New Physics: Sae Mulli (The Korean Physical Society), DOI: 10.3938/NPSM.60.633

(1)

Volume 60, Number 6, 2010¸ 6 Z 4, pp. 633∼636

New Physics: Sae Mulli (The Korean Physical Society), DOI: 10.3938/NPSM.60.633

Æ

k È4 M X ê s; c" e P c lß f Ä T Ç S Ë mP ê s V R Ë= k8 ý R c lV ê s? 0 Ö ² z ºY Ñ ÷ M × D

» û Ba : @

Ø

æ Å Ò@ / < Æ § § ª < ÆÂ Ò, Ø æ Å Ò 380-702 (2010¸ 4 Z 4 27{ 9 ~ Ã Î6 £ §, 2010¸ 6 Z 4 10{ 9 > F S X & ñ )

s

f ç y © $ í ^ \ " f H ü @Â Ò l © s \ O H â Ä º\ ë ß © s 0 p x t ë ß , s f ç ì ø Íy © $ í ^ \ " f H ü

@Â Ò l © s > r F H â Ä º\ ¸ © s è ß . ü @Â Ò l © s > r F H â Ä º\ H y s

f ç ¸+ þ A_ $ í | 9 ` ¦ ¸ ú ¸Ø Ôl M :ë H \ s f ç ì ø Íy © $ í ^ _ © s \ ¦ s K H X < e # Q" f ´ ú § É r # Q 9¹ ¡ § s

e . : £ ¤ y e > l © \ " f H Á ºÃ ºy ´ ú § É r Ã º_ { © I [ þ t s > r F l M :ë H \ y s f ç ì

ø Íy © $ í ^ _ : £ ¤$ í ` ¦ Ã Ð½ ¨ H כ É r _ Ô ¦ 0 p x . e > l © \ " f Å Òl & h â > ¸| ` ¦ L × L y (L = 4 ∼ 10) s f ç ì ø Íy © $ í ^ _ { © I ' pà Ô Ðx s

c

(L) ü < l o m

c

(L)\ ¦ & ñ S X

> > í ß % i . \ P % i < Æ& h F G ô Ç (L → ∞)\ " f { © I ' pà Ô Ðx s

^c

= 0.4076(7)k

B

ü < { © I l

o m

^c

= 0.5457(42)\ ¦ % 3 % 3 .

Ù þ

d # Q: s f ç ì ø Íy © $ í ^ , e > l © , { © I , Å Òl & h â > ¸| ` ¦ y

Ground-state Entropy and Magnetization of the Square-lattice Ising Antiferromagnet in the Critical Magnetic Field

Seung Yeon Kim ^∗

School of Liberal Arts and Sciences, Chungju National University, Chungju 380-702 (Received 27 April, 2010 : accepted 10 June, 2010)

The introduction of an external magnetic field destroys the phase transition of the Ising ferromag- net whereas a nonzero uniform field does not destroy the transition of the Ising antiferromagnet.

Because most of the properties of the Ising model in an external magnetic field are not known, not even on a square lattice, it is very difficult to understand the phase transition of the Ising antiferromagnet. In particular, the Ising antiferromagnet has innumerable ground states in the critical magnetic field. Consequently, it is almost impossible to investigate the properties of the square-lattice Ising antiferromagnet in the critical magnetic field. We have exactly evaluated the ground-state entropy s

c

(L) and magnetization m

c

(L) of the Ising antiferromagnet in the critical magnetic field on a L × L square lattice (L = 4 ∼ 10) with fully periodic boundary conditions.

Finally, we have obtained s

c

= 0.4076(7)k

B

and m

c

= 0.5457(42) in the thermodynamic limit (L → ∞).

PACS numbers: 05.50.+q, 64.60.Cn, 75.10.Hk, 75.50.Ee

Keywords: Ising antiferromagnet, Critical magnetic field, Ground states, Square lattice with periodic bound- ary conditions

∗

E-mail: [email protected] -633-

(2)

-634- ô Ç² D GÓ ü t o < Æ rt “D hÓ ü t o ”, Volume 60, Number 6, 2010¸ 6 Z 4

© s (phase transitions)ü < e > & ³ © (critical phe- nomena) É r > \ " f © Ð¼ # & h & ³ © s . ç ß ô Ç

$ í ^ ¸+ þ A s f ç ¸+ þ A (Ising model) É r © s ü < e >

&

³ © _ s K \ e # Q" f © × æ כ ¹ô Ç % i ½ + É` ¦ K M ® o [1,2].

s

f ç ¸+ þ A É r Ä »^ > _ Ó o^ -l ^ © s ü < e > & ³ © ¸ & ñ S X

> l Õ ü t K ï r . 1944¸ \ : r (Onsager) ü @Â Ò

l © s \ O H â Ä º\ y (square lattice) s f ç ¸ + þ

A_ & ñ S X ô Ç K [3]\ ¦ ¹ 1 Ô Í Ç xÜ ¼ 9, : r _ K (Onsager solution) H © s ü < e > & ³ © ` ¦ ] j@ / Ð s K ½ + É Ã º e H U

´` ¦ \ P % 3 . s # Q" f 1952¸ \ H ª (Yang)s ü @Â Ò l

© s % ò Ü ¼ Ð H F G ô Ç\ " f y s f ç y © $ í ^ _

µ 1 Ï l o (spontaneous magnetization) < ÊÃ º\ ¦ ¹ 1 Ô Í Ç

x [4]. : r _ K µ 1 Ï| ) a s Ê ê t è ß 66¸ 1 l x î ß ü @ Â

Ò l © s e H â Ä º\ ¸ y s f ç ¸+ þ A_ & ñ S X ô

Ç K \ ¦ ¹ 1 Ôl 0 AK " f Ã º\ O s ´ ú § É r ¸§ 4 ` ¦ l Ö ¦% i 6 £ § \ ¸ Ô

¦ ½ ¨ ¦ & ñ S X ô Ç K \ ¦ ¹ 1 Ô H X < z ´J % i .

s

f ç y © $ í ^ _ â Ä º ü @Â Ò l © s \ O ` ¦ M :\ ë ß ©

s l M :ë H \ [5], l y Ã ºÖ ¦ (magnetic sus- ceptibility) \ @ /ô Ç ë H ] j\ ¦ ] jü @ ¦ H y s f ç ¸ + þ

A_ y © $ í ^ - © $ í ^ © s ü < e > & ³ © ` ¦ ¢ - a# 4 > s

K ½ + É Ã º e . ì ø Í s f ç ì ø Íy © $ í ^ _ â Ä º ü @Â Ò l

© s e ` ¦ M :\ ¸ © s l M :ë H \ [6–10], s f ç

ì ø Íy © $ í ^ _ © s ü < e > & ³ © ` ¦ : r & h Ü ¼ Ð s K

l 0 AK " f H ü @Â Ò l © s e H â Ä º\ s f ç ¸+ þ A_

&

ñ S X ô Ç K ü < $ í | 9 [ þ t \ @ /ô Ç & ñ Ð 9 כ ¹ . t ë ß ü @ Â

Ò l © s e H y s f ç ì ø Íy © $ í ^ _ â Ä º Å Ò

#

Q l © ° ú כ\ @ /K " f © s { 9 # Q H : r ¸ (

© l : r& h & ñ Ð\ K { © H) e > (critical line)_

&

ñ S X ô Ç ° ú כ ¸ · ú 94 R e t · ú § . & ³F t ü @Â Ò l © s e

H y s f ç ì ø Íy © $ í ^ _ e > \ @ /ô Ç @ /| Ä Ì

&

h Æ Ò& ñ u [ þ t ë ß s · ú 94 R e [11].

ü @Â Ò l © (B)s e H â Ä º þ j H] X s Ö © © ñ 6 x ( ½ + Ë[ jl J)õ N

s

> h_ & h ` ¦ s f ç ¸+ þ A É r A _

K x 9 Ðm î ß \ _ K " f & ñ _ ) a .

H = −J X

hi,ji

σ

i

σ

j

− B X

i

σ

i

(1)

d

(1)\ " f σ

ⁱ

H & h i(= 1, 2, ..., N

s

) \ " f s f ç ¸+ þ As

| 9 Ã º e H Û ¼ 2 ; ° ú כÜ ¼ Ð +1õ −1` ¦ 2 [½ + É Ã º e . Õ ª o

¦ ' Í P : ½ + Ë É r ¸ H N

b

> h_ þ j H] X s Ö ©[ þ t (nearest neighbors) \ @ /ô Ç ½ + Ës . þ j H] X s Ö © © ñ 6 x s ª Ã

º(J > 0){ 9 M : H d (1)s s f ç y © $ í ^ \ ¦ & ñ _ K Å Ò 9, þ

j H] X s Ö © © ñ 6 x s 6 £ § Ã º(J < 0){ 9 M : H s f ç ì ø Íy ©

$ í ^ \ ¦ & ñ _ K ï r . 7 £ ¤ ì ø Íy © $ í ^ \ " f H s Ö © ô Ç Û ¼

2 ; σ

_i

ü < σ

j

" f Ð ì ø Í@ / ~ ½ Ó ¾ ÓÜ ¼ Ð & ñ § > = 9 ¦ ô Ç .

σ

i

σ

j

= −1 (2)

Fig. 1. The approximate curve for the critical line B(T ) of the square-lattice Ising antiferromagnet (J < 0) in an external magnetic field B (in unit of −J ), as a function of temperature T (in unit of −J/k

_B

). Below the critical line, the antiferromagnetic ordered-states are preserved even though the magnetic field aligns all spins to its di- rection.

" f s f ç ì ø Íy © $ í ^ \ " f H K x 9 Ðm î ß ` ¦ 6 £ § õ ° ú s

æ ¼ ¼ # o .

E = H

(−J ) = X

hi,ji

σ

i

σ

j

− B (−J )

X

i

σ

i

(3)

ì

ø Íy © $ í ^ _ > r F \ ¦ % 6 £ § Ü ¼ Ð s K ô Ç 3 A q(N´eel) É r 1970¸ \ ¸6 \ Ó ü t o < Æ © ` ¦ Ã º © % i .

Figure 1 É r y s f ç ì ø Íy © $ í ^ _ e > B(T ) \ @ /ô Ç @ /| Ä Ì& h Æ Ò& ñ u [ þ t` ¦ 5 Å q& h / B G ( : r

¸ T _ < ÊÃ º)Ü ¼ Ð Ð# ï r . Õ ªa Ë >s Ð# Å Ò1 p w s e > B(T ) A Â Òì r \ " f H ü @Â Ò l © s > r F < Ê\ ¸ Ô ¦ ½ ¨

¦ l © ~ ½ Ó ¾ ÓÜ ¼ Ð & ñ § > =÷ &t · ú § É r ì ø Íy © $ í & ñ § > = © I

(ordered states) 0 p x . e > B(T ) 0 A Â Òì r _

â

Ä º ü @Â Ò l © ~ ½ Ó ¾ ÓÜ ¼ Ð & ñ § > =ô Ç y © $ í © I è

ß . Z } É r : r ¸\ " f H Û ¼ 2 ;[ þ t s Á º| 9 " f > C u ) a ©

$ í © I è ß . s Qô Ç s Ä »[ þ t ü ì r \ s f ç ì ø Íy ©

$ í ^ _ â Ä º e > B(T ) 6= 0\ " f © s ü < e > & ³

© s è ß .

y s f ç ì ø Íy © $ í ^ _ e > É r B(T

N

) = 0 \ K

{ © H 3 A q : r ¸ T

N

\ " f r ô Ç .

T

N

= 2J k

B

ln( √

2 − 1) = 2.2691853 (−J ) k

B

(4)

3 A q : r ¸ H y © $ í ^ _ e > & h Ç ©o (Curie) : r ¸ T

^C

ü <

1 l x{ 9 9, 3 A q : r ¸\ " f H : r _ K & h 6 x ) a . 7 £ ¤ : r

(3)

e

> l © \ " f y s f ç ì ø Íy © $ í ^ _ { © I ' pà Ô Ðx ü < l o – ^ 5 p x · Seung Yeon Kim -635-

Table 1. The exact integer values for the number of ground states Ω

_c

(L) and the ground-state entropy s

c

(L) = L

⁻²

ln Ω

c

(L) (in unit of k

B

) of the L × L square- lattice Ising antiferromagnet with fully periodic bound- ary conditions in the critical magnetic field B

c

= 4|J |.

L Ω

c

(L) s

c

(L)

4 743 0.41316850

6 2406862 0.40816207

8 213256442503 0.40759002

10 498819827260367617 0.40751021

_ K H ü @Â Ò l © s \ O H â Ä º y s f ç y ©

$ í ^ ü < ì ø Íy © $ í ^ \ ¦ 1 l x r \ [ O " î K ï r . ô Ç¼ # e >

É r B

c

(T = 0) = 4|J | \ K { © H e > l © (critical magnetic field) B

_c

\ " f = å Q s è ß . " f Ä ºo H y

s f ç ì ø Íy © $ í ^ \ @ /ô Ç e > _ r & h õ = å Q& h

`

¦ & ñ S X > · ú ¦ e . ì ø Í r & h õ = å Q& h s _ e

>

_ ° ú כ[ þ t É r H & h Æ Ò& ñ u [ þ t{ 9 ÷ r s .

e

> l © \ " f H ¿ º > h_ ì ø Íy © $ í { © I [ þ t (two antiferromagnetic ground-states) õ (ü @Â Ò l © ~ ½ Ó ¾ ÓÜ ¼

Ð & ñ § > = ) a) Ä »{ 9 ô Ç y © $ í { © I (the unique ferro- magnetic ground-state) 1 l x{ 9 ô Ç \ -t \ ¦ ° ú H .

" f e > l © \ " f H Á ºÃ ºy ´ ú § É r Ã º_ { © I [ þ t s

> r F 9, s t & h \ " f y s f ç ì ø Íy © $ í ^ \ ¦

Ã

Ð½ ¨ H כ É r _ Ô ¦ 0 p x . s 7 Hë H \ " f H e > l

© B

c

= 4|J | \ " f T = 0 ( © s { 9 # Q H t & h )

â Ä º y s f ç ì ø Íy © $ í ^ _ : £ ¤$ í ` ¦ ½ ¨ 9

¦ ô Ç . $ p è& ñ ï r ² ú ' § > = (microcanonical trans- fer matrix) [7–9] ~ ½ ÓZ O ` ¦ s 6 x # , e > l © \ " f Å Ò l

& h â > ¸| ` ¦ L × L y s f ç ì ø Íy © $ í

^

_ { © I _ Ã º Ω

c

(L) ü < 6 £ § õ ° ú s & ñ _ ) a {

© I l o (ground-state magnetization) m

^c

(L)\ ¦ & ñ S X

> > í ß % i .

m

c

(L) = m(L; T = 0, B = 4|J |) = h P

i

σ

_i

i L

²

(5)

Table 1 É r L = 4 ∼ 10 \ " f > í ß ô Ç L×L y s f ç ì

ø Íy © $ í ^ _ { © I _ Ã º Ω

^c

(L) _ & ñ S X ô Ç & ñ Ã º ° ú כ[ þ t

`

¦ Ð# ï r . > _ ß ¼l Ls 7 £ x < Ê\ { © I _ Ã

º / å L > 7 £ x < Ê` ¦ · ú Ã º e . 10×10 y s f ç

ì ø Íy © $ í ^ _ â Ä º { © I _ Ã º H @ /| Ä Ì 5 × 10

¹⁷

\ K

{ © ô Ç . { © I _ Ã º Ω

^c

(L) \ ÐÕ ª\ ¦ 2 [ & ñ S X

ô Ç { © I ' pà Ô Ðx ° ú כ` ¦ 6 £ § õ ° ú s % 3 ` ¦ Ã º e .

s

_c

(L) = k

_B

L

²

ln Ω

_c

(L) (6)

Table 2. The exact values for the ground-state magneti- zation m

c

(L; T = 0) = L

⁻²

h P

i

σ

i

i of the L × L square- lattice Ising antiferromagnet with fully periodic bound- ary conditions in the critical magnetic field B

c

= 4|J |.

L m

c

(L)

4 0.52355316

6 0.54239836

8 0.54598943

10 0.54668337

° ú כ (é ß 0 A: k

^B

)[ þ t ¸ Ð# Å Ò ¦ e . > _ ß ¼l Ls 7 £ x

É

r & ³F t · ú 94 R e t · ú §t ë ß , 0\ ] X H t · ú §Ü ¼o

Ð Ó

H á Ô (Metcalf)ü < ª (Yang) É r Å Òl & h â > ¸|

`

¦ Ä »ô Ç > \ " f & ñ ï r ² ú ' § > =` ¦ H & h Ü ¼ Ð > í ß

c

= 0.4075k

B

_ Æ Ò& ñ u \ ¦ µ 1 Ï³ ðÙ þ ¡ [12]. x 9 Ð[ jq u (Milo- sevic) 1 p x É r Ä » â > ¸| ` ¦ Ä »ô Ç > \ @ /K " f, e

>

Ð > í ß # s

c

= 0.407495k

_B

_ Æ Ò& ñ u \ ¦ Ð ¦Ù þ ¡ [13].

s

[ þ t Æ Ò& ñ u [ þ t É r ¸ í ß & ñ \ O s Ð ¦÷ &% 3 . Õ ªo ¦ s [

þ

t Æ Ò& ñ u [ þ t` ¦ ¶ ú ( R Ð Table 1\ e H 10 × 10 y

< Ê` ¦ · ú Ã º e . s 7 Hë H \ " f H Table 1 _ « Ñ ÐÂ Ò' BST ~ ½ ÓZ O [14]` ¦ s 6 x # \ P % i < Æ& h F G ô Ç(L → ∞)\ " f _

c

= 0.4076(7)k

B

\ ¦ % 3 % 3

¸ ú { 9 u ô Ç .

c

= 4|J | \ " f Å Òl & h â >

c

(L) _ & ñ S X ô Ç ° ú כ` ¦ Ð# ï r

c

(B = B

c

) _ & ñ S X ô Ç ° ú כ É r · ú 94 R e t · ú § . ì ø Í \

\ P

% i < Æ& h F G ô Ç\ " f B < B

^c

c

â

$ í

)s ) a .

e

l o\ @ /ô Ç 7 H _ \ ¦ Ð (Monte Carlo) ~ ½ ÓZ O ` ¦ s

(4)

-636- ô Ç² D GÓ ü t o < Æ rt “D hÓ ü t o ”, Volume 60, Number 6, 2010¸ 6 Z 4

6 x ô Ç Æ Ò& ñ u H m

c

(T = 0, B = B

c

) = 0.5472 s ¦ & ñ ï

r ² ú ' § > = ~ ½ ÓZ O ` ¦ s 6 x ô Ç Æ Ò& ñ u H m

c

= 0.5470 s

&

ñ u H m

c

= 0.5482(4) s 9 [15], s ° ú כ É r Ð Ó H á Ôü < ª _ m

c

= 0.5470 s 0.5472 Ð H H ° ú כs . s 7 Hë H \ " f

H BST ~ ½ ÓZ O ` ¦ s 6 x # Table 2_ « Ñ ÐÂ Ò' \ P % i < Æ& h F

c

= 0.5457(42)\ ¦

% 3

þ

t Ð ç ß É r ° ú כs 9 ¸ # 3 0 A ? /\ " f É r Æ Ò& ñ u [

þ

t` ¦ í < Êô Ç .

Y

c p w Ã U Ø ô

[1] M. H. Krieger, Constitutions of Matter: Mathemat- ically Modeling the Most Everyday of Physical Phe- nomena (The University of Chicago Press, Chicago, 1996).

[2] C. Domb, The Critical Point (Taylor and Francis, London, 1996).

[3] L. Onsager, Phys. Rev. 65, 117 (1944).

[4] C. N. Yang, Phys. Rev. 85, 808 (1952).

[5] T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952).

[6] C. Binek, Ising-type Antiferromagnets (Springer, Berlin, 2003).

[7] S. -Y. Kim, Phys. Rev. Lett. 93, 130604 (2004).

[8] S. -Y. Kim, Phys. Rev. E 71, 017102 (2005).

[9] S. -Y. Kim, Phys. Lett. A 358, 245 (2006).

[10] S. -Y. Kim, C. -O. Hwang and J. M. Kim, Nucl.

Phys. B 805, 441 (2008).

[11] F. Y. Wu, X. N. Wu and H. W. J. Bl¨ ote, Phys. Rev.

Lett. 62, 2773 (1989); X. N. Wu and F. Y. Wu, Phys. Lett. A 144, 123 (1990); J. L. Monroe, Phys.

Rev. E 64, 016126 (2001).

[12] B. D. Metcalf and C. P. Yang, Phys. Rev. B 18, 2304 (1978).

[13] S. Milosevic, B. Stosic and T. Stosic, Physica A 157, 899 (1989).

[14] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in Fortran 77 2nd Ed (Cambridge University Press, Cambridge, 1992), p. 104.

New Physics: Sae Mulli (The Korean Physical Society), DOI: 10.3938/NPSM.60.633

Volume 60, Number 6, 2010¸  6 Z 4, pp. 633∼636

New Physics: Sae Mulli (The Korean Physical Society), DOI: 10.3938/NPSM.60.633

Æ

k È4   M X ê s; c" e  P c lß f Ä  T Ç S Ë   mP ê s V R Ë= k8 ý  R c lV ê s? 0  Ö ² z ºY Ñ ÷  M × D

 » û Ba : @

Ø 

æ Å Ò@ / < Æ §  § ª  < ÆÂ Ò, Ø  æ Å Ò 380-702 (2010¸  4 Z 4 27{ 9  ~ Ã Î6 £ §, 2010¸  6 Z 4 10{ 9  > F  S X & ñ )

s

f ç y ©  $ í ^ \ " f  H ü @Â Ò  l  © s  \ O   H  â Ä º\ ë ß   ©   s  0 p x t ë ß , s f ç ì ø Íy ©  $ í ^ \ " f  H ü

@Â Ò  l  © s   > r F    H  â Ä º\  ¸  ©   s   è ß  . ü @Â Ò  l  © s   > r F    H  â Ä º\   H  y      s

f ç  ¸+ þ A_  $ í | 9 `  ¦ ¸ ú   ¸Ø Ôl  M :ë  H \  s f ç ì ø Íy ©  $ í ^ _   ©   s \  ¦ s K    H X < e  # Q" f ´ ú § É r # Q 9¹ ¡ § s

 e   . : £ ¤ y  e  >   l  © \ " f  H Á ºÃ ºy  ´ ú § É r Ã º_   {   © I [ þ t s   > r F  l  M :ë  H \   y      s f ç ì

(L) ü <  l  o m

(L)\  ¦ & ñ S X 

>  > í ß  % i  . \ P % i  < Æ& h F G ô  Ç (L → ∞)\ " f  {   © I   ' pà Ô Ðx  s

= 0.4076(7)k

ü <  {   © I    l

 o m

= 0.5457(42)\  ¦ % 3 % 3  .

Ù þ

d  # Q: s f ç ì ø Íy ©  $ í ^ , e  >   l  © ,  {   © I , Å Òl & h    â >  ¸| `  ¦    y     

Ground-state Entropy and Magnetization of the Square-lattice Ising Antiferromagnet in the Critical Magnetic Field

Seung Yeon Kim ∗

School of Liberal Arts and Sciences, Chungju National University, Chungju 380-702 (Received 27 April, 2010 : accepted 10 June, 2010)

The introduction of an external magnetic field destroys the phase transition of the Ising ferromag- net whereas a nonzero uniform field does not destroy the transition of the Ising antiferromagnet.

(L) and magnetization m

(L) of the Ising antiferromagnet in the critical magnetic field on a L × L square lattice (L = 4 ∼ 10) with fully periodic boundary conditions.

Finally, we have obtained s

= 0.4076(7)k

and m

= 0.5457(42) in the thermodynamic limit (L → ∞).

PACS numbers: 05.50.+q, 64.60.Cn, 75.10.Hk, 75.50.Ee

Keywords: Ising antiferromagnet, Critical magnetic field, Ground states, Square lattice with periodic bound- ary conditions

E-mail: [email protected] -633-

-634- ô  Ç² D GÓ ü t o  < Æ rt  “D hÓ ü t o ”, Volume 60, Number 6, 2010¸  6 Z 4



©   s  (phase transitions)ü < e  >  & ³ ©  (critical phe- nomena) É r    > \ " f  ©   Ð¼ # & h    & ³ © s  . ç ß   ô  Ç



$ í ^   ¸+ þ A  s f ç  ¸+ þ A (Ising model) É r  ©   s ü < e  > 



&

³ © _  s K \  e  # Q" f  ©  ×  æ כ ¹ô  Ç % i ½ + É`  ¦ K  M ® o  [1,2].

s

f ç  ¸+ þ A É r Ä »^ > _  Ó  o^ -l ^   ©   s ü < e  >  & ³ ©  ¸ & ñ S X

 >  l Õ ü t K ï  r . 1944¸  \   : r    (Onsager) ü @Â Ò



l  © s  \ O   H  â Ä º\   y      (square lattice) s f ç  ¸ + þ

A_  & ñ S X  ô  Ç K  [3]\  ¦ ¹ 1 Ô Í Ç xÜ ¼ 9,  : r   _  K  (Onsager solution)  H  ©   s ü < e  >  & ³ © `  ¦ ] j@ / Ð s K ½ + É Ã º e    H U

´`  ¦ \ P % 3  . s # Q" f 1952¸  \   H  ª  (Yang)s  ü @Â Ò  l 



© s  % ò Ü ¼ Ð   H F G ô  Ç\ " f  y      s f ç y ©  $ í ^ _ 



µ 1 Ï  l  o (spontaneous magnetization)  < ÊÃ º\  ¦ ¹ 1 Ô  Í Ç

x  [4].  : r   _  K  µ 1 Ï|  ) a s Ê ê t è ß  66¸  1 l x î ß  ü @ Â

Ò  l  © s  e    H  â Ä º\  ¸  y      s f ç  ¸+ þ A_  & ñ S X  ô

 Ç K \  ¦ ¹ 1 Ôl  0 AK " f Ã º\ O s  ´ ú § É r  ¸§ 4 `  ¦ l Ö  ¦% i 6 £ § \  ¸ Ô

 ¦ ½ ¨  ¦ & ñ S X  ô  Ç K \  ¦ ¹ 1 Ô  H X < z ´J  % i  .

s

f ç y ©  $ í ^ _   â Ä º ü @Â Ò  l  © s  \ O `  ¦ M :\ ë ß   © 



 s    l  M :ë  H \  [5],  l  y  Ã ºÖ  ¦ (magnetic sus- ceptibility) \  @ /ô  Ç ë  H ] j\  ¦ ] jü @  ¦  H  y      s f ç  ¸ + þ

A_  y ©  $ í ^ - ©  $ í ^   ©   s ü < e  >  & ³ © `  ¦ ¢ - a# 4 >  s

K ½ + É Ã º e   . ì ø Í  s f ç ì ø Íy ©  $ í ^ _   â Ä º ü @Â Ò  l 



© s  e  `  ¦ M :\  ¸  ©   s    l  M :ë  H \  [6–10], s  f ç

ì ø Íy ©  $ í ^ _   ©   s ü < e  >  & ³ © `  ¦  : r  & h Ü ¼ Ð s K 

l  0 AK " f  H ü @Â Ò  l  © s  e    H  â Ä º\  s f ç  ¸+ þ A_ 

&

ñ S X  ô  Ç K ü < $ í | 9 [ þ t \  @ /ô  Ç & ñ  Ð  9 כ ¹  . t ë ß  ü @ Â

Ò  l  © s  e    H  y      s f ç ì ø Íy ©  $ í ^ _   â Ä º Å Ò

#

Q   l  ©  ° ú כ\  @ /K " f  ©   s  { 9 # Q   H  : r  ¸  (



©  l  : r& h   & ñ  Ð\  K { ©    H) e  >   (critical line)_ 

&

ñ S X  ô  Ç ° ú כ ¸ · ú  94 R e  t  · ú § .  & ³F  t  ü @Â Ò  l  © s  e 

  H  y      s f ç ì ø Íy ©  $ í ^ _  e  >   \  @ /ô  Ç @ /| Ä Ì

&

h   Æ Ò& ñ u [ þ t ë ß s  · ú  94 R e    [11].

Volume 60, Number 6, 2010¸ 6 Z 4, pp. 633∼636

k È4 M X ê s; c" e P c lß f Ä T Ç S Ë mP ê s V R Ë= k8 ý R c lV ê s? 0 Ö ² z ºY Ñ ÷ M × D

» û Ba : @

Ø

æ Å Ò@ / < Æ § § ª < ÆÂ Ò, Ø æ Å Ò 380-702 (2010¸ 4 Z 4 27{ 9 ~ Ã Î6 £ §, 2010¸ 6 Z 4 10{ 9 > F S X & ñ )

f ç y © $ í ^ \ " f H ü @Â Ò l © s \ O H â Ä º\ ë ß © s 0 p x t ë ß , s f ç ì ø Íy © $ í ^ \ " f H ü

@Â Ò l © s > r F H â Ä º\ ¸ © s è ß . ü @Â Ò l © s > r F H â Ä º\ H y s

f ç ¸+ þ A_ $ í | 9 ` ¦ ¸ ú ¸Ø Ôl M :ë H \ s f ç ì ø Íy © $ í ^ _ © s \ ¦ s K H X < e # Q" f ´ ú § É r # Q 9¹ ¡ § s

e . : £ ¤ y e > l © \ " f H Á ºÃ ºy ´ ú § É r Ã º_ { © I [ þ t s > r F l M :ë H \ y s f ç ì

(L) ü < l o m

(L)\ ¦ & ñ S X

> > í ß % i . \ P % i < Æ& h F G ô Ç (L → ∞)\ " f { © I ' pà Ô Ðx s

ü < { © I l

o m

= 0.5457(42)\ ¦ % 3 % 3 .

d # Q: s f ç ì ø Íy © $ í ^ , e > l © , { © I , Å Òl & h â > ¸| ` ¦ y

Seung Yeon Kim ^∗

-634- ô Ç² D GÓ ü t o < Æ rt “D hÓ ü t o ”, Volume 60, Number 6, 2010¸ 6 Z 4

© s (phase transitions)ü < e > & ³ © (critical phe- nomena) É r > \ " f © Ð¼ # & h & ³ © s . ç ß ô Ç

$ í ^ ¸+ þ A s f ç ¸+ þ A (Ising model) É r © s ü < e >

³ © _ s K \ e # Q" f © × æ כ ¹ô Ç % i ½ + É` ¦ K M ® o [1,2].

f ç ¸+ þ A É r Ä »^ > _ Ó o^ -l ^ © s ü < e > & ³ © ¸ & ñ S X

> l Õ ü t K ï r . 1944¸ \ : r (Onsager) ü @Â Ò

l © s \ O H â Ä º\ y (square lattice) s f ç ¸ + þ

A_ & ñ S X ô Ç K [3]\ ¦ ¹ 1 Ô Í Ç xÜ ¼ 9, : r _ K (Onsager solution) H © s ü < e > & ³ © ` ¦ ] j@ / Ð s K ½ + É Ã º e H U

´` ¦ \ P % 3 . s # Q" f 1952¸ \ H ª (Yang)s ü @Â Ò l

© s % ò Ü ¼ Ð H F G ô Ç\ " f y s f ç y © $ í ^ _

µ 1 Ï l o (spontaneous magnetization) < ÊÃ º\ ¦ ¹ 1 Ô Í Ç

x [4]. : r _ K µ 1 Ï| ) a s Ê ê t è ß 66¸ 1 l x î ß ü @ Â

Ò l © s e H â Ä º\ ¸ y s f ç ¸+ þ A_ & ñ S X ô

Ç K \ ¦ ¹ 1 Ôl 0 AK " f Ã º\ O s ´ ú § É r ¸§ 4 ` ¦ l Ö ¦% i 6 £ § \ ¸ Ô

¦ ½ ¨ ¦ & ñ S X ô Ç K \ ¦ ¹ 1 Ô H X < z ´J % i .

f ç y © $ í ^ _ â Ä º ü @Â Ò l © s \ O ` ¦ M :\ ë ß ©

s l M :ë H \ [5], l y Ã ºÖ ¦ (magnetic sus- ceptibility) \ @ /ô Ç ë H ] j\ ¦ ] jü @ ¦ H y s f ç ¸ + þ

A_ y © $ í ^ - © $ í ^ © s ü < e > & ³ © ` ¦ ¢ - a# 4 > s

K ½ + É Ã º e . ì ø Í s f ç ì ø Íy © $ í ^ _ â Ä º ü @Â Ò l

© s e ` ¦ M :\ ¸ © s l M :ë H \ [6–10], s f ç

ì ø Íy © $ í ^ _ © s ü < e > & ³ © ` ¦ : r & h Ü ¼ Ð s K

l 0 AK " f H ü @Â Ò l © s e H â Ä º\ s f ç ¸+ þ A_

ñ S X ô Ç K ü < $ í | 9 [ þ t \ @ /ô Ç & ñ Ð 9 כ ¹ . t ë ß ü @ Â

Ò l © s e H y s f ç ì ø Íy © $ í ^ _ â Ä º Å Ò

Q l © ° ú כ\ @ /K " f © s { 9 # Q H : r ¸ (

© l : r& h & ñ Ð\ K { © H) e > (critical line)_

ñ S X ô Ç ° ú כ ¸ · ú 94 R e t · ú § . & ³F t ü @Â Ò l © s e

H y s f ç ì ø Íy © $ í ^ _ e > \ @ /ô Ç @ /| Ä Ì

h Æ Ò& ñ u [ þ t ë ß s · ú 94 R e [11].

ü @Â Ò l © (B)s e H â Ä º þ j H] X s Ö © © ñ 6 x ( ½ + Ë[ jl J)õ N

> h_ & h ` ¦ s f ç ¸+ þ A É r A _

K x 9 Ðm î ß \ _ K " f & ñ _ ) a .

d

(1)\ " f σ

H & h i(= 1, 2, ..., N

) \ " f s f ç ¸+ þ As

| 9 Ã º e H Û ¼ 2 ; ° ú כÜ ¼ Ð +1õ −1` ¦ 2 [½ + É Ã º e . Õ ª o

¦ ' Í P : ½ + Ë É r ¸ H N

> h_ þ j H] X s Ö ©[ þ t (nearest neighbors) \ @ /ô Ç ½ + Ës . þ j H] X s Ö © © ñ 6 x s ª Ã

º(J > 0){ 9 M : H d (1)s s f ç y © $ í ^ \ ¦ & ñ _ K Å Ò 9, þ

j H] X s Ö © © ñ 6 x s 6 £ § Ã º(J < 0){ 9 M : H s f ç ì ø Íy ©

$ í ^ \ ¦ & ñ _ K ï r . 7 £ ¤ ì ø Íy © $ í ^ \ " f H s Ö © ô Ç Û ¼

" f Ð ì ø Í@ / ~ ½ Ó ¾ ÓÜ ¼ Ð & ñ § > = 9 ¦ ô Ç .

" f s f ç ì ø Íy © $ í ^ \ " f H K x 9 Ðm î ß ` ¦ 6 £ § õ ° ú s

æ ¼ ¼ # o .

ø Íy © $ í ^ _ > r F \ ¦ % 6 £ § Ü ¼ Ð s K ô Ç 3 A q(N´eel) É r 1970¸ \ ¸6 \ Ó ü t o < Æ © ` ¦ Ã º © % i .

Figure 1 É r y s f ç ì ø Íy © $ í ^ _ e > B(T ) \ @ /ô Ç @ /| Ä Ì& h Æ Ò& ñ u [ þ t` ¦ 5 Å q& h / B G ( : r

¸ T _ < ÊÃ º)Ü ¼ Ð Ð# ï r . Õ ªa Ë >s Ð# Å Ò1 p w s e > B(T ) A Â Òì r \ " f H ü @Â Ò l © s > r F < Ê\ ¸ Ô ¦ ½ ¨

¦ l © ~ ½ Ó ¾ ÓÜ ¼ Ð & ñ § > =÷ &t · ú § É r ì ø Íy © $ í & ñ § > = © I

(ordered states) 0 p x . e > B(T ) 0 A Â Òì r _

â

Ä º ü @Â Ò l © ~ ½ Ó ¾ ÓÜ ¼ Ð & ñ § > =ô Ç y © $ í © I è

ß . Z } É r : r ¸\ " f H Û ¼ 2 ;[ þ t s Á º| 9 " f > C u ) a ©

$ í © I è ß . s Qô Ç s Ä »[ þ t ü ì r \ s f ç ì ø Íy ©

$ í ^ _ â Ä º e > B(T ) 6= 0\ " f © s ü < e > & ³

© s è ß .

y s f ç ì ø Íy © $ í ^ _ e > É r B(T

) = 0 \ K

{ © H 3 A q : r ¸ T

\ " f r ô Ç .