Ground state properties of a homogeneous 2D system of Bosons with dipolar interactions.

1
Ground state properties of a homogeneous 2D
system of Bosons with
dipolar interactions.
Departament de Fìsica i Enginyeria
Nuclear Universitat Politècnica de Catalunya,
Barcelona, Spain
G. E. Astrakharchik J. Boronat J. Casulleras I.
L. Kurbakov Yu. E. Lozovik 16-20 July,
2007 RPMBT-14
2
MODEL HAMILTONIAN
We consider a (quasi-) two dimensional
homogeneous Bose system with a dipole-dipole
interaction Vint (z) Cdd / r3. Such a system
is described by the following Hamiltonianwhere
m is mass and N is number of dipoles. An
important parameter in a homogenous system is a
dimensionless density n r02 , with r0mCdd /4p?2.
By expressing distances in units of r0 and energy
in units of ?2 /m r02 Hamiltonian takes a
simple form In the dilute regime we will use
gas parameter na2 for comparison to perturbative
results of a weakly interacting Bose gas. Here a
denotes the swave scattering length
3
QUANTUM MONTE CARLO
1) The Variational Monte Carlo (VMC) method
makes it possible to calculate multidimensional
averages of physical quantities over the N-body
trial wave function ?T The VMC calculation
gives an upper bound for the ground-state energy.
2) The Diffusion Monte Carlo (DMC) method
solves the Schrödinger equation in the imaginary
time at T0. It permits to find the ground state
energy E of a bosonic system exactly (in
statistical sense). It allows to find nS/n and
local quantities (e.g. g2(z), Sk, etc.) in a
pure (non-depending on the choice of trial
w.f.) way. An extrapolation procedure can be used
for predictions of non-local quantities (e.g.
g1(z), nk, etc.)
4
TRIAL WAVEFUNCTION
We construct the trial wave function in the
following form where ri, i1,,N are particle
coordinates and rllatt , l1,,M are coordinates
of triangular lattice sites. The two-body term is
chosen as
i.e. 1)
short distances solution of 2-body scattering
problem at zero energy. Thus, it describes
effects of pair-collisions relevant for short
distances. 2) large distances hydrodynamic
solution (sound) symmetrized, so that it has
zero derivative at L/2. Solutions (1) and (2) are
matched in a continuous way fixing constants
C1, C2, C3, f2(L/2)0. Localization width a is
optimized variationally. Trial w.f. is
symmetric with respect to interchange of any two
particles, while it allows to describe particle
localization close to lattice sites.
5
TRIAL WAVEFUNCTION
In the regime of high density we find two minima
in the variational energy as a function of the
localization width a.

E / N
7000
nd
2
minimum

SOLID
1) a0 translational invariance is preserved.
Density profile is flat. This minimum corresponds
to a gas/liquid state.
6900
0
200
400
600
800
a
2) a gt0 translational invariance is broken.
Density profile has crystal symmetry. This
minimum corresponds to a solid state.
6
RESULTS GROUND STATE ENERGY
Ground state energy per particle in units of (?2
/Mr02)/(nr02)3/2 as a function of parameter nr02
symbols DMC results, lines best fit. Inset
energy per particle in units of ?2 /Mr02.
7
PHASE TRANSITION
At small density the liquid phase is
energetically favorable and solid phase is
metastable. At larger densities the system
crystallizes and triangular lattice is formed.
We fit our data points with dependence E/N
a1(n r02) 3/2a2(n r02) 5/4 a3(n r02) ½
a14.536(8), a24.38(4), a31.2(3) - liquid
a14.43(1), a24. 80(3), a32.5(2) - solid
Classical crystal limit is recovered at large
density Etriang/N 4.446 (n r02)
3/2.Transition point is estimated as ncr02
290?30. PIMC estimation 1 ncr02
320?140GFMC estimation 2 ncr02 230?20
nr02 ?256
1 H.P. Büchler, E.Demler, M.Lukin, A.Micheli,
N.Prokof'ev, G.Pupillo, P.Zoller,
Phys.Rev.Lett. 98, 060404 (2007) 2 C. Mora, O.
Parcollet, X. Waintal, cond-mat/0703620
nr02 ? 358
8
PAIR DISTRIBUTION LIQUID PHASE
Pair distribution function
9
MEAN-FIELD THEORY
In the dilute regime the equation of state is
expected to be universal. It depends only on
the density n and the scattering length as.2D
Gross-Pitaevskii equation has coupling constant
(see Ref. 1)leading to the following
ground-state energy (same as in Ref. 2
)Condensate fraction 21 E. Lieb, R.
Seiringer, J. Yngvason Commun. Math. Phys. 224,
17 (2001)2 M. Schick, Phys. Rev. A 3, 1067
(1971)
10
ENERGY FAILURE OF MF-GPE
Behavior of energy in the dilute regime is
universal and is the same for hard-disks of
diameter as 1. At the same time this universal
behavior is not completely reproduced for
densities nasgt10-250 (1 error).1 S. Pilati,
J. Boronat, J. Casulleras, S. Giorgini, PRA 71,
023605 (2005).
11
ONE-BODY DENSITY MATRIX GAS
One-body density matrix
for different densities. Finite asymptotic value
is a manifestation of Off-Diagonal Long-Range
Order
12
CONDENSATE FRACTION
Condensate fraction n0/n as a function of gas
parameter na2.
13
SUPERSOLID?
There are several ways to define a
supersolid 1) Spatial order of a solid
finite superfluid density 2) Spatial order of a
solid (diagonal order in OBDM) off-diagonal
long-range order (finite long-range asymptotic
of one-body density matrix)
Literature overview1 H.P.
Büchler, E.Demler, M.Lukin, A.Micheli,
N.Prokof'ev, G.Pupillo, P.Zoller, Phys.Rev.Lett.
98, 060404 (2007)- Low-temperature simulation
(PIMC) shows that gas phase is completely
superfluid, no superfluid fraction is found in
crystal phase. Presence of (a possible)
supersolid can be masked by much smaller critical
temperature in a crystal. 2 C. Mora, O.
Parcollet, X. Waintal, cond-mat/0703620
Zero-temperature method is used with a
symmetrized w.f. No conclusions are drawn for
presence/absence of a supersolid due to an
unsufficient overlap of trial w.f. with the
actual ground state.
14
WINDING NUMBER
Diffusion coefficient of center of masses D as a
function of imaginary time t in a crystal at
critical density na2290 for different system
size.
15
SUPERFLUID FRACTIONADDING VACANCIES
Superfluid fraction nS/n as a function of
concentration of vacancies for N30 particles in
crystal phase.
16
ONE-BODY DENSITY MATRIX SOLID
One-body density matrix in a crystal at critical
density na2290 for different system sizes,
symmetrized and non-symmetrized w.f.
17
CONDENSATE AND SUPERFLUID FRACTIONS
Superfluid fraction (blue) and condensate
fraction (red) as a function of vacancy
concentration.
18
CONCLUSIONS
Diffusion Monte Carlo method was used to study
the properties of a dipolar two-dimensional Bose
system at T0.-) the ground state energy, pair
distribution function, one-bode density matrix
are calculated in a wide range of densities -)
fit to the energy (10-100ltnr02 lt1024.) can be
used for LDA-) gas-solid quantum phase
transition is found at density nr02 290(30).
-) limitations (failure) of mean-field
description are discussed in universal
low-density regime-) existence of the
off-diagonal long-range order was shown in
one-body density matrix and the condensate
fraction was found in - gas
phase - finite-size
crystal close to phase transition-) finite
superfluid fraction is found in crystal phase -)
we observe supersolid behavior in a finite-size
crystal, signal is increased in presence of
vacancies.
1 G.E.A., J. Boronat, I.L. Kurbakov, Yu.E.
Lozovik, Phys. Rev. Lett. 98, 060405 (2007)2
G.E.A., J. Boronat, J. Casulleras, I.L. Kurbakov,
Yu.E. Lozovik , Phys. Rev. A 75, 063630
(2007) 3 Yu. E. Lozovik, I. L. Kurbakov, G.
E.A., J. Boronat, M. Willander, in print.
19
THANK YOU FOR YOUR ATTENTION!
20
INTRODUCTION
Why low-dimensional systems are interesting?-
Role of correlations and quantum fluctuations is
increased superfluid-normal phase transition
occurs at a finite-temperature and follows
the peculiar scenario of Berezinskii-Kosterlitz-Th
ouless Bose-Einstein condensation is absent
in 2D homogeneous systems at finite
temperatures Two dimensional crystals are
possible candidates for a supersolid Why
dipolar systems are interesting ?- Long-range
dipolar forces compete with short-range s-wave
scattering and extend to larger distances-
relative ease of tuning the effective strength of
interactions, which makes the system highly
controllable. - dipole particles are also
considered to be a promising candidate for the
implementation of quantum-computing schemes
21
ATOMS
Cold bosonic atoms with a large dipole
moment and confined in a very tight pancake trap.
If the energy of atoms is not enough to excite
levels of the tight confinement, the system is
dynamically two-dimensional. a) If permanent
magnetic dipoles are aligned by a magnetic field,
the coupling constant is Cdd M2, where M is the
magnetic dipole moment. b) If the dipoles are
induced by an electric field E then the
interaction constant is Cdd E2a2, where a is a
static polarizability. c) Polar molecules a
static electric field E coupling of lowest
rotor states by microwave field gives possibility
of shaping the potential refer to H.P.Büchler et
al.Phys. Rev. Lett. 98 (2007)
22
EXCITONS
The phenomenon of the Bose condensation can be
observed in a system of composite bosons, formed
by two fermions. Bound electron-hole pairs
(excitons) in semiconductors at low temperatures
(T 1 ?) may form a sort of quantum liquid
degenerated bose gas and might experience Bose
condensation. Spatial separation between electron
and hole increases exciton lifetime. If the
size of an exciton pair is much smaller than the
distance between exciton such a pair acts as a
dipole. In this case Cdd e2 D2 / e, where e
is the charge of an electron, e is the
dielectric constant of the semiconductor, and D
is the separation between electron and hole.
23
2-BODY SCATTERING PROBLEM
The dipole-dipole interaction potential
Vint(r)1/r3 is slowly decaying, but still it is
not a long-range in 2D. I.e. it decays faster
than 1/r2 and the interaction potential is
integrable at large distances Two-body
scattering problem can be solved analytically for
scattering at zero energy where K0(r) is
modified Bessel function of the second kind,
?0.577 Long-range behavior defines the
scattering length a The scattering length on a
2D dipole-dipole potential is
24
OPTIMIZATION OF PARAMETERS
Variational energy has two minima for densities
nr02 ?8 1) a 0 no localization, i.e. liquid
2) a gt 0 localized system, i.e. crystal
25
FINITE SIZE DEPENDENCE
Energy (solid phase) for nr02 256 as a function
1/N. Symbols DMCtail, solid line best fit,
dashed line extrapolation to thermodynamic
limit.
26
RESULTS GROUND STATE ENERGY
Ground state energy per particle in units of ?2
/Mr02 as a function of nr02 symbols DMC
results, solid lines best fit, dashed
classical crystal.
27
LINDEMANN RATIO
The Lindemann ratio gives a quantitative
description to particle diffusion from lattice
sites and is defined as where aL is the lattice
length. We estimate the thermodynamic Lindemann
ratio at the transition density to be equal to ?
0.230(6). Comparison to other two-dimensional
systems ? 0.279 hard-disks, L. Xing, Phys.
Rev. B 42, 8426 (1990), ? 0.235(15) 2D Yukawa
bosons, W. R. Magro and D. M. Ceperley, Phys.
Rev. B 48, 411 (1993) ? 0.24(1) 2D Coulomb
bosons, W. R. Magro and D. M. Ceperley, Phys.
Rev. Lett. 73, 826 (1994) In three-dimensional
system value of ? at transition is generally
larger, for example ? 0.28 for 3D Yukawa
potential, D. Ceperley, G. V. Chester, and M.
H. Kalos, Phys. Rev. B 17, 1070 (1978)
28
CORRELATION FUNCTIONS
The pair distribution function gives the
possibility to find a particle at a distance r
from another particle The static structure
factor is related to the pair distribution
function
29
PAIR DISTRIBUTION FUNCTION LIQUID PHASE
Pair distribution functions at densities (liquid
phase)
30
PAIR DISTRIBUTION FUNCTIONSOLID PHASE
Pair distribution functions at densities nr02
384, 512, 768 (solid phase)
31
STATIC STRUCTURE FACTOR
The static structure factor is a continuous
function in the liquid phase. In the solid phase
a d-peak appears at a momentum, corresponding to
the inverse lattice spacing.
32
STATIC STRUCTURE FACTOR
Static structure factor (symbols). Behavior
linear, small k (dashed lines)
, c is speed of sound, weakly-interacting regime
(solid lines) obtained from Bogoliubov excitation
spectrum
33
STATIC STRUCTURE FACTOR
The static structure factor is a continuous
function in the liquid phase. In the solid phase
a d-peak appears at a momentum, corresponding to
the inverse lattice spacing.
34
EXCITATION SPECTRUM
Upper bound to the excitation spectrum obtained
from Sk by Feynman relation
Roton minimum appears for nr02?16
35
ONE-BODY DENSITY MATRIX
Important correlation properties can be extracted
from the one-body density matrix. In the case of
a zero temperature Bose gas the one-body reduced
density matrix possess an eigenvalue of order
of the total number of particles N. This behavior
is a manifistation of the Bose-Einstein
condensation and for a homogeneous systems
implies the asymptotic condition
g1(r1,r1)?constgt0 as r1-r1 ??. The
off-diagonal long-range order (ODLRO) is present
in the system. In the above expression ?(r) and
?(r) denote the creation (annihilation) operator
of spin-up particles.
36
GROUND STATE ENERGY (DILUTE GAS)
Ground state energy per particle in units of ?2
/Mr02 as function of nr02. Symbols DMC results,
green line mean field prediction.
37
EOS MEAN-FIELD GPE
In the dilute regime the equation of state is
expected to be universal. It depends only on
the density n and the scattering length as.The
leading term is given by the mean-field
contribution1 M.Schick, Phys.Rev.A 3, 1067
(1971)Lieb et al. (2002) rigorously prove that
2D Gross-Pitaevskiiequation has coupling
constantthus recovering MF energy
38
EOS BEYOND MEAN-FIELD
A number of beyond mean-field corrections exist
in literature. Iterating Schicks expression for
thechemical potential0)1) 2)Such
corrections are obtained in D.Hines, N.Frankel,
D.Mitchell, Phys.Rev.Lett. 68A,12 (1978)
E.Kolomeisky and J.Starley, Phys.Rev.B 46,11749
(1992) A.A. Ovchinnikov, J.Phys.Cond.mat. 5,
8665 (1993)3) Subsequent corrections are
obtained in J.O.Andersen Eur.Phys. J B 28, 389
(2002)
but
39
TESTING ln ln 1/na2 TERM IN EXPANSION
Beyond MF terms red line
,blue line fit
40
TESTING ln ln 1/na2 TERM IN EXPANSION
Analytic expansions1)2)3)where a is a
cut-off lengthNumerical fit1) Dipoles,
na210-100-10-102) Hard spheres 1,
na210-8-10-21 S.Pilati et al., PRA 71,
023605 (2005)
41
GROUND STATE ENERGY (DILUTE GAS)
Analytic expansions vs DMC data. Results for
hard-disks, soft-disks, pseudopotential are taken
from S.Pilati et al., PRA 71, 023605 (2005)
42
ENERGY DEPENDENT SCATTERING LENGTH
In order to improve further the accuracy, we
consider (potential specific) energy-dependent
scattering length. The scattering length is found
as the first node of analytic continuation of the
2-body scattering solution from the region where
the interaction potential is absent.- for the
hard-disks it is constant
43
GROUND STATE ENERGY (DILUTE GAS)
Analytic expansions vs DMC data. Results for
hard-disks, soft-disks, pseudopotential are taken
from S.Pilati et al., PRA 71, 023605 (2005)
44
SOME TYPICAL NUMBERS (EXCITONS)
I Spatially separated indirect excitons. r0
m e2 D2 / (e ?2), 1) Timofeev et al.,
n 2.5 1010 cm-2, m0.22 me, D 14 nm r06
10-6 cm, n r02 0.8 2) D.W. Snoke et
al, n 5 109 cm-2, m0.14 me, D 10.5
nm r02 10-6 cm, n r02 0.03 3)
Butov et al, n 1 1010 cm-2 - 3 1010 cm-2,
m0.27 me, D 12.5 nm me is mass of a
free electron r0 6 10-6 cm, n r02 0.4 -
1.2
45
SOME TYPICAL NUMBERS (ATOMIC GASES)
II Atoms with permanent moments r0m M2 /?2
. For example, for 52Cr has a
relatively large magnetic moment M 6 µB.
Assuming density n 5 109 cm-2 (corresponding to
3D density 3 1014 cm-3 ) one finds r0 2 10-7
cm, n r02 2 10-3 In addition s-wave
scattering is present with as 2.8 10-7 cm. Thus
the ratio of the characteristic energies ?2 /m
r02 and ?2 /m as2 is of the order of one. Note
that for 87Rb the same ratio is lt0.001.
For heteronuclear molecules with an electric
dipole moment the dipolar coupling can be
increased by a factor of 100 with respect to the
value of Cr.
46
SUPERSOLID

• Superfluid density was calculated using Path
  Integral Monte Carlo
• method in 1. Superfluid fraction was found to
  be equal to
• unity in gas phase
• zero in solid phase
• Still, absence of a superfluid fraction in a
  solid (i.e. supersolid) is not
• conclusive as a critical temperature of a
  (possible) supersolid can be
• much smaller compared to transition temperature
  of a gas phase.
• Diffusion Monte Carlo method is a strictly
  zero-temperature method and
• is free of this problem. Recent preliminary
  results give a fraction of
• 0.0007 in a solid phase. This result have to be
  checked and confirmed.
• Adding vacancies (one,two, lattice sites
  unoccupied) increase a lot the
• superfluid signal.
• 1 H.P. Büchler, E. Demler, M. Lukin, A.
  Micheli, N. Prokof'ev,
• G. Pupillo, P. Zoller, Phys. Rev. Lett.98, 060404
  (2007)

47
CONCLUSIONS
Diffusion Monte Carlo method was used to study
the properties of a dipolar Bose system at
T0.-) the ground state energy is calculated in
a wide range of densities The constructed fit
(10-100ltnr02 lt1024.) can be used for local
density approximation. -) quantum phase
transition from liquid to crystal is found at
density nr02 290(30). -) Lindemann ratio at
transition point is ? 0.230(6)-) pair
distribution function g2 was found for different
values of the interaction strength. -) static
structure factor Sk has peak in solid phase.-)
existence of the off-diagonal long-range order
was shown in one-body density matrix and the
condensate fraction was found. Agreement with
predictions of a weakly interacting Bose gas is
found at small densities. -) beyond mean-field
expansion is discussed in details in the weakly
interacting regime
48
DIFFUSION MONTE CARLO METHOD
Time-dependent Schrödinger equation in imaginary
time for the function At large
times Observables extracted from averages over
drift force
local energy
Ground state of bosons - probability
distribution
Fermions or excited state if nodes of ? and ?T
coincide
49
TAIL ENERGY
Homogeneous system at a given density n is
modeled by N particles in asimulation box Lx ?
Ly with periodic boundary conditions n N /
(LxLy). In order to avoid double counting of
image a cut-off is introduced both in the
potential energy and in the trial w.f. at
distance ri-rjL/2
Finite size effects can be significantly reduced
by adding the tail energy where the pair
distribution function g2 (r) can be approximated
by its limiting value g2 (r) ? n, r gt L/2. ?
Etail / N 1 / N1/2
50
GROUND STATE ENERGY
Ground state energy per particle in units of ?2
/mr02 as a function of the characteristic
parameter nr02 red squares - DMC results, solid
line - best fit 8.595 exp1.35 ln(nr02)0.0120
ln(nr02) 2
51
DILUTE GAS EXPANSION OF EOS
In the dilute regime the equation of state is
expected to be universal and it should depend
only on the density and the scattering
length. 1) The leading term is given by the
mean-field contribution1 M.Schick,
Phys.Rev.A 3, 1067 (1971) 2) The beyond
mean-field correction was expressed in
perturbative form2 D.Hines, N.Frankel,
D.Mitchell, Phys.Rev.Lett. 68A,12 (1978)
E.Kolomeisky and J.Starley, Phys.Rev.B 46,11749
(1992) 3) Substitution of density in (1) by
chemical potential gives
52
EXTRAPOLATION TECHNIQUE
The OBDM is a non-diagonal quantity and the
choice of the trial w.f. matters. The bias can be
reduced by extrapolation
53
SCATTERING IN A QUASI-2D GEOMETRY
Mean-field energy in 2D system The mean-field
approximation for the coupling constantPresen
ce of the tight confinement leads to
renormalization of the coupling constant,
D.S.Petrov, M.Holzmann, and G.V.Shlyapnikov,
Phys.Rev.Lett. 84, 2551 (2000)
54
BEYOND MEAN-FIELD CORRECTION
Using diagramatic approach one can derive
following expression
(1)
for the dimensionless
energyFormula (1) should be solved in a
self-consistent way.We do it iteratively,
taking as starting approximation the mean-field
expressionThe resulting formula leads to
series expansion reported in the talk
55
MELTING AND FREEZING POINTS
Construction of the Maxwell double tangent
construction shows that the region of phase
coexistence is very small and freezing and
melting points are indistinguishable within error
bars of or calculation
56
CONDENSATE FRACTION (HIGH DENSITY)
Condensate fraction n0/n as a function of
dimensionless density nr02.Condensate depletion
is large for considered densities.
57
TESTING ln ln 1/na2 TERM IN EXPANSION
Beyond MF terms red line
,blue line fit
58
GROUND STATE ENERGY (DILUTE GAS)
Analytic expansions vs DMC data. Results for
hard-disks, soft-disks, pseudopotential are taken
from S.Pilati et al., PRA 71, 023605 (2005)
59
ENERGY DEPENDENT SCATTERING LENGTH
In order to improve further the accuracy, we
consider (potential specific) energy-dependent
scattering length. The scattering length is found
as the first node of analytic continuation of the
2-body scattering solution from the region where
the interaction potential is absent.- for the
hard-disks it is constant
60
GROUND STATE ENERGY (DILUTE GAS)
Analytic expansions vs DMC data. Results for
hard-disks, soft-disks, pseudopotential are taken
from S.Pilati et al., PRA 71, 023605 (2005)
61
SOME TYPICAL NUMBERS (EXCITONS)
I Spatially separated indirect excitons. r0
m e2 D2 / (e ?2), 1) Timofeev et al.,
n 2.5 1010 cm-2, m0.22 me, D 14 nm r06
10-6 cm, n r02 0.8 2) D.W. Snoke et
al, n 5 109 cm-2, m0.14 me, D 10.5
nm r02 10-6 cm, n r02 0.03 3)
Butov et al, n 1 1010 cm-2 - 3 1010 cm-2,
m0.27 me, D 12.5 nm me is mass of a
free electron r0 6 10-6 cm, n r02 0.4 -
1.2
62
PHASE TRANSITION CRITICAL DENSITY
II Atoms with permanent moments r0m Cdd
/?2 . For example, for 52Cr has a
relatively large magnetic moment M 6 µB.
Assuming density n 5 109 cm-2 (corresponding to
3D density 3 1014 cm-3 ) one finds r0 2 10-7
cm, n r02 2 10-3 In addition s-wave
scattering is present with as 2.8 10-7 cm. Thus
the ratio of the characteristic energies ?2 /m
r02 and ?2 /m as2 is of the order of one. Note
that for 87Rb the same ratio is lt0.001.
For heteronuclear molecules with an electric
dipole moment the dipolar coupling can be
increased by a factor of 100 with respect to the
value of Cr.
63
SOME TYPICAL NUMBERS (ATOMIC GASES)
II Atoms with permanent moments r0m M2 /?2
. For example, for 52Cr has a
relatively large magnetic moment M 6 µB.
Assuming density n 5 109 cm-2 (corresponding to
3D density 3 1014 cm-3 ) one finds r0 2 10-7
cm, n r02 2 10-3 In addition s-wave
scattering is present with as 2.8 10-7 cm. Thus
the ratio of the characteristic energies ?2 /m
r02 and ?2 /m as2 is of the order of one. Note
that for 87Rb the same ratio is lt0.001.
For heteronuclear molecules with an electric
dipole moment the dipolar coupling can be
increased by a factor of 100 with respect to the
value of Cr.
64
SUPERSOLID

• Superfluid density was calculated using Path
  Integral Monte Carlo
• method in 1. Superfluid fraction was found to
  be equal to
• unity in gas phase
• zero in solid phase
• Still, absence of a superfluid fraction in a
  solid (i.e. supersolid) is not
• conclusive as a critical temperature of a
  (possible) supersolid can be
• much smaller compared to transition temperature
  of a gas phase.
• Diffusion Monte Carlo method is a strictly
  zero-temperature method and
• is free of this problem. Recent preliminary
  results give a fraction of
• 0.0007 in a solid phase. This result have to be
  checked and confirmed.
• Adding vacancies (one,two, lattice sites
  unoccupied) increase a lot the
• superfluid signal.
• 1 H.P. Büchler, E. Demler, M. Lukin, A.
  Micheli, N. Prokof'ev,
• G. Pupillo, P. Zoller, Phys. Rev. Lett.98, 060404
  (2007)

65
CONCLUSIONS
Diffusion Monte Carlo method was used to study
the properties of a dipolar Bose system at
T0.-) the ground state energy is calculated in
a wide range of densities The constructed fit
(10-100ltnr02 lt1024.) can be used for local
density approximation. -) quantum phase
transition from liquid to crystal is found at
density nr02 290(30). -) Lindemann ratio at
transition point is ? 0.230(6)-) pair
distribution function g2 was found for different
values of the interaction strength. -) static
structure factor Sk has peak in solid phase.-)
existence of the off-diagonal long-range order
was shown in one-body density matrix and the
condensate fraction was found. Agreement with
predictions of a weakly interacting Bose gas is
found at small densities. -) beyond mean-field
expansion is discussed in details in the weakly
interacting regime