WORM ALGORITHM APPLICATIONS
Nikolay Prokofiev, Umass, Amherst
Boris Svistunov, Umass, Amherst
Igor Tupitsyn, PITP
Vladimir Kashurnikov, MEPI, Moscow
Massimo Boninsegni, UAlberta, Edmonton
Many thanks to collaborators
NASA
Les Houches, June 2006
Matthias Troyer, ETH
Lode Pollet, ETH
Anatoly Kuklov, CSI, CUNY
Worm Algorithm
No critical slowing down(efficiency)
Better accuracyLarge system sizeFinite-size scalingCritical phenomenaPhase diagrams
Reliably!
( , )G r
New quantities, more theoretical tools to address physics
Grand canonical ensembleOff-diagonal correlations“Single-particle” and/or condensate wave functionsWinding numbers and
Examples from: superfluid-insulator transition, spin chains, helium solid & glass, deconfined criticality, holes in the t-J model, resonant fermions, …
S ( )r
( )N
20( )j i i i i
ij ij i
H t b b U n n
1, ( , )in
Superfluid-insulator transitionin disordered bosonic system
For any finite the sequence is always
SF - Bose glass - Mott insulator
Fisher, Fisher,Weichman,Grinstein ‘89
Not found in helium films
“Disproved” in numerical simulations (many, 1D and 2D)
New theories to support direct SF-MI transition have emerged
0
0S
0
0, (gapless)S
0
0, (gap)S
?
/ 0.2U - The data look as a perfect direct SF-MI transition ( )
/ 0.4U - Up to the data look as a direct SF-MI transition, but …
2 2160 L 2 240 L
/t U
10
160
/t U
10
160
40
For small the Bose glass state is dominated by rare (exponentially) statisticalfluctuations resulting in hole-rich and particle-rich regions
/U
( / )ct U
“Wave function” of the added particle
160L
Complete phase diagram
/t U
Gap in theIdeal system
/U
It is a theorem that for the compressibility is finite
GAPE
Quantum spin chainsmagnetization curves, gaps, spin wave spectra
[ ( ) ]H x jx ix jy iy z jz iz izij i
J S S S S J S S H S
S=1/2 Heisenberg chain
Bethe ansatz
MC data
Line is for the effective fermion
theory with spectrum
2 2( 2 / )p n L cp
0.4105(1)
2.48(1)c
Lou, Qin, Ng, Su, Affleck ‘99deviations are due to magnon-magnon interactions
'
†
2( )†
1''
= T (x, ) (0)( , )
G
ipxI
E Ep JG
G e dx S S
S e Z e
p
One dimensional S=1 chain with / 0.43z xJ J
0.02486(5) Spin gap
Z -factor 0.980(5)Z
Energy gaps:
( )
( , 1)1
e eG p Z
e
Kosterlitz-Thouless scaling:
( )=A exp
cz z
B
J J
Is (red curve)
an exact answer ?
( ) 0czJ
Superfluid (XY) – insulator transition in the one dimensional S=1 Heisenberg chain
Density matrix close to
First principles simulations of helium: 2
( )2
H ii j
i i j
pU r r N
m
CT ( ) (0) ( ) /n r r n
64
2048 0( ) exp4 S
mTn r n
r
Superfluid hydrodynamics(Bogoliubov)
Finte-size scaling
64
2048
SL
m
Better then 1% agreement at all Tafter finite-size scaling
calculated
experiment
2.193CT 2.177CT
Exponential decay of the single-particle density matrix
0.2 , 800T K N
3o
0.0292An
3o
0.0359An
Insulating hcp crystals of He-4
near melting
T ( , ) ( ,0)( 0, ) kG k k
Activation energies for vacancies and interstitials:
, of course
Melting density, N=800, T=0.2 KE(N+1)-E(N) can not be done with this accuracy
Large activation energies at all Pressures (thermodynamic limit)
In fact, the vacancy gas, even if introduced “by hand”, is absolutely unstable and phase Separates (grand canonical simulations with ) V
Superglass state of He-4
Single-particle density matrix density-density correlator
30.0359 , 100 0.2 , 800o
n A T K K N
0.07(2)S ODLRO,
Monte Carlo temperature quench from normal liquid
Condensate wave function maps reveal broken translation symmetry
10 slices across the z-axis
4 10 1 3
4 410 10 10 srelax Dt JA rough estimate of metastability:
Superglass state of He-4
0 ( )r density of points
Condensate maps
simulation box
the
4 sl
ices
across x-axis
across y-axis
across z-axis
Each of the 8 cubes is a randomly oriented crystallite (24 interfaces)
Superfluid ridges and interfaces in He-4
Worm Algorithm
No critical slowing down(efficiency)
Better accuracyLarge system sizeFinite-size scalingCritical phenomenaPhase diagrams
Reliably!
( , )G r
New quantities, more theoretical tools to address physics
Grand canonical ensembleOff-diagonal correlations“Single-particle” and/or condensate wave functionsWinding numbers and
classical stat. mech. models [Ising, lattice field theories, polymers],quantum lattice spin and particle systems,continuous space quantum particle systems(high-T series, Feynman diagrams in either momentumor real space, path-integrals, whatever loop-like …)
S ( )r
( )N
- Extended configuration space for WZ Z G
( , ) ( , )S S local moves of source/drain or etc. operators
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