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