6 Z 4, pp. 622∼632

 6 Z 4, pp. 622∼632

ø m

nj ˜ m X N Ë M X ê s5 ; c" e8 ý w Š ¹ ÅX ì Äß Ã Å ­ Žä à ŠŒ ˜ m6 Kª Ž®  o­ Ž { ¢¨ | 8 ý X N ˽  ʔ X ¢  M × D V R Ë Ä Z Ø



¡) o £ Ó

Ø 

æ· ¡ ¤ @ /† < Ɠ § Ó ü t o † < Æõ  x 9  BK21 Ó ü t o  á Ԗ ÐÕ ªÏ þ ›, ' õ AÅ Ò 361-763 (2010¸   4 Z 4 7{ 9  ~ à Î6 £ §, 2010¸   6 Z 4 10{ 9  > F  S X ‰& ñ )

Stratonovich   ¨ 8 Š _  “ ¦„  & h “  + þ AI ü < Laplace ~ ½ ÓZ O `  ¦  6   x # Œ { 9 ì ø Í& h “    l  © œ? /\ " f_  “ ¦„  & h 

“

  Û ¼— 2 ; ì ø ÍX <Ø Ôµ 1 ÏÛ ¼ — ¸4 S q_   l  o $ í ì  r`  ¦ & ñ S X ‰ >  ½ ¨ % i  . — ¸4 S q_  — ¸Ž  H % ò % i \  @ /K " f | 9 " fB > h



 Ã º σ

0

 ë ß –7 á ¤   H ~ ½ Ó& ñ d ” “   Langevin † < Êà º_  + þ AI   © œ{ © œy    + þ A÷ &“ ¦ Õ ª כ _  _ p • ¸ & ñ  l  © œ? /

\

" f Ô  ¦ì  r" î K f ” `  ¦ ˜ Ѐ Œ ¤ . XY x 9  Heisenberg F G ô  Ç\ " f  H d ” t # Q Æ Ò† ½ Ós  Ò q t|   . — ¸Ž  H  l  o $ í ì

 r`  ¦ σ

0

_  † < Êà º– Ð" f ½ ¨ % i   H X <, : £ ¤ y   l  © œs    t €   | 9 " fB > h† < Êà º σ

⁰

ü <  l  o_  › ' a >  Ó ü t o 

&

h Ü ¼– Ð " î S X ‰ K  f ” `  ¦ ^  ¦ à º e ” % 3  .

Ù þ

˜d ” # Q:  l  o, ì  r C † < Êà º, { 9 ì ø Í& h “    l  © œ

Exact Magnetization Components of the Classical Spin van der Waals Model in Arbitrary Static Magnetic Fields

Suhk Kun Oh ^∗

BK21 Physics Program and Department of Physics, Chungbuk National University, Cheongju 361-763

(Received 7 April, 2010 : accepted 10 June, 2010)

By utilizing the classical version of the Stratonovich transformation and the method of Laplace, we obtained the magnetization components for the classical spin van der Waals model in arbitrary magnetic fields. The equation for the magnitude of the usual order parameter, σ

0

, for all limits of the model in the form of a Langevin function is greatly modified, its meaning is found to be rather obscure in the presence of static magnetic fields, and it even contains an extra term in the XY and the Heisenberg limits. All the magnetization components are given as a function of σ

0

. In the zero-field limit, the relations between σ

0

and the magnetization, which can also be used as the order parameter, are explicitly manifested.

PACS numbers: 05.50.+q, 05.70.Ce, 75.10.Hk

Keywords: Magnetization, Partition function, Arbitrary magnetic field

I. " e  ] Ø

Ã

º z  ¸  1 l x î ß –     Û ¼— 2 ;— ¸4 S q“ É r ] X ƒ    $ í Ó ü t| 9 _   $ í

`

 ¦ s K    H X < j þ t — ¸ e ” % 3   [1–7]. s  Qô  Ç — ¸4 S q[ þ t _ 

∗

E-mail: [email protected]

ì

 r C † < Êà º\  ¦ ½ ¨   H כ s  B Ä º # Q 9Ä ºÙ ¼– Ð @ /> h   H  & h Ü ¼

–

Ð  À Ò# Q”     e ”  . Õ ªX O t ë ß – & ñ S X ‰ ô  Ç   õ   H   _  1 p x

@

/% ƒ! 3  — ¸Ž  H   H   ~ ½ ÓZ O _  & h ½ + Ë$ í `  ¦ µ 1 ß) € ï  r  . ˜ Ðl \  ¦ [ þ t

€

 ,     Û ¼— 2 ;— ¸4 S q_  e ” > ‰ & ³ © œ\  › ' a ô  Ç F ½ ©   oç  H s  : r

`

 ¦ s 6   x ô  Ç ˜ м # $ í \  › ' a ô  Ç  7 H _   H t F K  t  ´ ú §“ É r : Ÿ x > % i 

†

< Æ ƒ  ½ ¨ _  s 3 l q`  ¦  H   e ”   [8]. Õ ªX O t ë ß – F ½ ©   o

-622-

ç

 H s  : r \  ½ + ËZ O $ í `  ¦  Ò# Œô  Ç  כ “ É r 2 " é ¶ Ising — ¸4 S q`  ¦ q  2

Ÿ

¤ ô  Ç # Œ Q & ñ S X ‰ >  Û  ¦ o   H — ¸4 S q[ þ t s % 3   [9–11]. Õ ª Q Ù

¼– Ð & ñ S X ‰ ô  Ç   õ   H † ½ Ó © œ • ¸¹ ¡ § s  ÷ & 9 ¢ ¸ô  Ç ´ ú §“ É r u 

e ”

 .

s

p  ƒ  / å L ô  Ç  ü < ° ú  s , & ñ S X ‰ >  Û  ¦ o   H    — ¸4 S q– Ð



 H Ising — ¸4 S qõ  Õ ª כ Ü ¼– Ð Â Ò'   Ò q t ) a  כ [ þ t s  e ”  . Ô  ¦' Ÿ 

> • ¸ s [ þ t — ¸4 S q“ É r | 9 " f B > h  à º Hamiltonianõ  “ §

¨ 8

Š 0 p x$ í `  ¦ t Ù ¼– Ð “ ¦Ä »_  1 l x§ 4 † < Æ& h  $ í | 9 `  ¦ t t 

· ú

§  H  .   " f 1 l x§ 4 † < Æ& h  $ í | 9 _  ƒ  ½ ¨\  ¦ 0 AK " f  H | 9 " f B

> h  à º_  " é ¶ s  1 ˜ Ð   H 7 ˜'  Û ¼— 2 ;— ¸4 S q`  ¦ “ ¦ 9K   ë

ß – ô  Ç . „  + þ A& h “   7 ˜'  Û ¼— 2 ;— ¸4 S q– Ѝ  H XY x 9  Heisenberg

—

¸4 S qs  e ”   [6,12,13].  © œ ç ß –é ß –ô  Ç 1 l x§ 4 † < Æ& h  Ó ü t o | ¾ Ós 



l  os Ù ¼– Ð, 7 ˜'  Û ¼— 2 ;— ¸4 S q_  1 l x§ 4 † < Æ`  ¦ ƒ  ½ ¨ l  0 A ô

 Ç ' Í é ß –> – Ð" f & ñ  l  © œs     9 e ” `  ¦ M :_  ì  r C † < Êà º\  ¦

½

¨K   ô  Ç . Õ ª Q
 > í ß – © œ_  # Q 9¹ ¡ § M :ë  H \  & ñ  l  © œ s

    9 e ” `  ¦ M : Û  ¦ o   H 7 ˜'  Û ¼— 2 ;— ¸4 S q“ É r ™  ¥ t  · ú § . Õ ª

  Û  ¦ 2 ; — ¸4 S q• ¸ 1      " é ¶ \ " f ½ ¨ô  Ç כ Ü ¼– Ð  © œ„  s 

‰

&

³ © œ`  ¦ ˜ Ðs t  · ú §  H  .

Õ

ªo  # Œ  © œ„  s  { 9 # Q± ú ˜ M :_  “ ¦Ä » 1 l x§ 4 † < Æ& h  $ í | 9 

`

 ¦ % 3 l  0 AK " f Û ¼— 2 ; ì ø ÍX <Ø Ô µ 1 ÏÛ ¼ — ¸4 S q (SVW) [14–16]s 



“ ¦ Ô  ¦ o Ä º  H Á ºô  Ç# 3 0 A  © œ  ñ Œ •6   x`  ¦   H 7 ˜'  Û ¼— 2 ;— ¸ 4

S qs  • ¸{ 9 ÷ &% 3  . ¨ î + þ A$ í | 9 \  › ' a ô  Çô  Ç SVW  H ¨ î ç  H  © œ Ä » + þ

A_  — ¸4 S qs t ë ß – Õ ª כ _  1 l x§ 4 † < Ɠ É r ¨ î ç  H  © œ — ¸4 S q_  Õ ª כ õ

  Ø Ô>  „  ) €  ™ è t  · ú §“ É r 1 l x§ 4 † < Æ`  ¦ ”   . z  ´] j– Ð s

 — ¸4 S q“ É r à º¨ î  l  © œõ  à ºf ”   l  © œ_  ” > r F \  ¦ ½ ¨ì  r ½ + É Ã

º e ” Ü ¼Ù ¼– Ð, s  Qô  Ç  © œ5 Å q \ " f F p e ”   H 1 l x§ 4 † < Æ& h  $ í | 9  _

 s \  ¦ ˜ Ð# Œ×  ¦  כ `  ¦ l @ /½ + É Ã º e ”  .

Õ

ªX O t ë ß – s X O >  ç ß –é ß –ô  Ç — ¸4 S q\  @ /K " f• ¸ > í ß – © œ_  # Q



9¹ ¡ § Ü ¼– Ð “   # Œ { 9 ì ø Í& h “    l  © œs     9 e ” `  ¦ M :_  & ñ

§

4 † < Æ& h  x 9  1 l x§ 4 † < Æ& h  $ í | 9 s  · ú ˜ 94 R e ” t  · ú § . z  ´] j– Ð SVW _  & ñ § 4 † < Æ& h  $ í | 9 “ É r à ºf ”  © œ_   â Ä º\ ë ß – · ú ˜ 94 Re ” 



 [17]. Õ ª! 3 \ • ¸ Ô  ¦ ½ ¨ “ ¦, “ ¦„  & h “   Û ¼— 2 ; ì ø ÍX <Ø Ôµ 1 ÏÛ ¼

—

¸4 S q (CSVW) [18]s  “ ¦  ÒØ Ô  H S = ∞ F G ô  Ç`  ¦ 2 [ €   SVW _  q  “ § ¨ 8 Š 0 p x$ í ë  H ] j\  ¦ x ½ + É Ã º e ” # Q" f  8¹ ¡ ¤  8

· ú

¡Ü ¼– Ð
 ° ú ˜ à º e ”  .

Õ

ª QÙ ¼– Ð ‘ : r  7 Hë  H \ " f  H { 9 ì ø Í& h “    l  © œs     9e ”   H CSVW _  & ñ  l  o\  ¦ ½ ¨K ˜ Ðl – Ð ô  Ç . Õ ª   õ – Ð" f & ñ



l  © œs     9 e ” t  · ú §“ É r  â Ä º\   H s „  _    õ d ” õ  ° ú  

“ É

r ~ ½ Ó& ñ d ” `  ¦ % 3 t ë ß –, { 9 ì ø Í& h “   & ñ  l  © œs     9 e ”   H  â Ä

º\   H Langevin ~ ½ Ó& ñ d ” s    + þ A÷ &  H כ ü @\ • ¸ „  ) € \ V



© œ t  · ú §€ Œ ¤~   Æ Ò† ½ Ós 
 
>  H † d`  ¦ µ 1 Ï|  >  | ¨ c  כ s

 . Õ ªo “ ¦,  l  © œs  \ O `  ¦  â Ä º\   H Ä ºo  % 3 “ É r ~ ½ Ó& ñ d ”

Ü ¼– Ð Â Ò'  Langevin ~ ½ Ó& ñ d ” \ 
 
  H | 9 " f B > h   Ã

º_  ß ¼l ü <  l  o $ í ì  r  s _  › ' a > \  ¦ f ” › ' a& h Ü ¼– Ð s  K

½ + É Ã º e ” >  | ¨ c  כ s  .

II. CSVW { ¢¨ | 

CSVW   H 8 ú x Û ¼— 2 ;$ í ì  r S α (α = x, y, z) _  † < Êà º– Ð" f   6

£ § Hamiltonian

H = − J

4N (S _x ² +S _y ² )− J z

4N S _z ² − h x

2 S _x − h y

2 S _y − h z

2 S _z (1) Ü

¼– Ð l Õ ü t ÷ &  H X <, J ü < J ^z   H y Œ •y Œ •   ½ + Ë © œÃ º, h ^α (α = x, y, z)  H α ~ ½ ӆ ¾ Ó_  & ñ  l  © œ $ í ì  r s  . 8 ú x Û ¼— 2 ;$ í ì  r S α



 H S α = P N

i=1 s ^α _i – Ð & ñ _    H X <, s ^α i   H     © œ_  0 Au  i \  Z  ~“   “ ¦„  Û ¼— 2 ;s  . F G y Œ • (polar angle) θ i ü < ~ ½ Ó0 A y

Œ

•(azimuthal angle) φ ⁱ _  † ½ ÓÜ ¼– Ð" f ³ ð‰ & ³ €   y Œ •y Œ • s ^x i = sin θ i cos φ i , s ^y _i = sin θ i sin φ i , s ^z _i = cos θ i s  .

#

Œl " f  l  r„  q Ö  ¦ (gyromagnetic ratio) “ É r  l  © œ

$ í

ì  r h α \  Ÿ í† < Ê % i  ..

III. Ä Z Ø9 0] K ¤• ¤

CSVW _  ì  r C † < Êà º_  > í ß –`  ¦ ~ 1 >  l  0 A # Œ €  $  { 9

ì ø Í & ñ  l  © œs     9 e ”   H  © œ  ñ Œ •6   x`  ¦ t  · ú §  H N > h _

 “ ¦„  & h  Û ¼— 2 ;Ü ¼– Ð ½ ¨$ í  ) a — ¸4 S q_  ì  r C † < Êà º\  ¦ “ ¦ 9K 

˜

Ð  [19–22]. s   â Ä º\  d ”  (1)\ " f J = J ^z = 0 s  . Õ ª o

 # Œ

Z 0 (a, b, c) = Π ^N _i=1 Z 2π

0

Z π 0

sin θ i dθ i dφ i e ^{−β P}

^Ni=1

^[

^hx2

^s

^xi

⁺

^hy2

^s

^yi

⁺

^hz2

^s

^zi

^]

= Π ^N _i=1 Z 2π

0

Z π 0

sin θ i dθ i dφ i e ^β[

^hx2

^s

^xi

⁺

^hy2

^s

^yi

⁺

^hz2

^s

^zi

^]

= [ Z 2π

0

Z π 0

e ^{c cos θ}

ⁱ

^{+a sin θ}

ⁱ

^{cos φ}

ⁱ

^{+b sin θ}

ⁱ

^{sin φ}

ⁱ

sin θ i dθ i dφ i ] ^N

= (4π) ^N sinh ^N √

a ² + b ² + c ²

(a ² + b ² + c ² ) ^N/2 (2)

“

 X <, a = βh x /2, b = βh _y /2, c = βh _z /2 s “ ¦, ⍠ H (] X @ /

“

: r • ¸) × (Boltzmann  © œÃ º)_  % i à ºs  9,  6 £ § d ”  [23]

R 2π 0

R π

0 f (c cos θ + a sin θ cos φ + b sin θ sin φ) sin θdθdφ

= 2π R π 0 f ( √

a ² + b ² + c ² cos α) sin αdα = 2π R 1

−1 f (( √

a ² + b ² + c ² z)dz (3)

`

 ¦ s 6   x % i “ ¦, ፠ H " é ¶ A _  ý a³ ð> \  ¦ D h– Ðî  r ý a³ ð> _  z» ¡ ¤ \  @ / # Œ 3 " é ¶  © œ\ " f  r„  r †   # Q‹ "  y Œ •• ¸s  .

Ñ ü

t P :– Ð Stratonovich   ¨ 8 Š [24] _  “ ¦„  + þ AI 

e

^γ24α

= r α π

Z ∞

−∞

e ^−αx

²

^+γx dx (4)

\

 ¦ s 6   x % i   H X <, ㍠ H # Q‹ "  ƒ  í ß – s “ ¦ ፠ H z  ´Ã º  © œÃ ºs 



.  t } Œ •Ü ¼– Ð Laplace ~ ½ ÓZ O  [25]`  ¦  6   x   H X <, # Q‹ "  ¸ ú ˜ '

Ÿ 1 l x   H † < Êà º f(z), g(z)ü < &  ê ø Í N° ú כ\  › ' a K " f Z ∞

−∞

g(z)e ^{N f (z)} dz ≈

s 2π

−N f ⁰⁰ (z 0 ) g(z 0 )e ^{N f (z}

⁰

⁾ (5) s

 . # Œl " f, f(z)\  ¦

f (z) = f (z 0 )+(z −z 0 )f ⁰ (z 0 )+ 1

2 (z −z 0 ) ² f ⁰⁰ (z 0 )+· · · (6) ü

< ° ú  s  „  > h   H X < f ⁰ (z ₀ ) = 0 s  “ ¦ & ñ ô  Ç .

Ä

ºo   H CSVW _  [ j F G ô  Ç Ising, XY x 9  Heisenberg F G ô

 Ç`  ¦ “ ¦ 9 l – Ð ô  Ç .

Case 1. Ising ; ³ ø 5  d ”

 (1)_  Hamiltonian\ " f J = 0“   Ising F G ô  Ç\ " f ì  r C

† < Êà º  H

Z I (a, b, c) = Π ^N _i=1 Z 2π

0

Z π 0

sin θ i dθ i dφ i e

^βJz4N

^S

^2z

^+aS

^x

^+bS

^y

^+cS

^z

(7)

_

 + þ AI \  ¦ ”   . d ”  (4)\  α = N/βJ ^z ü < γ = S ^x \  ¦ u 

¨ 8

Š €  

Z I (a, b, c) = s

N πβJ z

Z ∞

−∞

e ⁻

^βJzN

^z

²

Z 0 (a, b, c + z)dz,

= s

N πβJ _z

Z ∞

−∞

e ⁻

^βJzN

^z

²

(4π) ^N sinh ^N pa ² + b ² + (c + z) ²

[a ² + b ² + (c + z) ² ] ^N/2 dz,

≡ (4π) ^N s

N πβJ z

Z ∞

−∞

e ^{N f}

^I

^(z) dz, (8)

f _I (z) = − 1 βJ z

z ² + ln sinh pa ² + b ² + (c + z) ²

pa ² + b ² + (c + z) ² (9)

s

 .  l  © œs  \ O `  ¦ M : ° ú  “ É r — ¸4 S q`  ¦   É r ~ ½ ÓZ O Ü ¼– Ð > í ß – ô

 Ç  â Ä ºü < ° ú  “ É r   õ 
š ¸• ¸2 Ÿ ¤ σ = 2z/βJ z \  ¦ • ¸{ 9  

#

Œ d ”  (8)`  ¦  r  æ ¼€  

Z _I (a, b, c) = (4π) ^N s

N πβJ z

Z ∞

−∞

e ^{N f}

^I

^(σ) dσ, (10)

f _I (σ) = − βJ _z 4 σ ² + ln

sinh q

a ² + b ² + (c + ^βJ ₂

^z

σ) ² q

a ² + b ² + (c + ^βJ ₂

^z

σ) ² (11)

s

 .

t

F K Laplace _  ~ ½ ÓZ O `  ¦ + ‹" f &  ê ø Í N\  @ /K " f & h   H

&

h “   & h ì  r`  ¦ €  

Z I (a, b, c) ≈ (4π) ^N

s βJ z

−2f _I ⁰⁰ (σ ₀ ) e ^{N f}

^I

^(σ

⁰

⁾ , (12)

f _I ⁰ (σ ₀ ) = βJ _z

2 [−σ ₀ + c + ^βJ ₂

^z

σ ₀ q

a ² + b ² + (c + ^βJ ₂

^z

σ 0 ) ² coth

r

a ² + b ² + (c + βJ _z 2 σ ₀ ) ²

− c + ^βJ ₂

^z

σ 0

a ² + b ² + (c + ^βJ ₂

^z

σ ₀ ) ² ] = 0, (13) f _I ⁰⁰ (σ ₀ ) = βJ _z

2 [− c c + ^βJ ₂

^z

σ 0

−

βJ

z

2 (c + ^βJ ₂

^z

σ ₀ ) ² {a ² + b ² + (c + ^βJ ₂

^z

σ 0 ) ² }

³²

×{coth r

a ² + b ² + (c + βJ z

2 σ 0 ) ² − 2 q

a ² + b ² + (c + ^βJ ₂

^z

σ ₀ ) ²

}]. (14)

Case 2. XY ; ³ ø 5  d ”  (1)\ " f J ^z = 0“   XY F G ô  Ç\ " f ì  r C † < Êà º  H

Z XY (a, b, c) = Π ^N _i=1 Z 2π

0

Z π 0

sin θ i dθ i dφ i e

^4NβJ

^(S

^2x

^+S

^y2

^)+aS

^x

^+bS

^y

^+cS

^z

(15)

_

 + þ AI \  ¦ ”   . α = N/βJ, γ = S x Õ ªo “ ¦ δ = S y \  ¦



6 £ § \  @ /{ 9 ô  Ç .

e

^{γ2 +δ24α}

= α π

Z ∞

−∞

e ^−αx

²

^+γx dx Z ∞

−∞

e ^−αy

²

^+δy dy. (16) Õ

ªo  # Œ ì  r C † < Êà º  H  6 £ § õ  ° ú   .

Z XY (a, b, c) = N πβJ

Z ∞

−∞

Z ∞

−∞

e ⁻

^βJN

^(x

²

^+y

²

⁾ Z 0 (a + x, b + y, c)dxdy

= N πβJ

Z ∞

−∞

Z ∞

−∞

e ⁻

^βJN

^(x

²

^+y

²

⁾ × (4π) ^N sinh ^N p(a + x) ² + (b + y) ² + c ² [(a + x) ² + (b + y) ² + z ² ] ^N/2 dxdy,

≡ (4π) ^N N πβJ

Z ∞

−∞

Z ∞

−∞

e ^{N f}

^XY

^(x,y) dxdy, (17)

f XY (x, y) = − 1

βJ (x ² + y ² ) + ln sinh p(a + x) ² + (b + y) ² + c ²

p(a + x) ² + (b + y) ² + c ² . (18)

t

F K D h– Ðî  r   à º X = a + x ü < Y = b + y\  ¦ • ¸{ 9  

#

Œ s    õ d ” `  ¦ F G ý a³ ð › ' a > d ”  R = √

X ² + Y ² ü < Θ =

arctan (Y /X) _  † ½ ÓÜ ¼– Ð" f  r  ³ ð‰ & ³ €  

Z XY (a, b, c) = (4π) ^N N

πβJ e ⁻

^βJN

^(a

²

^+b

²

⁾ Z ∞

0

e ⁻

^βJN

^R

²

sinh ^N √ R ² + c ² (R ² + c ² ) ^N/2 RdR

Z 2π 0

e

^2NβJ

(a cos Θ+b sin Θ)

dΘ,

= 2(4π) ^N N

βJ e ⁻

^βJN

^(a

²

^+b

²

⁾ Z ∞

0

e ⁻

^βJN

^R

²

sinh ^N √ R ² + c ² (R ² + c ² ) ^N/2 I ₀ ( 2N

βJ R p

a ² + b ² )RdR (19)

s

  ) a  . # Œl " f  H Gradshteyn [23] _   6 £ §d ”  Z 2π

0

e p cos x+q sin x

dx = 2πI 0 ( p

p ² + q ² ) (20)

\

 ¦  6   x % i Ü ¼ 9, I 0 (x)  H modified Bessel function of or- der 0\  ¦
  · p .

Ä

ºo _  > í ß –  õ   l  © œs  \ O `  ¦ M :_  ° ú  “ É r — ¸4 S q`  ¦  

 É

r ~ ½ ÓZ O Ü ¼– Ð > í ß – % i `  ¦ M :ü < ° ú  >  ÷ &• ¸2 Ÿ ¤ σ = 2R/βJ \  ¦

•

¸{ 9  # Œ d ”  (19)\  ¦  6 £ § õ  ° ú  s  ³ ð‰ & ³  .

Z XY (a, b, c) = N (4π) ^{N −}

¹²

r βJ

2 e ⁻

^βJN

^(a

²

^+b

²

⁾ Z ∞

0

e ^{N f}

^XY

^(σ) σdσ, (21)

f XY (σ) = − βJ 4 σ ² + ln

sinh q

c ² + ( ^βJ ₂ σ) ² q

c ² + ( ^βJ ₂ σ) ² + 1

N ln I 0 (N σ p

a ² + b ² ). (22)

Laplace _  ~ ½ ÓZ O `  ¦ + ‹" f &  ê ø Í N° ú כ\  @ /K " f & h   H& h Ü ¼

–

Ð & h ì  r`  ¦ €    6 £ § õ  ° ú   .

Z _XY (a, b, c) ≈ N (4π) ^{N −}

¹²

r βJ

2 e ⁻

^βJN

^(a

²

^+b

²

⁾ σ ₀

s βJ

−2f _XY ⁰⁰ (σ 0 ) e ^{N f}

^XY

^(σ

⁰

⁾ , (23)

f _XY ⁰ (σ 0 ) = βJ 2 [−σ 0 +

βJ 2 σ 0

q

c ² + ( ^βJ ₂ σ ₀ ) ² coth

r

c ² + ( βJ 2 σ 0 ) ² −

βJ 2 σ 0

c ² + ( ^βJ ₂ σ 0 ) ²

+ 2 βJ

p a ² + b ² I ₁ (N σ ₀ √

a ² + b ² ) I ₀ (N σ ₀ √

a ² + b ² ) ]

≈ βJ 2 [−σ 0 +

βJ 2 σ 0

q

c ² + ( ^βJ ₂ σ 0 ) ² coth

r

c ² + ( βJ 2 σ 0 ) ² ,

−

βJ 2 σ 0

c ² + ( ^βJ ₂ σ 0 ) ² + 2 βJ

p a ² + b ² ] = 0. (24)

#

Œl " f  H &  ê ø Í x° ú כ\  › ' a K " f $ í w n    H & h   H& h “   d ”  [26]

I ν (x) ≈ e ^x

√

2πx [1 − 4ν ²

8z + · · ·] (25) ü

< I 0 ⁰ (x) = I 1 (x)\  ¦ s 6   x % i “ ¦ I ¹ (x)  H modified Bessel

function of order one s  . ¢ ¸ô  Ç

f _XY ⁰⁰ (σ ₀ ) = βJ 2 [−1 −

βJ 2

c ² + ( ^βJ ₂ σ 0 ) ² + 2( ^βJ ₂ ) ³ σ ² ₀ [c ² + ( ^βJ ₂ σ 0 ) ² ] ² +{

βJ 2

q

c ² + ( ^βJ ₂ σ ₀ ) ²

− ( ^βJ ₂ ) ³ σ ² ₀

[c ² + ( ^βJ ₂ σ ₀ ) ² ]

³²

} coth r

c ² + ( βJ

2 σ 0 ) ² ]. (26)

Case 3. Heisenberg ; ³ ø 5  d ”

 (1)_  Hamiltonian\ " f J ^z = J Heisenberg F G ô  Ç\ 

"

f, ì  r C † < Êà º  H

Z H (a, b, c) = Π ^N _i=1 Z 2π

0

Z π 0

sin θ i dθ i dφ i e

^βJ4N

^(S

^2x

^+S

^2y

^+S

^2z

^)+aS

^x

^+bS

^y

^+cS

^z

(27)

_

 + þ AI \  ¦ t   H X <,

e

^{γ2 +δ2 +ε24α}

= α π

Z ∞

−∞

e ^−αx

²

^+γx dx Z ∞

−∞

e ^−αy

²

^+δy dy.

Z ∞

−∞

e ^−αz

²

^+εz dz (28)

\

 α = N/βJ, γ = S x , δ = S _y Õ ªo “ ¦ ε = S z \  ¦ @ /{ 9  

€

 

Z H (a, b, c) = ( N πβJ )

³²

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

e ⁻

^βJN

^(x

²

^+y

²

^+z

²

⁾ Z 0 (a + x, b + y, c + z)dxdydz,

= ( N πβJ )

³²

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

e ⁻

^βJN

^(x

²

^+y

²

^+z

²

⁾

×(4π) ^N sinh ^N p(a + x) ² + (b + y) ² + (c + z) ²

[(a + x) ² + (b + y) ² + (c + z) ² ] ^N/2 dxdydz,

≡ (4π) ^N ( N πβJ )

³²

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

e ^{N f}

^H

^(x,y,z) dxdydz, (29)

f H (x, y, z) = − 1

βJ (x ² + y ² + z ² ) + ln sinh p(a + x) ² + (b + y) ² + (c + z) ²

p(a + x) ² + (b + y) ² + (c + z) ² (30)

s

 . s  d ” `  ¦   à º X = a + x, Y = b + y x 9  Z = c + z _  † ½ ÓÜ ¼– Ð" f
 ? /“ ¦,  r  F G ý a³ ð    o

d ”

 R = √

X ² + Y ² + Z ² , Φ = arctan (Y /X) x 9  Θ = arctan √

X ² + Y ² /Z † ½ ÓÜ ¼– Ð" f + ‹˜ Ѐ  

Z H (a, b, c) = (4π) ^N N

πβJ e ⁻

^βJN

^(a

²

^+b

²

^+c

²

⁾ Z ∞

0

e ^{−N (}

^βJ1

^R

²

^+ln

^{sinh RR}

⁾ R ² dR

× Z 2π

0

Z π 0

e

^{2N RβJ}

(a sin Θ cos Φ+b sin Θ sin Φ+c cos Θ)

sin ΘdΘdΦ,

= 2(4π) ^N ( N

πβJ )

³²

e ⁻

^βJN

^(a

²

^+b

²

^+c

²

⁾ Z ∞

0

e ^{N (−}

^βJ1

^R

²

^+ln

^{sinh RR}

⁾ sinh ( ^{2N R} _βJ √

a ² + b ² + c ² )

2N R βJ

√

a ² + b ² + c ² R ² dR (31)

`

 ¦ % 3   H  . Ä ºo _  > í ß –s   l  © œs  \ O `  ¦ M :   É r ~ ½ ÓZ O  Ü

¼– Ð > í ß –ô  Ç   õ ü < ° ú  • ¸2 Ÿ ¤ σ = 2R/βJ \  ¦ • ¸{ 9  # Œ d ” 

(29)\  ¦  r  æ ¼€  

Z H (a, b, c) = 2(4π) ^{N −}

³²

(N βJ )

³²

e ⁻

^βJN

^(a

²

^+b

²

^+c

²

⁾

× Z ∞

0

e

N [−

^βJ₄

σ

²

+ln

^sinh(

βJ 2σ) βJ

2 σ

] sinh(N σ √

a ² + b ² + c ² ) N σ √

a ² + b ² + c ² σ ² dσ,

= 2(4π) ^{N −}

³²

N

¹²

(βJ )

³²

1

√

a ² + b ² + c ² e ⁻

^βJN

^(a

²

^+b

²

^+c

²

⁾

× Z ∞

0

e

N [−

^βJ₄

σ

²

+ln

^sinh(

βJ 2σ) βJ

2 σ

]

sinh(N σ p

a ² + b ² + c ² )σdσ

≡ 2(4π) ^{N −}

³²

N

¹²

(βJ )

³²

1

√ a ² + b ² + c ² e ⁻

^βJN

^(a

²

^+b

²

^+c

²

⁾ Z ∞

0

e ^{N f}

^H

^(σ) σdσ, (32)

f _H (σ) = − βJ

4 σ ² + ln sinh( ^βJ ₂ σ)

βJ

2 σ + 1

N ln sinh(N σ p

a ² + b ² + c ² ). (33)

Õ

ª Q€   Laplace_  ~ ½ ÓZ O `  ¦ + ‹" f &  ê ø Í N° ú כ\  @ /K " f 0 A _

 & h ì  r`  ¦ & h   H& h Ü ¼– Ð > í ß –K  ˜ Ѐ  

Z H (a, b, c) ≈ 2(4π) ^{N −}

³²

N

¹²

(βJ )

³²

1

√

a ² + b ² + c ² e ⁻

^βJN

^(a

²

^+b

²

^+c

²

⁾ σ 0

s βJ

−2f _H ⁰⁰ (σ 0 ) e ^{N f}

^H

^(σ

⁰

⁾ , (34) f ⁰ (σ ₀ ) = βJ

2 [−σ ₀ + coth( βJ

2 σ ₀ ) − 2

βJ σ ₀ + 2 βJ N

p a ² + b ² + c ² tanh(N σ ₀ p

a ² + b ² + c ² )],

≈ βJ

2 [−σ 0 + coth( βJ

2 σ 0 ) − 2 βJ σ 0

+ 2 βJ

p a ² + b ² + c ² ],

= 0 (35)

s

“ ¦

f _H ⁰⁰ (σ 0 ) = βJ

2 [−1 − βJ

2 csch ² ( βJ

2 σ 0 ) + 2

βJ σ ² ₀ ] (36)

“

 X <, &  ê ø Í x° ú כ\  @ /K " f sinh x ≈ e ^x /2e ” `  ¦ s 6   x % i 



.

IV. X N ˽  ʔ X ¢  M × D V R Ë Ä Z Ø

CSVW _  y Œ • F G ô  Ç\ " f & h ] X ô  Ç ¼ # p ì  r`  ¦ † < ÊÜ ¼– Ð+ ‹   l

 o $ í ì  r`  ¦ ½ ¨K ˜ Ð . + þ Ad ” & h Ü ¼– Ð CSVW_  ì  r C † < Êà º



 H

Z(a, b, c) = Π ^N _i=1 Z 2π

0

Z π 0

sin θ _i dθ _i dφ _i e

^4NβJ

^(S

^2x

^+S

^2y

⁾⁺

^βJz4N

^S

^z2

^+aS

^x

^+bS

^y

^+cS

^z

∝ e ^{N f (σ}

⁰

⁾ (37)

_

 + þ AI – Ð Å Ò# Q”   . Õ ªo  # Œ y Œ •y Œ •_   l  o $ í ì  r“ É r   6

£ § d ” Ü ¼– Ð Â Ò'  % 3 `  ¦ à º e ”  .

m x = lim

N →∞

∂

∂a 1

N ln Z = ∂

∂a f (σ)| σ=σ

₀

, (38) m y = lim

N →∞

∂

∂b 1

N ln Z = ∂

∂b f (σ)| σ=σ

₀

, (39) m _z = lim

N →∞

∂

∂c 1

N ln Z = ∂

∂c f (σ)| _σ=σ

₀

. (40)

Case 1. Ising ; ³ ø 5 

d ”

 (12)ü < (38) Õ ªo “ ¦

σ ₀ = c + ^βJ ₂

^z

σ ₀ q

a ² + b ² + (c + ^βJ ₂

^z

σ 0 ) ² [coth

r

a ² + b ² + (c + βJ z

2 σ ₀ ) ² − 1 q

a ² + b ² + (c + ^βJ ₂

^z

σ 0 ) ²

] (41)

\

 ¦ s 6   x €  

m x = aσ 0

c + ^βJ ₂

^z

σ ₀ = h x σ 0

h z + J z σ 0

. (42)

m y   H d ”  (12), (39) x 9  (41)Ü ¼– Ð Â Ò'  m y = bσ 0

c + ^βJ ₂

^z

σ ₀ = h y σ 0

h z + J z σ 0

(43)

–

Ð Å Ò# Qt  9, d ”  (12), (40) x 9  (41)– Ð Â Ò'  m z   H

m z = σ 0 (44) e ”

`  ¦ · ú ˜ à º e ”  .



l  © œs  \ O   H  â Ä º\   H m x = m y = 0, m z = σ 0 = coth( βJ z

2 σ 0 ) − 2 βJ z σ 0

(45)

“

 X <, : £ ¤ y  m ^z \  › ' a ô  Ç d ” “ É r í  H à ºô  Ç Ising F G ô  Ç\ " f_  | 9 

"

f B > h  à º_  ~ ½ Ó& ñ d ” õ  { 9 u ô  Ç .

¢

¸   É r < É ª p e ”   H  â Ä º  H transverse Ising F G ô  Ç_   â Ä º

“

 X <, J = 0, h y = h _z = 0 s  . s M : d ”  (41)“ É r Å Ò# Q”  

“

: r • ¸\ " f x-~ ½ ӆ ¾ Ó  l  © œs  e ” >   l  © œ h ^x,c ˜ Ð   H Á º

| 9

" f o % ò % i \ " f  H σ 0 = 0\  ¦ ë ß –7 á ¤ # Œ m _x = coth( βh _x

2 ) − 2 βh x

,

m y = m z = 0 (46) s

 9, e ” >   l  © œ h ^x,c   H Å Ò# Q”   “ : r • ¸\ " f h x,c

J = coth( βh x,c

2 ) − 2

βh _x,c (47) _

 d ” Ü ¼– Ð & ñ _ ô  Ç . ì ø ̀  \  x-~ ½ ӆ ¾ Ó  l  © œs  e ” >   l 



© œ h x,c ˜ Ð   Œ •“ É r  â Ä º\   H σ 0 6= 0 s Ù ¼– Ð | 9 " f o % ò % i \ 

"

f

m _x = h _x J z

, m y = 0,

m z = σ 0 , (48)

σ 0 =

βJ

_z

2 σ 0

q

a ² + ( ^βJ ₂

^z

σ ₀ ) ² [coth

r

a ² + ( βJ z

2 σ 0 ) ²

− 1

q

a ² + ( ^βJ ₂

^z

σ 0 ) ²

] (49)

\

 ¦ ë ß –7 á ¤ ô  Ç .

Case 2. XY ; ³ ø 5  d ”

 (22), (38) Õ ªo “ ¦

σ 0 =

βJ 2 σ 0

q

c ² + ( ^βJ ₂ σ ₀ ) ² [coth

r

c ² + ( βJ

2 σ 0 ) ² − 1 q

c ² + ( ^βJ ₂ σ ₀ ) ² ] + 2

βJ

p a ² + b ² (50)

\

 ¦ s 6   x €   m x = a(− 2

βJ + σ 0

√ a ² + b ² ) = − h x

J + h x σ 0

q h ² _x + h ² _y

. (51)

q

5 p w >  m y   H d ”  (22), (39) x 9  (46)Ü ¼– РÒ'  m _y = b(− 2

βJ + σ 0

√ a ² + b ² ) = − h y

J + h y σ 0

q h ² _x + h ² _y (52)

`

 ¦ % 3   H  . d ”  (22), (40), (46)Ü ¼– РÒ'  m ^z   H

m z = 2c

βJ [1 − 2 βJ σ ₀

p a ² + b ² ) = h z

J [1 −

q h ² _x + h ² _y J σ ₀ ].

(53) s

 . d ”  (47)õ  ((48)\ " f (m _x + h _x

J ) ² + (m _y + h _y

J ) ² = σ ² ₀ . (54)

`

 ¦ % 3   H X <, σ ⁰ _  Ó ü t o & h  _ p \  ¦ f ” › ' a& h Ü ¼– Ð · ú ˜ à º \ O  .

Õ

ªX O t ë ß –  l  © œs  \ O Ü ¼€   m z = 0 s “ ¦ σ 0 = coth( βJ

2 σ 0 ) − 2 βJ σ ₀ = q

m ² _x + m ² _y (55) e ”

`  ¦ · ú ˜ à º e ”   H X <, s  כ “ É r í  H à º XY F G ô  Ç\ " f_  | 9 " f B 

>

h  à º ~ ½ Ó& ñ d ” õ  { 9 u ô  Ç . Õ ªo “ ¦ s  d ” “ É r ¨ î + þ A\ " f   l

 o σ 0 _  ß ¼l \  ¦ t “ ¦ XY -¨ î €   © œ_  e ” _ _  ~ ½ ӆ ¾ Ó

`

 ¦ o ( ” `  ¦ _ p ô  Ç .

¢

¸   É r F p e ”   H  â Ä º  H longitudinal XY F G ô  Çs  .

s

  â Ä º\  h x = h _y = 0 s  . d ”  (46)“ É r z- ~ ½ ӆ ¾ Ó  l  © œs  e ”

>   l  © œ h z,c ˜ Ð   H Á º| 9 " f o % ò % i \ " f σ 0 = 0 _  K 

\

 ¦ t   H X <, s M : m _x = 0, m y = 0, m _z = coth( βh _z

2 ) − 2 βh z

(56)

“

 X <, e ” >   l  © œ h z,c   H Å Ò# Q”   “ : r • ¸\ " f h z,c

J = coth( βh z,c

2 ) − 2 βh z,c

(57) _

 d ” Ü ¼– Ð & ñ _ ô  Ç . ô  Ǽ # , z-~ ½ ӆ ¾ Ó  l  © œs  e ” >   l  © œ

h z,c ˜ Ð   Œ •“ É r | 9 " f o % ò % i \ " f  H σ 0 6= 0 s  9

coth r

c ² + ( βJ

2 σ ₀ ) ² − 1 q

c ² + ( ^βJ ₂ σ 0 ) ²

= q

c ² + ( ^βJ ₂ σ 0 ) ²

βJ 2

(58)

s

 $ í w n  Ù ¼– Ð

σ ₀ = coth( βJ

2 σ ₀ ) − 2

βJ σ ₀ = q

m ² _x + m ² _y

m z = h z /J (59)

s

 .

Case 3. Heisenberg ; ³ ø 5  d ”

 (34),(38) x 9 

σ 0 = coth ( βJ

2 σ 0 ) − 2 βJ σ 0

+ 2 βJ

p a ² + b ² + c ² (60)

Ü

¼– Ð Â Ò' 

m x = a[− 2

βJ + σ 0

√

a ² + b ² + c ² ] = − h x

J + h x σ 0

q h ² _x + h ² _y + h ² _z

(61)

\

 ¦ % 3   H  . m y   H d ”  (34), (39) x 9  (52)\ " f ½ ¨½ + É Ã º e ”  .

m _y = b[− 2

βJ + σ ₀

√ a ² + b ² + c ² ] = − h _y

J + h _y σ ₀ q h ² _x + h ² _y + h ² _z

. (62)

d ”

 (34), (40) x 9  (49)– Ð Â Ò'  loose 1 equation

m _z = c[− 2

βJ + σ ₀

√ a ² + b ² + c ² ] = − h _z

J + h _z σ ₀ q h ² _x + h ² _y + h ² _z

(63)

e ”

`  ¦ · ú ˜ à º e ”  .

Õ ªo  # Œ

(m _x + h _x

J ) ² + (m _y + h _y

J ) ² + (m _z + h _z

J ) ² = σ ₀ ² (64) e ”

`  ¦ · ú ˜>  ÷ & 9, σ 0 _  Ó ü t o & h  _ p  f ” › ' a& h Ü ¼– Ð ˜ Ðs t 

· ú

§  H  .

Õ

ª Q
  l  © œs  \ O   H  â Ä º\   H σ 0 = coth( βJ

2 σ 0 ) − 2 βJ σ ₀ = q

m ² _x + m ² _y + m ² _z (65)

 $ í w n    H X <, s  כ “ É r í  H à º Heisenberg F G ô  Ç\ " f_  | 9 

"

f B > h  à º ~ ½ Ó& ñ d ” õ  { 9 u   9, ¨ î + þ A\ " f  l  o_  ° ú כ s

 σ 0 s “ ¦ 3 " é ¶  © œ_  e ” _ _  ~ ½ ӆ ¾ Ó`  ¦ o ( ” `  ¦ _ p ô  Ç .

V. + s Ç Â ] Ø

‘

: r  7 Hë  H \ " f  H { 9 ì ø Í& h “    l  © œs     9 e ” `  ¦ M : CSVW _

 ì  r C † < Êà º\  ¦ Stratonovich   ¨ 8 Š _  “ ¦„  & h  + þ AI ü < La- pace ~ ½ ÓZ O `  ¦  6   x # Œ ½ ¨ % i  . s  כ Ü ¼– Ð Â Ò'   l  o

$ í

ì  r`  ¦ y Œ • F G ô  Ç\ " f & ñ S X ‰ >  > í ß – % i  . Õ ª   õ – Ð" f

| 9

" fB > h  à º ~ ½ Ó& ñ d ” _  — ¸€ ª œs  B Ä º ² ú ˜ t “ ¦,  â Ä º\ 



 " f  H D h– Ðî  r † ½ Ó[ þ t s  Æ Ò H † d`  ¦ µ 1 Ï|  % i  .

Õ

ªo “ ¦, Ä ºo   H { 9 ì ø Í& h “    l  © œs     9 e ” `  ¦ M : σ ⁰ _  Ó

ü

t o & h  _ p \  ¦ · ú ˜ à º \ O t ë ß –,  l  © œs    t €   Õ ª כ _

 _ p  " î S X ‰ K f ” `  ¦ ˜ Ð% i  . 7 £ ¤, XY F G ô  Ç\ " f  H σ 0 _  ß

¼l \  ¦ t “ ¦ XY -¨ î €   © œ_  e ” _ _  ~ ½ ӆ ¾ Ó`  ¦ o v “ ¦, Heisenberg F G ô  Ç\ " f  H σ 0 _  ß ¼l \  ¦ t “ ¦ 3 " é ¶  © œ  © œ _

 e ” _ _  ~ ½ ӆ ¾ Ó`  ¦ o ( ” `  ¦ · ú ˜ à º e ” % 3  .

P

c p 8 ý ò k >

s

  7 Hë  H“ É r 2009¸  • ¸ Ø  æ· ¡ ¤ @ /† < Ɠ § † < ÆÕ ü tƒ  ½ ¨t " é ¶  \ O _ 

ƒ

 ½ ¨q t " é ¶ \  _  # Œ ƒ  ½ ¨÷ &% 3 _ þ v m  .

Y

c p w Š à U Ø ”  ô

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