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GwnØæÅÒ@/<Ƨ §ªõ&ñÂÒ, ØæÅÒ 380-702
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æz´@/<Ƨ Óüto<Æõ x9 ìr[O>½¨G'p', "fÖ¦ 156-743 (2009¸ 4Z4 20{9 ~ÃÎ6£§)
L × L y sfç¸+þA(L = 4 ∼ 20)\"f 7H_ \¦Ð ~½ÓZOܼР%3Ér H&h ìrC<Êú H[þt`¦
^
>&hs¦ ½¨^&hܼР¶ú(RФ. >_ ß¼l &4R¸ Óüto&hܼР_p eH 'Í P: Hõ Õª ÅÒ 0
A\"fH 7H_ \¦Ð ~½ÓZO_ H&h õ &ñSX<Ê`¦ ·ú ú e%3. \P%i<Æ&h FGôÇ\"f 7H_ \¦
Ð «ÑÐÂÒ' e>&h °úכܼР0.4209(78)`¦ %3%3ܼ 9 \P»¡¤'tú °úכܼР0.992(55)`¦ %3%3. %3x9ôÇ
õ[þtõ q§KÐ s[þt °úכ[þtÉr 1 % ?/ü@_ ¸ \¦ Ð%i.
PACS numbers: 05.50.+q, 05.70.−a, 64.60.Cn, 75.10.Hk Keywords: ìrC<Êú H, 7H_ \¦Ð ~½ÓZO, sfç¸+þA
I. "e Â]Ø
>\"f © м#&h &³© ©sü< e>&³©
`
¦ [O"î l 0AôÇ ìrC<Êú H(partition function zeros) s
:rÉr 1952¸\ ª(Yang)õ o(Lee)\ _K %6£§Ü¼Ð ]
jîß÷&%3 [1]. 1964¸\ x39(Fisher) 4¤èú :r¸
¨ î
(complex temperature plane)\"f_ ìrC<Êú H_
>
h¥Æ`¦ ]jîß ¦ [2] 1983¸\ ìrC<Êú H_ Ä»ôÇ-ß¼l
»
¡
¤'(finite-size scaling) ZOgË:s ¸{9÷&"f [3], ©s ü
< e>&³©\ 'aôÇ ªôÇ s:r[þt ׿\"f ìrC<Êú H s
:rs © ÅÒ3lq~ÃÎl r %i. þjH\ [þt#Qü<"f ( É
Ó' ×¼J?#Q x9 7H_ \¦Ð(Monte Carlo) ~½ÓZOs Ø
Ô> µ1Ï # 4¤èú :r¸ ¨î\"f_ ìrC<Êú H[þt
`
¦ z´6 x&hܼР>íß H כ s 0px > ÷&"f ìrC<Ê Ã
º H s:r`¦ s6 xôÇ ½¨ 7HëH[þt_ ú q&hܼРZþt
#
Qz¤ [4–30]. ìrC<Êú H s:r_ :x&h &h6 xìr
:x>Óütoü< 6£x|9Óüto ü@\¸ ÙþÓüto [11–13], {9Óüt o
[14–21] x9 ÒqtÓüt<Æ [22–30] ìr 1px\"f¸ ìrC<Êú H s
:rs Ö¸µ1Ï > 6 x÷&¦ e. þjH_ 6£x6 x\"f © Z
tîr &hÉr z´+«>Óüto<Æ($í^ [8–10] x9 "é¶Ùþ [11,12]
1 p
x)\"f %3#Q X<s'_ ìr$3\"f¸ ìrC<Êú H s:r s
6 x÷&¦ eH &hs.
ì
rC<Êú H[þt`¦ ½¨ l 0AK"f © ´ú§s s6 x÷&¦ e
H ~½ÓZOÉr þjH\ µ1ÏôÇ &³@/&h 7H_ \¦Ð ~½ÓZO
∗§ $: [email protected]
s
[4–6,15–28]. &ñSXôÇ ìrC<Êú H[þt`¦ %3l 0AK"f
H $ ìrC<Êú\¦ %3x9 > ·ú ôÇ. tëß 7H _
\¦Ð ~½ÓZO`¦ s6 x ìrC<Êú\¦ H&hܼÐëß ·ú Ã
º e. "f 7H_ \¦Ð ~½ÓZOܼÐH &ñSXôÇ ìrC<Ê Ã
º H[þt`¦ ·ú ú \O.
1984¸\ oo(Marinari)H 7H_ \¦Ð ~½ÓZO`¦ s
6 x # éßíH{9~½Ó(simple-cubic lattice) sfç¸ +
þ
A(Ising model)_ ìrC<Êú H[þt`¦ >íß %i [4]. éß í
H{9~½Ó sfç¸+þA_ âĺ 4 × 4 × 4 \"f %3x9 ô
Ç ìrC<Êú ·ú94R e [31]. ooH 4 × 4 × 4
\"f &ñSXôÇ ìrC<Êú H[þtõ 7H_ \¦Ð ~½ÓZOܼÐ
% 3
Ér H&h ìrC<Êú H[þt`¦ q§ # (Óüto&hܼР_
p eH) ª_ z´Ãº»¡¤\ © îr ìrC<Êú H - :
x©&hܼР'Í P: H(first zero) ¢¸H ¸ H(leading zero)s¦ Ô¦aË> - Ér "fÐ _ {9u<Ê`¦ ·úÍÇx. 7£¤ 4 × 4 × 4 éßíH{9~½Ó sfç¸+þA\"fH 7H_ \¦Ð ~½Ó Z
O
ܼР%3Ér H&h 'Í P: H`¦ ø@½+É Ãº e. t
ëß 4 × 4 × 4 éßíH{9~½Ó sfç¸+þA\"f_ õëßܼ
Ð Ér ¸H Óüto>\"f 7H_ \¦Ð ~½ÓZOܼР%3Ér H
&h ìrC<Êú H[þt_ õ[þt`¦ &ñ{©o½+É ÃºH \O.
s
7HëH\"fH :r(Onsager)_ &ñSXôÇ K(exact solution) [32] >rF H ÅÒl&h â>¸| (periodic boundary conditions)`¦ L × L y(square lat- tice) sfç¸+þA(L = 4 ∼ 20)_ ìrC<Êú H[þt`¦ Mg-êøÍ
ĺ(Wang-Landau) 7H_ \¦Ð ~½ÓZO [33]ܼР½¨ 9¦ -667-
ô
Ç. sXO> %3Ér H&h ìrC<Êú H[þt_ õ[þt`¦
% 3
x9ôÇ õ[þtõ q§ # 7H_ \¦Ð ~½ÓZOܼР%3Ér H
&h ìrC<Êú H[þt`¦ s6 xôÇ ©sü< e>&³©
½
¨[þt\ @/ôÇ ø@¸\¦ &ñ|¾Ó&hܼР0puKЦ s[þt
½
¨[þt\ @/ôÇ ½+Ëo&h &ñ{©$í`¦ ÂÒ# 9¦ ôÇ.
II. TÇSË{¢]kù
þ
jH]X sÖ© © ñ6 x(½+Ë[jl J)õ ÅÒl&h â>¸
|
`¦ L × L y sfç¸+þAÉr H = −JX
hi,ji
σiσj (1)
_
Kx9Ðmîß\ _K"f &ñ_)a. #l\"f σiH
&
h
i(= 1, 2, ..., L2)\"f |9 ú eH Û¼2; °úכܼР+1õ
−1`¦ 2[½+É Ãº eܼ 9, ½+ËÉr ¸H 2L2>h_ þjH]X sÖ© [
þ
t(nearest neighbors)\ @/ôÇ כ s. %ò ¢¸H ª_ &ñú
° ú
כëß`¦ 2[½+É Ãº eH \-t E\¦ 6£§õ °ú s &ñ_ E = 1
2 X
hi,ji
(1 − σiσj) (2)
K
x9Ðmîß`¦ H = 2J(E − L2)ü< °ú s jþt ú e.
d
(2)\¦ s6 x ¸H 0pxôÇ ©I[þt\ @/ôÇ ½+Ë ìr C
<Êú
Z = X
{σn}
e−βH (3)
\
¦(βH kBT _ %iús 9 kBH ^¦ÞÔëß ©Ãº, T H :r¸s
) 6£§õ °ú s jþt ú e.
Z(y) = e2βJL2
2L2
X
E=0
Ω(E)yE (4)
#
l\"f Ω(E)H ©Ix9¸(density of states)s¦ yH e−2βJÐ &ñ_)a ªs. ú yH T = 0\"f 0s¦ T =
∞\"f 1sl M:ëH\ ÁºôÇôÇ ^ :r¸ %ò%i`¦ 0õ 1 s
_ Ä»ôÇôÇ ½¨çßܼР³ð&³K ïr.
'
pàÔÐxH ©Ix9¸ Ω(E)\ ÐÕª\¦ 2[ %3
# Q.
S(E) = kBln Ω(E) (5)
"f ìrC<Êú Z(y)H 6£§õ °ú s ³ð&³)a.
Z(y) = e2βJL2
2L2
X
E=0
exph S(E) kB
i
yE (6)
III. ÃXØ?_õu§zº Uês0nÉ
Å
Òl&h â>¸| `¦ y sfç¸+þA\"fH :r
K(Onsager solution)ÐÂÒ' &ñSXôÇ ©Ix9¸\¦ 6 x s
> %3`¦ ú e [34]. s 7HëH\"fH Mg-êøÍĺ 7H _
\¦Ð ~½ÓZO`¦ s6 x # H&h ©Ix9¸\¦ >íß
9¦ ôÇ. Mg-êøÍĺ ~½ÓZO\"fH p(E1→ E2) = min
"
Ω(E1) Ω(E2), 1
#
(7) _
sSXÒ¦(transition probability)`¦ s6 xôÇ [5, 33].
#
l\"f \-t E1H ×þ)a Û¼2;`¦ +'|9l(spin flip)
_ >_ \-ts¦ E2H +'|9Ér Êê_ \-ts.
$
¸H ©I\¦ Áº¸| )6 x H ½¨¸ús l(random walk) ·ú¦o1pu`¦ z´' # þjí_ ©Ix9¸ Ω0(E)\¦ Ò
q
t$íôÇ [5,33].
M
g-êøÍĺ ~½ÓZO_ ÙþdÉr ©Ix9¸\¦ Ìqt½+É M: ú&ñ
(modification factor) f (> 1)\¦ 6£§õ °ú s
Ω(E) → f Ω(E) (8)
6 xôÇH &hs. 'Í P: Mg-êøÍĺ 7H_ \¦Ð r Ð
3
x?/l(simulation)\"fH ú&ñ\¦ f1= eÐ ×þ 9, M
g-êøÍĺ 7H_ \¦Ð rÐ3x?/l ìøÍ4¤Hd\ ú&ñ
(i = 2, 3, ..., 30)\¦ fi=p
fi−1 (9)
Ð ×þôÇ [33]. s 7HëH\"fH 30r_ Mg-êøÍĺ 7H_
\¦Ð rÐ3x?/l\¦ z´'Ùþ¡Ü¼Ù¼Ð þj7áx&h ú&ñH f30= exp³ 1
229
´
= 1.00000000186 (10) s
. "f t}\H ú&ñ_ ´òõ _ \O#Q
"
f C0lw(Bhanot)_ 7H_ \¦Ð ~½ÓZO [5]õ Ä»K .
Ã
º&ñ f1\¦ s6 xôÇ Mg-êøÍĺ 7H_ \¦Ð rÐ3x
?
/l =åQ ©Ix9¸ Ω1(E)\¦ %3> ÷& 9 Ω1(E)H Ω0(E)Ð >h)a õs. s õ&ñ`¦ >5Åq ìøÍ4¤
© >h)a þj7áx ©Ix9¸ Ω30(E)\¦ %3`¦ ú e.
Ã
º&ñ\¦ 6 x ©Ix9¸\¦ >íß l 0AôÇ Ér 7H _
\¦Ð ~½ÓZO[þtÐ s` ØÔ> "é¶ H õ\¦ %3`¦ Ã
º e.
s
7HëH\"fH L×L y sfç¸+þA\ @/K"f 31×
L2×106_ 7H_ \¦Ð íß`¦ ú' %i. 7£¤ L = 6\
"
fH 1.12 × 109_ íß`¦ %i¦ L = 12\"fH 4.46 × 109_ íß`¦ %iܼ 9 L = 20\"fH 1.24 × 1010 _
íß`¦ z´' %i. Fig. 1Ér 14 × 14 y sfç
¸+þA\ @/K"f :r KÐÂÒ' %3Ér &ñSXôÇ 'pàÔÐx ü
< Mg-êøÍĺ 7H_ \¦Ð ~½ÓZO`¦ s6 x # %3Ér H&h
'pàÔÐx\¦ <Êa Ð#ÅÒ¦ e. ÕªaË>`¦ Ð ¿º 'p à
ÔÐx _ {9u<Ê`¦ ·ú ú e.