Transcript N. Prokofiev "Worm Algorithm for Many

WORM ALGORITHM: LIQUID & SOLID HE-4
Nikolay Prokofiev, Umass, Amherst
Boris Svistunov, Umass, Amherst
Masha
Massimo Boninsegni, UAlberta
Matthias Troyer, ETH
Ira
Lode Pollet, ETH
Anatoly Kuklov, CSI CUNY
NASA
RMBT14, Barcelona July 2007
Why bother with worm algorithm?
Efficiency
New quantities to address physics
PhD while still young
Better accuracy
Large system size
More complex systems
Finite-size scaling
Critical phenomena
Phase diagrams
Grand canonical ensemble N ()
Off-diagonal correlations G(r , )
condensate wave functions (r )
Winding numbers and S 
Examples from: helium liquid & solid
lattice bosons/spins, classical stat. mech.
disordered systems, deconfined criticality,
resonant fermions, polarons …
NP, B. Svistunov, I. Tupitsyn, ‘97
Worm algorithm idea
Feynman
path integrals for
Consider:
H RPMBT 14
pi2

 V (ri  rj )
i 2m
i j
- configuration space = closed loops
P   /
Z   dR ...dR    ( R , Ri ,)
P
i 1
- each cnf. has1 a weight
factor
W
i 1
cnf
P
A W

W
cnf


- quantity of interest
A
cnf
cnf
cnf
cnf
P
ri ,1
ri ,2 
Ri  (ri,1,ri,2 ,..., ri, N )
2
What is the best updating strategy?
1
P
“conventional”
sampling scheme:
local shape change
No sampling of
topological classes
can not
evolve to
(non-ergodic)
Critical slowing down
(large loops are related to
critical modes)
Add/delete small loops
 auto
 N updates

d
L


z

L


dynamical critical exponent
z  2 in many cases
NP, B. Svistunov, I. Tupitsyn, ‘97
Worm algorithm idea
draw and erase:
Masha
Ira
Ira
Masha
+
Masha
Masha
or
keep
drawing
All topologies are sampled (whatever you can draw!)
No critical slowing down in most cases
Disconnected loop is related to the
off-diagonal correlation function and
is not merely an algorithm trick!
GC ensemble N ()
Green function G(r , )
winding numbers S 
condensate wave func.(r ),etc.
Z
G
  (r ', t ')
 (r , t )
(open/close update)
Z
G
Ira
Masha
(insert/remove update)
G
Ira
Ira
Masha
(advance/recede update)
G
Masha
Ira
(swap update)
Ira
Path integrals + Feynman diagrams for V (r )  0
e
V ( rij )
 1  (e
V ( rij )
 1)  1  pij
Masha
statistical interpretation
ignore
Ira
V (rij )
Account for
: stat. weight 1
V (rij ) : stat. weight p
pij
i
j
10 times faster than conventional scheme, scalable (size independent) updates
with exact account of interactions between all particles (no truncation radius)
Grand-canonical calculations: n( , T ) , compressibility N 2  TV  ,
phase separation, disordered/inhomogeneous systems, etc.
Matsubara Green function:
G (r , r ',   ')  T  † (r , ) (r ', ')
Probability density of Ira-Masha
distance in space time
lim G ( p, )  Z p e E ( p )
G(r ,  0)  n(r )
Energy gaps/spectrum,
quasi-particle Z-factors
One-body density matrix,
Cond. density n0  n(r  )
 
Winding numbers:
ns 
m
W2
d 2
dTL
Winding number exchange cycles
lim G ( r , r ',  / 2)   ( r ) ( r ')
|r  r '|
 
particle “wave funct.” at
superfluid density
maps of local superfluid response
At the same CPU price as energy in conventional schemes!

2D He-4
superfluid density &
critical temperature
(n  0.0432 A2 )
TC  0.65(1)
Critical temp.
“Vortex diameter” d  9 A
Ceperley, Pollock ‘89
TC  0.72(2), d  3.5 A
3D He-4 at P=0
superfluid density &
critical temperature
64
experiment
2048
Pollock, Runge ‘92
?
TCAziz  2.193vs TCexp  2.177
3D He-4 at P=0
Density matrix &
condensate fraction
N=64
N=64
(Bogoliubov)
N=2048
n(r )  n0 emT / 4 rns
n0  0.024
N=2048
3D He-4 liquid near
the freezing point,
T=0.25 K, N=800
Calculated from 
Weakly interacting Bose gas,
pair product approximation;
TC (V )
Ceperley, Laloe ‘97
3
3
( na  5 10 example)
Nho, Landau ‘04
TC / T0  1.057(2)?
T / T0
TC / T0  1.078(1)?
Worm algorithm: Pilati, Giorgini, NP
underestimated
wrong
oferror
slices
20number
discrepancy
! bars
+ too small
(5 vs system
15)
size
100,000
Solid (hcp) He-4
Density matrix
T  0.2K ,N  800
Exponential decay
Insulator
near melting
o 3
n  0.0292 A 
o 3
n  0.0359 A
Solid (hcp) He-4
Green function
Exponential decay
Insulator
T  0.25K ,N  800
E

i,v
is
not
required!
EN 

E
G( p,|  |
)

Z

e
1
N
Energy subtraction
melting density
in the solid phase
Large vacancy / interstitial gaps at all P
Supersolid He-4
“… ice cream”
“… transparent honey”, …
A network of SF grain boundaries,
dislocations, and ridges
with superglass/superfluid pockets (if any).
GB
SF/SG
Disl
All “ice cream ingredients”
are confirmed to have
superfluid properties
Dislocations network (Shevchenko state) at
Ridge
He-3
He-3
T  T  TC
1
Frozen vortex tangle; relaxation time  ~ T T / T 
a
  T
l
where TC ~ T 
8 K 1
vs exp. timescale
Supersolid phase of He-4
Is due to extended defects:
metastable liquid
grain boundaries
screw dislocation, etc.
Pinned atoms
T  (0.25 K ,n  0.0287
N  384 1536
“physical” particles
screw dislocation axis
Supersolid phase of He-4
Is due to extended defects:
metastable liquid
grain boundaries
screw dislocation, etc.
Screw dislocation has a superfluid core:
nS  1 A1,gLutt.Liq.  5
T  (0.25 K ,n  0.0287
N  384 1536
Side (x-axis) view
Top (z-axis) view
T (nliquid  nsolid )  1.5(1)K
Maps of exchange cycles with non-zero winding number
superfluid grain boundaries
anisotropic stress
T (@ solid densities)
domain walls
+ superfluid glass phase (metastable)
Lattice path-integrals for bosons/spins (continuous time)
H  H0  H1  Uij ni n j   i ni   t (ni , n j )bj bi
ij
ij 
i
GIM =Tr T bM† ( M ) bI ( I )e- H


imaginary time
imaginary time
Z= Tr e-  H
Masha
Ira
0
0
lattice site
lattice site
I
I
M
M
I
I
At T ~ t one can simulate cold atom experimental
6
system “as is” for as many as N  10 atoms!
Classical models: Ising, XY, 
4
Ising model (WA is the best possible algorithm)

H
 K  i  j (  1)
T
ij 
Z   e H / T
GIM   e H / T I M
{i }
{i }
closed loops
Ira
Masha
I=M
I
M
M
M
M
Complete algorithm:
- If
I  M , select a new site for I  M at random
- otherwise, propose to move M in randomly selected direction
R
min(1, tanh( J / T ))for nbond  0nbond  1
min(1, tanh 1 ( J / T ))for nbond  1nbond  0
Easier to implement then single-flip!
Conclusions
extended configuration space Z+G
Worm Algorithm =
all updated are local & through end points exclusively
At no extra cost you get
A method of choice for
no critical slowing down
Grand Canonical ensemble
off-diagonal correlators
superfluid density
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
Superfluid grain boundaries in He-4
GB (periodic BC)
GB
atoms each
Maps of exchange-cycles
with non-zero winding numbers
nS 
two cuboids
N  12 12  7
mT W 2
dL
Lz
Ly
3a
XY-view
Lx
XZ-view
Lx
Superfluid grain boundaries in He-4
12
12
7
7
ODLRO’
(TC )GB
max  1.5K
TKT  0.6K
Continuation of the
-line to solid densities