• 검색 결과가 없습니다.

Study on Partition Function Zeros by Using the Monte Carlo Method

N/A
N/A
Protected

Academic year: 2021

Share "Study on Partition Function Zeros by Using the Monte Carlo Method"

Copied!
6
0
0

로드 중.... (전체 텍스트 보기)

전체 글

(1)

à X

Ø? _ õ u §z º U ê s0 n Éù p § T “ Ó Þ” X ¢ Ä Z Ø9 0] K ¤• ¤ ¿ R < T  ] Ø Ž ì ŏ Œ

™

»ŠûBa:@

² D

GwnæÅÒ@<Ɠ§ “§€ªœõ&ñÂÒ, ØæÅÒ 380-702

ƒ

‘

šø¶B)Ö< · ™».> *>

Õ ü

æz´@<Ɠ§ Óüto<Æõ x9 ìr[O>¨G'p', "¦ 156-743 (2009¸ 4Z4 20{9 ~ÃÎ6£§)

L × L yŒ• sf痸+þA(L = 4 ∼ 20)\"f 7H_…\¦–Ð ~½ÓZO¼–Ð %3Ér H&h ìrC<Êú H[þt`¦

^

‰>&hs¦ ½¨^‰&h¼–Ð ¶ú˜(ЀŒ¤. >_ ß¼l &4¸ Óüto&h¼–Ð _p e”H 'Í P:  Õª ÅÒ 0

A\"H 7H_…\¦–Ð ~½ÓZO_ H&h  &ñSX‰†<Ê`¦ ·ú˜ ú e”%3. \P%i<Æ&h FÇ\"f 7H_…\¦

–

Ð іÐÂÒ' e”>&h °úכܼ–Ð 0.4209(78)`¦ %3%3¼ 9 \P¡¤'‘tº °úכܼ–Ð 0.992(55)`¦ %3%3. %3x9Ç





[þ q§KЀ s[þt °úכ[þÉr €• 1 % ?@_ š¸ \¦ ˜Ð%i.

PACS numbers: 05.50.+q, 05.70.−a, 64.60.Cn, 75.10.Hk Keywords: ìrC<Êú H, 7H_…\¦–Ð ~½ÓZO, sf痸+þA

I. "e Â]Ø



>\"f ©œ ˜Ð¼#&h ‰&³©œ“ ©œ„s< e”>&³©œ

`



¦ [O"î l 0Ç ìrC<Êú H(partition function zeros) s

:Ér 1952¸\ €ªœ(Yang)õ o(Lee)\ _K %ƒ6£§Ü¼–Ð ]

ߖ÷&%3 [1]. 1964¸\ x39(Fisher) 4Ÿ¤™èú “:¸

¨ î

€(complex temperature plane)\"f_ ìrC<Êú H_

>

Æ`¦ ]ߖ ¦ [2] 1983¸\ ìrC<Êú H_ Ä»ôÇ-ß¼l

»

¡

¤'‘(finite-size scaling) ZOgË:s •¸{9"f [3], ©œ„s ü

< e”>&³©œ\ ›'Ç ªœôÇ s:r[þt ׿\"f ìrC<Êú H s

:rs ©œ ÅÒ3lq~ÃÎl rŒ• %i. þjH\ [þt#<"f (Ž É

Ó' ¼J?#Q x9 7H_…\¦–Ð(Monte Carlo) ~½ÓZOs  Ø

Ô> µ1τ #Œ 4Ÿ¤™èú “:¸ ¨î€\"f_ ìrC<Êú H[þt

`



¦ z´6 x&h¼–Ð >ߖ H כ s 0px > ÷"f ìrC<Ê Ã

º H s:r`¦ s6 Ç ƒ¨ 7H[þt_ ú q•&h¼–Ð Zþt

#

QzŒ¤ [4–30]. ìrC<Êú H s:r_ „:Ÿx&h &h6 r

“



 :Ÿx>üto< 6£x|9üto ü@\•¸ Ùþ˜Óüto [11–13], {9üt o

 [14–21] x9 Òqü<Æ [22–30] ìr 1px\"¸ ìrC<Êú H s

:rs Ö¸µ1Ï > 6 ¦ e”. þjH_ 6£x6 x\"f ©œ Z



tr &hÉr z´+«üto<Æ($í^‰ [8–10] x9 "é¶þ˜ [11,12]

1 p

x)\"f %3# X<s'_ ìr$3\"¸ ìrC<Êú H s:r s

 6 ¦ e”H &hs.

ì



rC<Êú H[þt`¦ ½¨ l 0AK"f ©œ ´ú§s s6 ¦ e

”

H ~½ÓZOÉr þjH\ µ1τÇ ‰&³@/&h 7H_…\¦–Ð ~½ÓZO

“§’ $: [email protected]

s

 [4–6,15–28]. &ñSX‰ôÇ ìrC<Êú H[þt`¦ %3l 0AK"f





H €$ ìrC<Êú\¦ %3x9 > ·ú˜ ôÇ. tߖ 7H _

…\¦–Ð ~½ÓZO`¦ s6 x  ìrC<Êú\¦ H&h¼–Ðëߖ ·ú˜ Ã

º e”. "f 7H_…\¦–Ð ~½ÓZO¼–ЍH &ñSX‰ôÇ ìrC<Ê Ã

º H[þt`¦ ·ú˜ ú \O.

1984¸\ oo(Marinari)H 7H_…\¦–Ð ~½ÓZO`¦ s

6 x #Œ éߖíH{9~½Ó(simple-cubic lattice) sf痸 +

þ

A(Ising model)_ ìrC<Êú H[þt`¦ >ߖ %i [4]. éߖ í



H{9~½Ó sf痸+þA_ âĺ 4 × 4 × 4 \"f %3x9 ô



Ç ìrC<Êú ·ú˜94R e” [31]. ooH 4 × 4 × 4





\"f &ñSX‰ôÇ ìrC<Êú H[þ 7H_…\¦–Ð ~½ÓZO¼–Ð

% 3

Ér H&h ìrC<Êú H[þt`¦ q§ #Œ (Óüto&h¼–Ð _

p e”H) €ªœ_ z´Ãº»¡¤\ ©œ r ìrC<Êú H - :

Ÿ

x©œ&h¼–Ð 'Í P: H(first zero) ¢¸H ‚¸ H(leading zero)s¦ Ô¦aË> - “Ér "Ð _ {9u<Ê`¦ ·ú˜Çx. 7£¤ 4 × 4 × 4 éߖíH{9~½Ó sf痸+þA\"H 7H_…\¦–Ð ~½Ó Z

O

¼–Ð %3Ér H&h 'Í P: H`¦ ’+É Ãº e”.  t

ߖ 4 × 4 × 4 éߖíH{9~½Ó sf痸+þA\"f_ ߖܼ

–

Ð Ér —¸ŽH Óüto>\"f 7H_…\¦–Ð ~½ÓZO¼–Ð %3Ér H



&h ìrC<Êú H[þt_ [þt`¦ &ñ{©œ+É ÃºH \O.

s

 7H\"H “:r(Onsager)_ &ñSX‰ôÇ K(exact solution) [32] ”>rF H ÅÒl&h â>¸| (periodic boundary conditions)`¦ ” L × L yŒ•(square lat- tice) sf痸+þA(L = 4 ∼ 20)_ ìrC<Êú H[þt`¦ Mg-êøÍ



º(Wang-Landau) 7H_…\¦–Ð ~½ÓZO [33]ܼ–Ð ½¨ ¦ -667-

(2)

ô



Ç. sXO> %3Ér H&h ìrC<Êú H[þt_ [þt`¦

% 3

x9Ç [þ q§ #Œ 7H_…\¦–Ð ~½ÓZO¼–Ð %3Ér H



&h ìrC<Êú H[þt`¦ s6 Ç ©œ„s< e”>&³©œ ƒ

½

¨[þt\ @Ç ’¸\¦ &ñ|¾Ó&h¼–Ð 0puKГ¦ s[þt ƒ

½

¨[þt\ @Ç ½+Ëo&h &ñ{©œ$í`¦ ÂÒ#Œ ¦ ôÇ.

II. TÇSË{¢]kù

þ

jH]X s© ©œ ñŒ•6 x(+Ë[jl J)õ ÅÒl&h â>¸

|

`¦ ” L × L yŒ• sf痸+þÉr H = −JX

hi,ji

σiσj (1)

_

 Kx9Ðmߖ\ _K"f &ñ_)a. #Œl\"f σiH 

&

h

 i(= 1, 2, ..., L2)\"f |9 ú e”H Û¼—2; °úכܼ–Ð +1õ

−1`¦ 2+É Ãº e”¼ 9, ½+˓Ér —¸ŽH 2L2>h_ þjH]X s© [

þ

t(nearest neighbors)\ @Ç כ s. %ò ¢¸H €ªœ_ &ñú

° ú

כëߖ`¦ 2+É Ãº e”H \-t E\¦ 6£§õ °ú s &ñ_  E = 1

2 X

hi,ji

(1 − σiσj) (2)

K

x9Ðmߖ`¦ H = 2J(E − L2< °ú s jþt ú e”.

d

”

 (2)\¦ s6 x  —¸ŽH 0pÇ ©œI[þt\ @Ç ½+˓ ìr C

<Êú

Z = X

n}

e−βH (3)

\



¦(βH kBT _ %iºs 9 kBH ^¦ÞÔëߖ ©œÃº, T H “:¸s



) 6£§õ °ú s jþt ú e”.

Z(y) = e2βJL2

2L2

X

E=0

Ω(E)yE (4)

#

Œl\"f Ω(E)H ©œIx9¸(density of states)s¦ yH e−2βJ–Ð &ñ_)a €ªœs. º yH T = 0\"f 0s¦ T =

∞\"f 1sl MH\ ÁºôÇôÇ „^‰ “:¸ %ò%i`¦ 0õ 1 s

_ Ä»ôÇôÇ ½¨çߖܼ–Ð ³ð‰&³K ïr.

 '

ԖÐxH ©œIx9¸ Ω(E)\ ÐÕª\¦ 2[  %3

# .

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>¸| `¦ ” yŒ• sf痸+þA\"H “:r



 K(Onsager solution)–ÐÂÒ' &ñSX‰ôÇ ©œIx9¸\¦ 6 x s

 > %3`¦ ú e” [34]. s 7H\"H Mg-êøÍº 7H _

…\¦–Ð ~½ÓZO`¦ s6 x #Œ H&h ©œIx9¸\¦ >ߖ 



¦ ôÇ. Mg-êøÍº ~½ÓZO\"H p(E1→ E2) = min

"

Ω(E1) Ω(E2), 1

#

(7) _

 „sSX‰Ò¦(transition probability)`¦ s6 Ç [5, 33].

#

Œl\"f \-t E1H ‚þ˜)a Û¼—2;`¦ +'|9l(spin flip)

„



_ >_ \-ts¦ E2H +'|9Ér Êê_ \-ts. €

$

 —¸ŽH ©œI\¦ Áº›¸|  )‡6 x H ¨¸úšs Žl(random walk) ·ú˜“¦o1pu`¦ z´' Ÿ#Œ þí_ ©œIx9¸ Ω0(E)\¦ Ò

q

t$íôÇ [5,33].

M



g-êøÍº ~½ÓZO_ Ùþ˜d”Ér ©œIx9¸\¦ Ìq+É M: ú&ñ

“



(modification factor) f (> 1)\¦ 6£§õ °ú s

Ω(E) → f Ω(E) (8)



6 ÇH &hs. 'Í P: Mg-êøÍº 7H_…\¦–Ð r Ð

3

x?/l(simulation)\"H ú&ñ“\¦ f1= e–Ð ×þ˜  9, M



g-êøÍº 7H_…\¦–Ð r3x?/l ìøÍ4Ÿ¤H†d\  ú&ñ

“



(i = 2, 3, ..., 30)\¦ fi=p

fi−1 (9)

–

Ð ×þ˜ôÇ [33]. s 7H\"H 30r_ Mg-êøÍº 7H_…

\¦–Ð r3x?/l\¦ z´'ŸÙþ¡Ü¼Ù¼–Ð þj7áx&h ú&ñ“H f30= exp³ 1

229

´

= 1.00000000186 (10) s

. "f t}Œ•\H ú&ñ“_ ´òõ _ \O#Q

"

f C0lw(Bhanot)_ 7H_…\¦–Ð ~½ÓZO [5]õ Ä»K ”.

Ã

º&ñ“ f1\¦ s6 Ç Mg-êøÍº 7H_…\¦–Ð r3x

?

/l =åQ ©œIx9¸ Ω1(E)\¦ %3> ÷& 9 Ω1(E)H 0(E)˜Ð >)a s. s õ&ñ`¦ >5Åq ìøÍ4Ÿ¤ 

€



 ©œ >)a þj7áx ©œIx9¸ Ω30(E)\¦ %3`¦ ú e”.

Ã

º&ñ“\¦ 6 x  ©œIx9¸\¦ >ߖ l 0Ç Ér 7H _

…\¦–Ð ~½ÓZO[þÐ s Ô> "éH \¦ %3`¦ Ã

º e”.

s

 7H\"H L×L yŒ• sf痸+þA\ @/K"f 31×

L2×106_ 7H_…\¦–Ð ƒߖ`¦ ú'Ÿ %i. 7£¤ L = 6\

"

H 1.12 × 109_ ƒߖ`¦ %i¦ L = 12\"H 4.46 × 109_ ƒߖ`¦ %i¼ 9 L = 20\"H 1.24 × 1010 _

 ƒߖ`¦ z´'Ÿ %i. Fig. 1“Ér 14 × 14 yŒ• sfç

—

¸+þA\ @/K"f “:r KÐÂÒ' %3Ér &ñSX‰ôÇ 'ԖÐx ü

< Mg-êøÍº 7H_…\¦–Ð ~½ÓZO`¦ s6 x #Œ %3Ér H&h

“



 'ԖÐx\¦ †<Êa ˜Ð#ŒÅғ¦ e”. ÕªaË>`¦ ˜Ð€ ¿º 'p à

ԖÐx _ {9u<Ê`¦ ·ú˜ ú e”.

수치

Fig. 1. The approximate and exact entropies S(E) = ln Ω(E) (in unit of k B ), as a function of energy E, of the 14 × 14 square-lattice Ising model with fully periodic boundary conditions
Table 1. The approximate (denoted as mc) and exact (denoted as ex) values of the first zeros y 1 (L) for the L×L square-lattice Ising models with fully periodic boundary conditions

참조

관련 문서

The index is calculated with the latest 5-year auction data of 400 selected Classic, Modern, and Contemporary Chinese painting artists from major auction houses..

The key issue is whether HTS can be defined as the 6th generation of violent extremism. That is, whether it will first safely settle as a locally embedded group

1 John Owen, Justification by Faith Alone, in The Works of John Owen, ed. John Bolt, trans. Scott Clark, &#34;Do This and Live: Christ's Active Obedience as the

First, we can approximate mismatch-induced offsets as low-frequency AC noise and hence substitute time- consuming Monte-Carlo simulations with small-signal noise consuming

¾ Since bipolar differential pair can be analyzed using half- circuit, its transfer function, I/O impedances, locations of poles/zeros are the same as that of the half

In gi ngi va,LCs are found i n oralepi thel i um ofnormalgi ngi va and i n smal l er amountsi nthesul cul arepi thel i um,buttheyareprobabl yabsentfrom thejuncti onal epi thel

[표 12] The true model is inverse-gaussian, out-of-control ARL1 and sd for the weighted modeling method and the random data driven

à For each subentity, create a table that includes the attributes of that entity set plus the primary key of the higher level