intro mc troyer

7/23/2019 Intro Mc Troyer

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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)



! /4



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



f = f p

p

:= f (

!

x) p(

!

x) p(

!

x)d !

x!"

d !

x!"



http://www.idquantique.com/




x n +1

= (ax n + c)modm



xn = x

n! p " xn! q modm



http://en.wikipedia.org/wiki/Normal_distribution

http://en.wikipedia.org/wiki/Normal_distribution

http://en.wikipedia.org/wiki/Infinite_monkey_theorem

http://en.wikipedia.org/wiki/Infinite_monkey_theorem

http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29

http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29

http://en.wikipedia.org/wiki/Birthday_paradox

http://en.wikipedia.org/wiki/Birthday_paradox

http://en.wikipedia.org/wiki/Poisson_distribution

http://en.wikipedia.org/wiki/Poisson_distribution



http://en.wikipedia.org/wiki/Craps

http://en.wikipedia.org/wiki/Craps

http://en.wikipedia.org/wiki/Wald-Wolfowitz_runs_test

http://en.wikipedia.org/wiki/Wald-Wolfowitz_runs_test

http://en.wikipedia.org/wiki/Exponential_distribution

http://en.wikipedia.org/wiki/Exponential_distribution



P[ x < y] = f (t )dt =: F ( y) !a

y

" P[u < F ( y)]

# x = F $1(u)



f ( x) =1

2!

exp " x 2( )

n1 = !2 ln(1 ! u

1) sinu

2

n2 = !2 ln(1 ! u1) cosu2



A =

A(c) p(c)dc"

#

p(c)dc" #

P[c]= p(c)

Z =

exp("# E (c))

Z



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)



P x, y= min 1, A y, x p( y)

A x, y p( x)

"

#

$ %

&

'



P = min 1,e!( E

new ! E

old

) /T

"# $%



p( x, t ) = p( x) +O(exp("# 2t ))

p( x) = p( y)W y, x y

"



" A=

Var A

M

" A =

Var A

M 1+ 2#

A( )

" A=

Ai+ t

Ai # A

2( )t =1

$

%

Var A



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



cV = E

2" E

2

T 2

U =m

4

m2

2



U "U =

1

M U

i

i=1

M

#

"U # 1

M ( M $1)U

i $U( )

2

i$1

M

%



U 0 =1

M f ( X i,Y i)

i=1

M

"



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 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 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

"



m4

m2

2



" 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 )



" 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 )



P = exp("2# J )

P =1" exp("2# J )



E c = E (T c) ∼ 0.74E 0



w(E ) = 1

g̃(E ) g̃(E ) = 1



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)



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



Z = e" E

c / k

BT

c

#

Z = Tr

e

"# H

$ pc

c%



Z = Tre−βH =

i

pi

|i1>

|i2>

|i3>

|i4>

|i1>



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



m ! (T c " T )

#

! "| T # T c

|#$



m ! (T c " T )

#

! "| T # T c

|#$

T >> T c



m ! (T c " T )

#

! "| T # T c

|#$

T >> T c

T " T c



m ! (T c " T )

#

! "| T # T c

|#$



" # T c$T ( )

$%

M " T c#T ( )

$

" M (T ) = M (# )$# %& /'

M (T , L) = M (" , L) = M (" / L)# "

$% /& L >>

" L$% /&

L << "

' ( )

A quantum antiferromagnet



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

intro mc troyer

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