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WORM ALGORITHM Nikolay Prokofiev, Umass, Amherst Boris Svistunov, Umass, Amherst Igor Tupitsyn, PITP, Vancouver Vladimir Kashurnikov, MEPI, Moscow Massimo Boninsegni, UAlberta, Edmonton NASA ISSP, August 2006 Evgeni Burovski, Umass, Amherst Matthias Troyer, ETH Masha Ira
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### Transcript of WORM ALGORITHM - issp.u-tokyo.ac.jp

Microsoft PowerPoint - ISSPworm.pptGrand canonical ensemble Off-diagonal correlations “Single-particle” and/or condensate wave functions Winding numbers and
Examples from : superfluid-insulator transition, spin chains, helium solid & glass, deconfined criticality, resonant fermions, holes in the t-J model, …
S ρ ( )r
Efficiency
PhD while still young Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams
Reliably!
- each cnf. has a weight factor c n fW
- quantity of interest c n fA c n f c n f
c n f
A W
A W
can not evolve to
Critical slowing down (large loops are related to critical modes)
Worm algorithm idea
draw and erase:
Topological are sampled (whatever you can draw!)
No critical slowing down in most cases
Disconnected loops are related to correlation functions and are not merely an algorithm trick!
Masha
Masha
H K
i j
i
K
j
M i
bN = iM even→ =
σ σ σ= ∑
G g
Worm algorithm cnf. space = Z G WZ Z G= + κ
Same as for generalized partition
Complete algorithm:
- If accept wit h prob.
I M=
M b
,I M
R = 1
min(1, tanh( ))
min(1, tanh ( ))
I=M
Z Z
Energy: either 1 2 (1)E JNd JNdg= − σ σ = −
Magnetization fluctuations: ( )22 ( )iM N g i= σ = ∑ ∑
or 2tanh( ) sinh ( )bondsE J K dN N K = − +
MC estimators
i i i
H t U
/H
i
expand 0
N
t Z d e
M M Me − = =→
Flux in = Flux out closed oriented loops of integer N-currents
( ) ( ) I M
Z ∗−− = = ψ ψ
(one open loop)
Same algorithm:
I=M
'
( 1) min 1,
= erase
M
M
M
M
( ) 2 2 2
; ; ;
[ ( )]a i a i a i ab a i b i a i a i i
i
ab
t U T
ν∗ +ν ν
1A+ 2A+
3A− 4A−
XY-VBS transition no DCP, always first-order
Neel-VBS transition, unknown !
Lattice path-integrals for bosons and spins are “diagrams” of closed loops!
10 ( , )ij i j i i ij
i j j i iji
< >
β
i j
im ag
in ar
y tim
H Ib bG β
The rest is conventional worm algorithm in continuo us time
(there is no problem to work with arbitrary number of continuous variables as long as an expansion is well defined)
( ) ( )1 2 1 2
; , , ,n nn n
A y d x d x d x D x x x y ξ
ξ ∞
K K
Diagram order
Same-order diagrams
Integration variables
M
Path-integrals in continuous space are consist of closed loops too!
2/ 1
1 1
=β τ +
P
1
2
P
1 2, , ,( , , ... , )i i i i NR r r r = 1,ir 2,ir P
βτ =
2
( ) 2
m < = + −∑ ∑
Feynman (space-time) diagrams for fermions with contact interaction (attractive)
1 1 1 1 2 2 2 2( , ) ( , ) ( , ) ( , )a r a r a r a rτ τ τ τ+ + ↑ ↓ ↓ ↑
U= −
perm(( ) )( ) 1n n n G G GD U drdG τ↑ ↑ ↓ ↓ −= −
rK K 2( ) ( ) ( , )t ( )de 0n n
i jnD x xG drdU ξ
ξ τ↑= − ≥∑ r r r sum over all possible connections2( !)n
Pair correlation function
= ε + −∑ ∑
Domain walls in 2D Ising model are loops!
G space is NOT necessarily physical
Disconnected loop is unphysical!
Quantum spin chains magnetization curves, gaps, spin wave spectra
[ ( ) ]H x jx ix jy iy z jz iz iz ij i
= − + + − ∑ ∑
theory with spectrum
2.48(1)c =
( ) '

S e Z e
0.02486(5) =Spin gap
Z -factor 0.980(5)Z =
e
Spin waves spectrum: One dimensional S=1 Heisenberg chain
Worm algorithm = cnf. space of + updates based on Z G Masha
Ira
Conclusions:
- different representations (path-integrals, Feynam di agrams,SSE)
- non-local solutions of balance Eqn. (clusters, dire cted loops, geom. worm)
Winding numbers
22 2
2 2
Ceperley Pollock ‘86
Homogeneous gauge in x-direction: , ( ) / , ( )iA x xt te A r L A r dx → = = ∫
( )2 /
=
i