WORM ALGORITHM FOR CLASSICAL AND QUANTUM

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WORM ALGORITHM FOR CLASSICAL AND QUANTUM
WORM ALGORITHM FOR CLASSICAL AND QUANTUM
STATISTICAL MODELS
Nikolay Prokofiev, Umass, Amherst
Many thanks to collaborators
on major algorithm developments
Boris Svistunov, Umass, Amherst
Igor Tupitsyn, PITP
Vladimir Kashurnikov, MEPI, Moscow
Evgeni Burovski, Umass, Amherst
Massimo Boninsegni, UAlberta, Edmonton
NASA
Les Houches, June 2006
Worm algorithm idea
Consider:
- configuration space = arbitrary closed loops
- each cnf. has a weight factor
Wcnf
A W

W
cnf
- quantity of interest
Acnf
A
cnf
cnf
cnf
cnf
“conventional”
sampling scheme:
local shape change
No sampling of
topological classes
Critical slowing down
Add/delete small loops
can not
evolve to
 Nupdates 
z

L


d
L


dynamical critical exponent
z  2 in many cases
Worm algorithm idea
draw and erase:
Masha
Ira
Ira
Masha
+
Masha
Masha
keep
drawing
or
Topological classes are (whatever you can draw!)
No critical slowing down in most cases
Disconnected loops relate to important
physics (correlation functions) and are
not merely an algorithm trick!
High-T expansion for the Ising model
Z   e H / T
{i }
H
  K  i  j (  1)
T
ij 


Nb 
K Nb
   
i  j  

{i }  b ij  Nb  0 N b !



K Nb  
Mi
  
   i 
{ Nb }  b ij  N b !  i  i 1

where
M i   N b ij   even
ij 
Nb


K
N
2



{ Nb }loops  b ij  Nb ! 
3
2
1
4
4
Nb  number of lines;
enter/exit rule  M i  even
4
2
Spin-spin correlation function:
g IM
GIM

, G   e H / T I M
Z
{ }
i
Nb




K
K Nb  
M i  iI iM
N
G   
 2
   i



N
!
{ Nb }  b ij 
{ Nb }loops  b ij  N b ! 
b  i  i 1

 IM worm
Worm algorithm cnf. space =
I
Z
Same as for generalized partition
1
3
4
ZW  Z  G
4
M
2
G
Getting more practical: since
eK12  cosh N ( K ) 1  tanh( K )12 


Nb
Z  cosh ( K )    tanh ( K ) 
{ Nb  0,1}  b

loops
dN
Complete algorithm:
- If
I  M , select a new site for I , M at random
- select direction to move
- If
Nb  0
1
accept
M , let it be bond b
Nb  1
0
with prob.
R
min(1, tanh( K ))
min(1, tanh 1 ( K ))
G(I  M )  G(I  M )  1
I=M
I
Z  Z  I ,M
M


N links  N links   I , M   N b 
 b

M
M
M
Correlation function:
Magnetization fluctuations:
Energy: either
or
g (i )  G (i ) / Z
M
2

  
i
2
 N  g (i )
E   JNd 12   JNdg (1)
E   J tanh( K )  dN  N links sinh 2 ( K ) 
Ising
lattice field theory

H
2
4
 t   i   i    i  U   i 
T
i ( x , y , z )
i
i
Z    d  i e H / T
i
expand

t Ni 
Z   

Ni  i Ni ! 
i
t N (i  i ) N

N!
N 0

t i i
e
  d 
i
M1i
i
 
 M 2i
i
2
e
 i U i
4

e i ( M 1  M 2 )  M 1  M 2  M
loops
 Q( M )
i
i
where Q( M ) 
0
if

M1  M 2
  dxx e
0
M x Ux 2
closed oriented loops

tabulated numbers
i


Ni
 
Flux in = Flux out
g (I  M ) 

i
i  '
 Ni ,
N i  ', '

i
closed oriented loops
of integer N-currents
G( I  M )
  I M
Z
N i
I
(one open loop)
M
Z-configurations have
I M
i
Same algorithm:
Z G
I=M
sectors, prob. to accept
 Q( M I  1) 
Rz G  min 1,

 Q( M I ) 
M M
N M   N M   1
draw

tQ( M M '  1) 
R  min 1,

(
N

1)
Q
(
M
)
M 
M' 

N M  ,  N M  ,  1 erase R  min 1, ( N M , )Q(M M  1) 


tQ( M )

Keep drawing/erasing …
M

M M
Multi-component gauge field-theory
(deconfined criticality, XY-VBS and Neel-VBS quantum phase transitions…

2
2
2
H
 t  a ,i   a ,i eiA (i )    a ,i  U ab  a ,i  b ,i   [  A (i )]2 
T
a ;i
a ;i
ab ;i

 A4
 A3
 A1
U11  U 22  U12
U11  U 22  U12
XY-VBS transition; understood (?)
no DCP, always first-order
Neel-VBS transition, unknown !
 A2
Winding numbers
Homogeneous gauge in x-direction:
t  teiA , Ax (r )   / L, Ax (r )dx 
Wx   ( Ni ,x  Ni  x ,x ) / L  1
Z  e
iWx
i
ZWx 
Wx
F  T ln Z  F (0)  Ld  S
 / L
eiA
2
eiA
2
e  iA
d 2
S  L
T Wx 2
 F
 d 2
2

L
2
Ceperley
Pollock ‘86

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