-
1777 Buffon 1942
(Fermi, Metropolis, Ulam, etc)1945 ENIAC (Pennsylvania Univ.)
1946 (Ulam )1952 MANIAC (Los Alamos) 1953 Metropolis, Rosenbluth2,
Teller2: 1955 Fermi-Pasta-Ulam 1957 Alder-Wainwright: 1959 Fosdick:
Ising model 1962 Handscomb: 1976 Suzuki: 1987 Swendsen-Wang:
-
Buffon
=
2pi
3408 =3.1415929 (Lazzerini 1901)
-
ENIAC
by Maria G. Mayer (1951)( )
:
)10( : logmin
irr
rd
i
i
-
1941 Ph.D. (Experimental Physica, University of Chicago)1943 Los
Alamos National Laboratory1945 Return to Chicago1946 ``Monte Carlo
method'' was named by S.Ulam1948 Return to Los Alamos1948 The first
actual Monte Carlo calculations on ENIAC1952 MANIAC became
operational1953 The Metropolis algorithm published
(Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and Edward
Teller)
1957 MANIAC II1957 Return to Chicago1965 Return to Los
Alamos
Metropolis
Metropolis, Nicholas Constantine (1915-1999)
-
( )( ) ( )
( )( ) ( )
( )
==
ii
iii
S
Si SW
SQSW
SW
SQSWQconstSP
( ) ( )( ) ( )
( )( )
1
=
i
ii
S
Sii
SQ
SW
SQSWQSWSP
Metropolis, Rosenbluth, Rosenbluth, Teller, Teller 1953
-
Markov
( ) ( ) ( ) ( )SWSSPSWSSP = | |
( )SW
Metropolis, Rosenbluth, Rosenbluth, Teller, Teller 1953
-
Alder-Wainwright
Metropolis, Rosenbluth, Rosenbluth, Teller, Teller 1953
-
Ising Model
Fosdick 1959Cu3Au
CF: Peierls Onsager
-
Fermi, Pasta, and Ulam 1955
-
Alder-Wainwright 1957
Kirkwood 1935,Born-Green 1959
-
g:
Kawashima-Young 1995
-
Handscomb's method (1962, modified by Lyklema in 1982)
( )
( )operator '``swap'
2
0
1,0
1
=
+=
>=
+
+
ij
iii
i
zii
ii
E
constEJH
JSHJH SSm
( ) ( ) ( )( )
( ) ( ) ( )( ) ),( , ,,,
Tr!
1Tr!
21
00
121
0
kkkn
ppppH
n
nCCn
n
n
nn
jippppCEEEEe
n
JCW
CWHn
Z
nn
=
=
==
=
=
=
L
L
"C'' C
-
Handscomb's method( ) ( )nkknkk ppppppCpppppC ,,,,,,, ,,,,,,
1new21121 LLLL ++
-
( )
( )
( )( )
=
=
=
+
=
S
S
M
kkk
SSLM
SSLd
SW
SSw
eDS
eDSZM
kkk
,
11
,
,
1
0
&
&
S :
Suzuki 1976
-
( ) ( )( )
=
SWSWSSP ' , 1min |'
-
Swendsen-Wang
S G S
Swendsen-Wang 1987
( ) ( ) ( ) ( )GWGSPSWSGP | | =
-
Evertz-Lana-Marcu 1993
( ) =2
JP
:
-
( ) =2
JP ( )2
J=
-
=
S
ii
2
1,
2
1S
)(Tr)(TrSLL P
=
1,1 21, 1,2 2,2
Split-spin trick
Kawashima-Gubenatis 1995Harada-Troyer-Kawashima 1998
-
An SU(N) model
( ) ( )( )
',
=rr
rJrJNJH
Representation:
Hamiltonian:
Santoro et al. (1999), presentAffleck (1985), Arovas-Auerbach
(1988)
Read-Sachdev (1990)
[ ] NJJJJ ,,2,1,,, , L==
-
Magnetic Correlation Function
Finite Neel order for N=2,3,4, but no Neel order for N>=5
( ) ( ) = r
ZZrd
d SrSLLM
S 0 )1( 2
,
,
pipipipi
( )
( )
)5(0004.0 :6)8(0003.0 :5
)6(158.0 )2(025.0 :4
)2(291.0 )1(085.0 :3
2
2
2
2
==
==
=
==
=
==
s
s
s
s
s
s
mNmNm
mNm
mN
-
Sandvik-Kurkijarvi 1991
( ) ( ) ( )
{ }( )
( ){ }
( ) ( )
( )
( ) ( ) ( )( )
=
=
=
=
=
=
=
+==
=
=
= = =
=
=
Cn
G SiLiLLi
n
Hk
n
G
L
kg
n
L
k
n
k
L
n n
L
k
n
nLL
n
nn
n
nn
SSSSGSCCW
SHSSHSSHSL
nL
HNggggG
HL
nL
HL
nL
Hn
Ln
eHn
Hn
Z
LL
k
k
k
kkk
k
,,, , ,
! !
1 , ,2 ,1 ,0,,,
Tr!
!
Tr!
!
1,0 Tr!
oTr!
Tr!
21
1211
0
21
1
1
0 : 1
100
11
L
L
L
L
-
( ) ( ) ( )
( )
( )
( ) ,,,
,,,
,
! ! 1
21
21
121 1
n
n
iLi
n
C
gggG
SSSS
GSC
SHSSHSLLCWCWZ
L
L
L
L
=
Sandvik-Kurkijarvi 1991
1
2
3
L
L-1
