worm algorithm: liquid & solid he-4 nikolay prokofiev, umass, amherst nasa rmbt14, barcelona...
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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
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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
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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?
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“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
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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
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( , )r t
( ', ')r t
ZG
(open/close update)
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Ira
Masha
(insert/remove update)
ZG
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Ira
Masha
(advance/recede update)
G
Ira
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Ira
(swap update)
G
Ira
Masha
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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
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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!
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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
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3D He-4 at P=0superfluid density &critical temperature
64
2048
experiment
exp2.193 2.177AzizC CT T vs
Pollock, Runge ‘92
?
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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)
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3D He-4 liquid near the freezing point,T=0.25 K, N=800
Calculated from
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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
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Solid (hcp) He-4Density matrix
0.2 , 800T K N
3o
0.0292An
3o
0.0359 An
near melting
InsulatorExponential decay
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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
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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
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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
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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
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+ superfluid glass phase (metastable)
anisotropic stress
(@ solid densities)T
domain walls
superfluid grain boundaries
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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
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M
II
II
M
At one can simulate cold atom experimental system “as is” for as many as atoms!
~T t610N
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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
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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!
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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
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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
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1212 7
7
0.6KTT K
ODLRO’
Superfluid grain boundaries in He-4
max( ) 1.5GBCT K
Continuation of the -line to solid densities