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)
updates z d
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
(positive definite, please)
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