%%

( 1 2 1 ! # 2 % *) W

F (∆Q; T ) − F (0; T ) = a(Q)|∆Q|2 + b|∆Q|4 + · · ·

&,I,

' ! &,I, * R * % 99

+,,-&

&,I ) % ! # &J W

Hph =

ωQbQ+bQ

&,I

( $ % ! W

Hint =

∆Qap++Qap

&,II

S # ! % ! Q ! !W

< H

>= ω

< b+b

+ b+ b

>= ω

|∆Q|2

N (EF )

∆

&,IJ

Q 2gQ2

−Q

5 # * # ! ! ! ( +,-

! Hint ! * ! . & &,,&

. & &,, [ ) a(Q)|∆Q|2 % . & &,, e

b|∆Q|4& G * ! # +,- % . & &,, [ * *! # # % 1/2 . & &,, e 1/4

& K ! # * * #

1 2 N ! # % 2& N ! a(Q) *

&, $ &I & G ( !* * ! ! # * ! Q = 2pF + q ωm = 0

ξp−2pF = −ξp W

[. & &,, [ ] = −T

2π G0(εnξp)G0(εn, −ξp + vF q) =

= N (EF ) − ln

πT

+ 2

+ 4πT

+ ψ

2 −

4πT − 2ψ

&,I

2γEF

ivF q

5 ! !W

λ − ln

+ 2

2 + 4πT

+ ψ

2 −

4πT − 2ψ

a(q) = N (EF )

πT

2γEF

ivF q

&,I

P F K . L - 2 H H D

. & &,,W 8 % 8 * $ R *

G (%% b * ! ! #) q

! !W

b = 2 T N (EF )

b = 4 [. & &,, e ] = 2 T

2π G02

(εnξp)G02(εn, −ξp)

&,IQ

2πi 2πi

n (iεn

ξp)2

(iεn

+ ξp)2 = −T N (EF )2πi

(2iεn)3 =

dξp

signεn

−

∞

N (EF )

7ζ(3)

N (EF )

16π2T 2

n=0

(n + 1/2)3

16π2T 2

&,IL

! * a(q) ! !* &,I & G ! q

& L * !W

a(q) = N (EF )

ln Tp0

+ 2 ψ

2 +

4πT

2 − 4πT − ψ

≈

ivF q

≈

Tp0

16π2T 2

N (E

)

T − Tp0

7ζ(3)

v2 q2

Tp0

N (E )

T − Tp0

ξ2(T )q2 &,IO

# # ! & O 1 2&

. P - >JK<== = E J<NU0 < J0 0 G/M< K V<1

8 * $ R * !

F (∆

; T )

F (0; T ) = N (E )

T − Tp0

∆

2 + N (E )ξ2(T

) ∆

−

Tp0

−

F 0

p0 |

+ 7ζ(3) N (EF )|∆Q|4 16π2T 2

&,J

S ) &I

# % * * ! T

Tp0 * ! &

a(q)

(@$+ ( ωm = 0+ > =

ivF q

−

ivF q

− ψ

(@ )$)+

a(q) = N (EF ) ln

Tp0

2πT τ

4πT

2πT τ

4πT

2 vF q 4πT =

a(q)

≈

N (E

)

T − Tp

v2 q2

(@ )$7+

16π2T 2

Tp " (@$?+

+ ψ

− ψ

= 0

(@ )$+

Tp0

2πTpτ

% B (@ )$7+ =

∞

ψ(2)

B =

= −

(@ )$$+

)n + 1 +

2πT τ

n=0

2πT τ *

2 ! - 8

(2)

(T ) = −

≈

ξ0

32π2T 2

2πT τ

7ζ(3)v2

4πT

(@ )$?+

16π2 T 2

≈ vF2 τ 2

τ1 4πT

M τ > τc =

= 2∆−1

πTp0

00 } .; %'' 1

, & 4 #

+ %

3 * # ^ ! ! # ! *

! \ / ! # ! !

! ! * # % *

! & ( ! ! !

% !

% * * ! ! !& 6

! 1 # 2 ( ! #

) ! 5 ! !

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