WORM ALGORITHM: LIQUID & SOLID HE-4

Nikolay Prokofiev, Umass, Amherst

NASA

RMBT14, Barcelona July 2007

Boris Svistunov, Umass, Amherst

Massimo Boninsegni, UAlberta

Matthias Troyer, ETH

Lode Pollet, ETH

Anatoly Kuklov, CSI CUNY

Masha

Ira

Why bother with worm algorithm?

PhD while still young

( , )G r

New quantities to address physics

Grand canonical ensembleOff-diagonal correlationscondensate wave functionsWinding numbers and

Examples from: helium liquid & solid lattice bosons/spins, classical stat. mech. disordered systems, deconfined criticality, resonant fermions, polarons …

S ( )r

( )N

Efficiency

PhD while still youngBetter accuracyLarge system sizeMore complex systemsFinite-size scalingCritical phenomena Phase diagrams

Worm algorithm idea

Consider:

- configuration space = closed loops

- each cnf. has a weight factor Wcnf

- quantity of interest

A WA

W

cnf cnfcnf

cnfcnf

NP, B. Svistunov, I. Tupitsyn, ‘97

P

1

2

P

1 2, , ,( , , ... , )i i i i NR r r r 1,ir 2,ir P

Feynman path integrals for 1

2

4 ( )2

ii j

iRPM

iB

jT

pH V r r

m

/

1 11

... ( , , )P

P i ii

Z dR dR R R

What is the best updating strategy?

“conventional” sampling scheme:

local shape change Add/delete small loops

can not evolve to

No sampling of topological classes(non-ergodic)

Critical slowing down(large loops are related tocritical modes)

zauto d

NL

L

updates dynamical critical exponent in many cases2z

Worm algorithm idea

draw and erase:

Masha

Ira

or

Masha

Ira+

keepdrawing

Masha

Masha

All topologies are sampled (whatever you can draw!)

No critical slowing down in most cases Disconnected loop is related to theoff-diagonal correlation function and is not merely an algorithm trick!

NP, B. Svistunov, I. Tupitsyn, ‘97

( , )G r GC ensembleGreen functionwinding numberscondensate wave func. ,etc.

S ( )r

( )N

( , )r t

( ', ')r t

ZG

(open/close update)

Ira

Masha

(insert/remove update)

ZG

Ira

Masha

(advance/recede update)

G

Ira

Ira

(swap update)

G

Ira

Masha

Ira

Masha

Path integrals + Feynman diagrams for ( ) 0V r

( ) ( )1 ( 1) 1ij ijV r V r

ije e p

ignore : stat. weight 1

Account for : stat. weight p

( )ijV r

statistical interpretation

( )ijV r

10 times faster than conventional scheme, scalable (size independent) updates with exact account of interactions between all particles (no truncation radius)

i j

ijp

Grand-canonical calculations: , compressibility , phase separation, disordered/inhomogeneous systems, etc.

( , )n T 2N TV

Matsubara Green function:†( , ', ') T ( , ) ( ', ' )G r r r r

Probability density of Ira-Masha distance in space time

( )lim ( , ) E ppG p Z e

Energy gaps/spectrum,quasi-particle Z-factors

( , 0) ( )G r n r

One-body density matrix,Cond. density

| ' |lim ( , ', / 2) ( ) ( ')

r rG r r r r

particle “wave funct.” at

Winding numbers: superfluid density2

2s d

mn W

dTL

0 ( )n n r

Winding number exchange cycles maps of local superfluid response

At the same CPU price as energy in conventional schemes!

Ceperley, Pollock ‘89

“Vortex diameter” 9d A

2D He-4 superfluid density &critical temperature

2( 0.0432 )n A 0.72(2), 3.5CT d A

Critical temp. 0.65(1)CT

3D He-4 at P=0superfluid density &critical temperature

64

2048

experiment

exp2.193 2.177AzizC CT T vs

Pollock, Runge ‘92

?

N=64N=64

N=2048

N=2048

0 0.024n

3D He-4 at P=0Density matrix &condensate fraction

/ 40( ) smT rnn r n e

(Bogoliubov)

3D He-4 liquid near the freezing point,T=0.25 K, N=800

Calculated from

Weakly interacting Bose gas, pair product approximation; ( example)( )CT V

0/T T

3 35 10n a

0/ 1.057(2) ?CT T

Ceperley, Laloe ‘97

0/ 1.078(1) ?CT T

Nho, Landau ‘04

20 discrepancy !wrong number of slices (5 vs 15)

underestimated error bars+ too small system size

Worm algorithm: Pilati, Giorgini, NP

100,000

Solid (hcp) He-4Density matrix

0.2 , 800T K N

3o

0.0292An

3o

0.0359 An

near melting

InsulatorExponential decay

Solid (hcp) He-4Green function 0.25 , 800T K N

melting density

( ,| | ) EG p Z e i, v

Large vacancy / interstitial gaps at all P

InsulatorExponential decay

in the solid phase

Energy subtraction is not required!1N NE E

Supersolid He-4 “… ice cream” “… transparent honey”, …

GB

Ridge He-3SF/SG

A network of SF grain boundaries, dislocations, and ridges

with superglass/superfluid pockets (if any).

Dislocations network (Shevchenko state) at where ~C

aT T T

l

All “ice cream ingredients”are confirmed to have superfluid properties

Disl

He-3

Frozen vortex tangle; relaxation time vs exp. timescale

CT T T

8 11 ~ /K

T T T

Supersolid phase of He-4 Is due to extended defects:metastable liquidgrain boundariesscrew dislocation, etc.

(0.25 , 0.0287

384 1536

T K n

N

Pinned atoms

“physical” particles

screw dislocation axis

Supersolid phase of He-4 Is due to extended defects:metastable liquidgrain boundariesscrew dislocation, etc.

(0.25 , 0.0287

384 1536

T K n

N

( ) 1.5(1)liquid solidT n n K

Screw dislocation has a superfluid core:1

. .1 , 5S Lutt Liqn A g

Maps of exchange cycles with non-zero winding number

Top (z-axis) view

Sid

e (

x-a

xis

) v

iew

+ superfluid glass phase (metastable)

anisotropic stress

(@ solid densities)T

domain walls

superfluid grain boundaries

Lattice path-integrals for bosons/spins (continuous time)

10 ( , )ij i j i iij

i j j iiji

H t n n b bH H U n n n

imag

inar

y ti

me

lattice site

-Z= Tr e H

0

† -M= Tr T ( ) ( ) eI M IM

HIb bG

imag

inar

y ti

me

lattice site

0

Ira

Masha

M

II

II

M

At one can simulate cold atom experimental system “as is” for as many as atoms!

~T t610N

Classical models: Ising, XY,

( 1)i jij

HK

T

4

/

{ }i

H TI MM IG e

/

{ }i

H TZ e

closed loops

Ising model (WA is the best possible algorithm)

Ira

Masha

I=M

M

I

M

M

M

Complete algorithm:- If , select a new site for at random

- otherwise, propose to move in randomly selected direction

I M

M

I M

R 1

min(1, tanh( / )) 0 1

min(1, tanh ( / )) 1 0

bond bond

bond bond

J T n n

J T n n

for

for

Easier to implement then single-flip!

Conclusions

no critical slowing downGrand Canonical ensembleoff-diagonal correlatorssuperfluid density

Worm Algorithm = extended configuration space Z+G

all updated are local & through end points exclusively

At no extra cost you get

Continuous space path integralsLattice systems of bosons/spins Classical stat. mech. (the best method for the Ising model !)Diagrammatic MC (cnfig. space of Feynman diagrams) Disordered systems

A method of choice for

GB

GB (periodic BC)

xL

yL

xL

zL

3a

XY-view

2

S

mT Wn

dL

XZ-view

Superfluid grain boundaries in He-4

12 12 7N

Maps of exchange-cycleswith non-zero winding numbers

two cuboids

atoms each

1212 7

7

0.6KTT K

ODLRO’

Superfluid grain boundaries in He-4

max( ) 1.5GBCT K

Continuation of the -line to solid densities