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f ( x)dxa
b
! =b " a
N f a + i
b " a
N
# $
% &
i=1
N
' + O(1/ N )
f ( x)dxa
b
! = b " a N
12
f (a) + f a + i b " a N # $ % &
i=1
N "1
' + 12
f (b)#
$ ( %
& ) + O(1/ N 2)
f ( x)dx
a
b
! =b " a
3 N
f (a) + (3" ("1)i) f a + i
b " a
N
#
$
%
&
i=1
N "1
' + f (b)#
$ (
%
& ) + O(1/ N
4)
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! /4
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f = f (!
x)d !
x!"
d !
x!"
f ! 1
M f (
!
x i)i=1
M
"
! = Var f M
" M #1/ 2
Var f = f 2 # f 2
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f = f p
p
:= f (
!
x) p(
!
x) p(
!
x)d !
x!"
d !
x!"
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x n +1
= (ax n + c)modm
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xn = x
n! p " xn! q modm
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P[ x < y] = f (t )dt =: F ( y) !a
y
" P[u < F ( y)]
# x = F $1(u)
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f ( x) =1
2!
exp " x 2( )
n1 = !2 ln(1 ! u
1) sinu
2
n2 = !2 ln(1 ! u1) cosu2
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A =
A(c) p(c)dc"
#
p(c)dc" #
P[c]= p(c)
Z =
exp("# E (c))
Z
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c1" c
2" ..." c
i" c
i+1" ...
" x, y #n : W n( )
x, y$ 0
W x, y
y
" = 1
0 =d dt
p( x) = p( y)W y, x y
" # p( x)W x, y y
" $ p( x) = p( y)W y, x y
"
W x, y
W y, x=
p( y)
p( x)
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P x, y= min 1, A y, x p( y)
A x, y p( x)
"
#
$ %
&
'
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P = min 1,e!( E
new ! E
old
) /T
"# $%
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P = min 1,e!( E
new ! E
old
) /T
"# $%
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P = min 1,e!( E
new ! E
old
) /T
"# $%
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p( x, t ) = p( x) +O(exp("# 2t ))
p( x) = p( y)W y, x y
"
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" A=
Var A
M
" A =
Var A
M 1+ 2#
A( )
" A=
Ai+ t
Ai # A
2( )t =1
$
%
Var A
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Ai
(l )=
1
2 A
2i"1
( l"1)+ A
2i
l
( )
"(l ) = Var A( l ) M (l ) l#$ % # % % " = (1+ 2& A )Var A M
& A= lim
l#$
1
2
2lVar A
(l )
Var A(0 ) '1
(
) *
+
, -
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0 2 4 6 8 10
L = 4L = 48
! ( l )
binning level l
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cV = E
2" E
2
T 2
U =m
4
m2
2
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U "U =
1
M U
i
i=1
M
#
"U # 1
M ( M $1)U
i $U( )
2
i$1
M
%
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U "U =
1
M U
i
i=1
M
#
"U # 1
M ( M $1)U
i $U( )
2
i$1
M
%
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U 0 =1
M f ( X i,Y i)
i=1
M
"
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U 0 =1
M f ( X i,Y i)
i=1
M
"
U 1=
1
M "1 f ( X i,Y i)
i=2
M
#
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U 0 =1
M f ( X i,Y i)
i=1
M
"
U 1=
1
M "1 f ( X i,Y i)
i=2
M
#
U j =
1
M "1 f ( X
i,Y
i)
i=1i# j
M
$
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U 0 =1
M f ( X i,Y i)
i=1
M
"
U 1=
1
M "1 f ( X i,Y i)
i=2
M
#
U j =
1
M "1 f ( X
i,Y
i)
i=1i# j
M
$
"U # M $1
M U
i $U( )
2
i$1
M
%
U "U 0 # ( M #1)(U #U 0)
U =
1
M U
i
i=1
M
"
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m4
m2
2
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" E = #2 J $ P = min 1,e#2" E /T ( ) =1
" E = +2 J # P = min 1,e$2" E /T ( ) = exp($2% J )
1" exp("2# J )
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" E = #2 J $ P = min 1,e#2" E /T ( ) =1
" E = +2 J # P = min 1,e$2" E /T ( ) = exp($2% J )
1" exp("2# J )
P = exp("2 J /T )
P =1" exp("2 J /T )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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P = exp("2# J )
P =1" exp("2# J )
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E c = E (T c) ∼ 0.74E 0
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w(E ) = 1
g̃(E ) g̃(E ) = 1
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w(E ) = 1
g̃(E ) g̃(E ) = 1
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w(E ) = 1
g̃(E ) g̃(E ) = 1
p(E 1 → E 2) = min
1, w(E 2)
w(E 1)
= min
1,
g̃(E 1)
g̃(E 2)
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w(E ) = 1
g̃(E ) g̃(E ) = 1
p(E 1 → E 2) = min
1, w(E 2)
w(E 1)
= min
1,
g̃(E 1)
g̃(E 2)
g̃(E ) = g̃(E ) · f
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w(E ) = 1
g̃(E ) g̃(E ) = 1
p(E 1 → E 2) = min
1, w(E 2)
w(E 1)
= min
1,
g̃(E 1)
g̃(E 2)
g̃(E ) = g̃(E ) · f
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w(E ) = 1
g̃(E ) g̃(E ) = 1
p(E 1 → E 2) = min
1, w(E 2)
w(E 1)
= min
1,
g̃(E 1)
g̃(E 2)
g̃(E ) = g̃(E ) · f
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Z = e" E
c / k
BT
c
#
Z = Tr
e
"# H
$ pc
c%
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Z = Tre−βH =
i
pi
|i1>
|i2>
|i3>
|i4>
|i1>
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A = Ai pi
i
" pi
i
" =
Ai sgn pi pi
i
" pi
i
"
sgn pi pi
i
" pi
i
" #
A $ sign p
sign p
sign =
i pii | pi|
= Z/Z | p| = e−βV (f −f |p|)
∆sign
sign = sign2 − sign2√
M sign ≈ eβV (f −f |p|)
√ M
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m ! (T c " T )
#
! "| T # T c
|#$
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m ! (T c " T )
#
! "| T # T c
|#$
T >> T c
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m ! (T c " T )
#
! "| T # T c
|#$
T >> T c
T " T c
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m ! (T c " T )
#
! "| T # T c
|#$
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" # T c$T ( )
$%
M " T c#T ( )
$
" M (T ) = M (# )$# %& /'
M (T , L) = M (" , L) = M (" / L)# "
$% /& L >>
" L$% /&
L << "
' ( )
A quantum antiferromagnet
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S (Q ) = L2m !L
2"z "#
! s "L
2#$
1
10
100
1000Staggered suscpetibilityStaggered structure
factor
2-! = 1.985 ± 0.025
2-z-! = 0.967 ± 0.005