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WORM ALGORITHM: LIQUID & SOLID HE-4. Nikolay Prokofiev, Umass, Amherst. Boris Svistunov, Umass, Amherst Massimo Boninsegni, UAlberta Matthias Troyer, ETH Lode Pollet, ETH Anatoly Kuklov, CSI CUNY. Masha. Ira. NASA. RMBT14, Barcelona July 2007. Why bother with worm algorithm?.

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slide1

WORM ALGORITHM: LIQUID & SOLID HE-4

Nikolay Prokofiev, Umass, Amherst

Boris Svistunov, Umass, Amherst

Massimo Boninsegni, UAlberta

Matthias Troyer, ETH

Lode Pollet, ETH

Anatoly Kuklov, CSI CUNY

Masha

Ira

NASA

RMBT14, Barcelona July 2007

slide2

Why bother with worm algorithm?

Efficiency

New quantities to address physics

PhD while still young

PhD while still young

Better accuracy

Large system size

More complex systems

Finite-size scaling

Critical phenomena

Phase diagrams

Grand canonical ensemble

Off-diagonal correlations

condensate wave functions

Winding numbers and

Examples from: helium liquid & solid

lattice bosons/spins, classical stat. mech.

disordered systems, deconfined criticality,

resonant fermions, polarons …

slide3

Worm algorithm idea

NP, B. Svistunov, I. Tupitsyn, ‘97

Feynman path integrals for

Consider:

- configuration space = closed loops

- each cnf. has a weight factor

P

- quantity of interest

2

What is the best updating strategy?

1

P

slide4

“conventional”

sampling scheme:

localshape change

Add/delete small loops

No sampling of

topological classes

(non-ergodic)

can not

evolve to

dynamical critical exponent

in many cases

Critical slowing down

(large loops are related to

critical modes)

slide5

Worm algorithm idea

NP, B. Svistunov, I. Tupitsyn, ‘97

draw and erase:

Masha

Ira

Ira

+

Masha

Masha

Masha

keep

drawing

or

All topologies are sampled (whatever you can draw!)

Disconnected loop is related to the

off-diagonal correlation function and

is not merely an algorithm trick!

No critical slowing down in most cases

GC ensemble

Green function

winding numbers

condensate wave func. ,etc.

slide10

Path integrals + Feynman diagrams for

statistical interpretation

ignore : stat.weight 1

Account for : stat. weight p

10 times faster than conventional scheme, scalable (size independent) updates

with exact account of interactions between all particles (no truncation radius)

slide11

Grand-canonical calculations: , compressibility ,

phase separation, disordered/inhomogeneous systems, etc.

Matsubara Green function:

Probability density ofIra-Masha

distance in space time

particle “wave funct.” at

Energy gaps/spectrum,

quasi-particle Z-factors

One-body density matrix,

Cond. density

Winding numbers: superfluid density

Winding number exchange cycles maps of local superfluid response

At the same CPU price as energy in conventional schemes!

slide12

2D He-4

superfluid density &

critical temperature

Ceperley, Pollock ‘89

Critical temp.

“Vortex diameter”

slide13

3D He-4 at P=0

superfluid density &

critical temperature

64

experiment

2048

Pollock, Runge ‘92

?

slide14

3D He-4 at P=0

Density matrix &

condensate fraction

N=64

(Bogoliubov)

N=64

N=2048

N=2048

slide15

3D He-4 liquid near

the freezing point,

T=0.25 K, N=800

Calculated from

slide16

Weakly interacting Bose gas,

pair product approximation;

( example)

Ceperley, Laloe ‘97

Nho, Landau ‘04

Worm algorithm: Pilati, Giorgini, NP

wrong number of slices

(5 vs 15)

underestimated error bars

+ too small system size

discrepancy !

100,000

slide17

Solid (hcp) He-4

Density matrix

Exponential decay

Insulator

near melting

slide18

Solid (hcp) He-4

Green function

Exponential decay

Insulator

Energy subtraction is not required!

melting density

in the solid phase

Large vacancy / interstitial gaps at all P

slide19

Supersolid He-4

“… ice cream”

“… transparent honey”, …

A network of SFgrain boundaries,

dislocations, and ridges

with superglass/superfluid pockets (if any).

GB

SF/SG

Ridge

He-3

Disl

He-3

All “ice cream ingredients”

are confirmed to have

superfluid properties

Dislocations network (Shevchenko state) at where

Frozen vortex tangle; relaxation time vs exp. timescale

slide20

Supersolid phase of He-4

Is due to extended defects:

metastable liquid

grain boundaries

screw dislocation, etc.

Pinned atoms

“physical” particles

screw dislocation axis

slide21

Supersolid phase of He-4

Is due to extended defects:

metastable liquid

grain boundaries

screw dislocation, etc.

Screw dislocation has a superfluid core:

Top (z-axis) view

Side (x-axis) view

Maps of exchange cycles with non-zero winding number

slide22

superfluid grain boundaries

anisotropic stress

domain walls

+ superfluid glass phase (metastable)

slide23

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

imaginary time

imaginary time

lattice site

lattice site

slide24

I

I

M

M

I

I

At one can simulate cold atom experimental

system “as is” for as many as atoms!

slide25

Classical models: Ising, XY,

Ising model (WA is the best possible algorithm)

closed loops

Ira

Masha

slide26

I=M

I

M

M

M

M

-If , select a new site for at random

Complete algorithm:

- otherwise, propose to move in randomly selected direction

Easier to implement then single-flip!

slide27

Conclusions

extended configuration space Z+G

Worm Algorithm =

all updated are local & through end points exclusively

no critical slowing down

Grand Canonical ensemble

off-diagonal correlators

superfluid density

At no extra cost you get

Continuous space path integrals

Lattice 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

slide28

Superfluid grain boundaries in He-4

GB (periodic BC)

two cuboids

GB

atoms each

Maps ofexchange-cycles

with non-zero winding numbers

3a

XY-view

XZ-view

slide29

Superfluid grain boundaries in He-4

ODLRO’

Continuation of the

-line to solid densities