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

Spin waves spectrum: One dimensional S=1 Heisenberg chain

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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