Departament de F ì sica i Enginyeria Nuclear Universitat Polit è cnica de Catalunya,

Departament de Fìsica i Enginyeria NuclearUniversitat Politècnica de Catalunya,

Barcelona, Spain

G. E. AstrakharchikJ. Boronat

J. CasullerasI. L. KurbakovYu. E. Lozovik

16-20 July, 2007RPMBT-14

Ground state properties of a homogeneous 2D system of Bosons

with dipolar interactions.

RPMBT14

MODEL HAMILTONIANWe consider a (quasi-) two dimensional homogeneous Bose system with a dipole-dipole interaction Vint (z) = Cdd / |r|3. Such a system is described by the following Hamiltonian

where m is mass and N is number of dipoles.

An important parameter in a homogenous system is a dimensionless density n r0

2 , with r0=mCdd /4π2. 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 s–wave scattering length

1dim ||

00 17.3)2exp( rra

QUANTUM MONTE CARLO1) The Variational Monte Carlo (VMC) method makes it

possible to calculate multidimensional averages of physical quantities over the N-body trial wave function ΨT

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 T=0. 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.)

rdrdrr

rdrdrrrr

...|),...,(|...

...|),...,(|),...,(A...A

TRIAL WAVEFUNCTIONWe construct the trial wave function in the following form:

where ri, i=1,…,N are particle coordinates and rllatt , l=1,…,M are

coordinates of triangular lattice sites. The two-body term is chosen as

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 α 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.

RrrLCrCC

RrrKCrf

,)/(/exp

),/2()(

lattkl

jijiNT rrrrfrr

2.21 })(exp{|)(|),...,(

TRIAL WAVEFUNCTION

0.00 0.02 0.04 0.06 0.08 0.100.0

In the regime of high density we find two minima in the variational energy as a function of the localization width α.

1) α=0 translational invariance is preserved. Density profile is flat. This minimum corresponds to a gas/liquid state.

2) α >0 translational invariance is broken. Density profile has crystal symmetry. This minimum corresponds to a solid state.

0 200 400 600 800

70001st minimum: LIQUID

2nd minimum: SOLID

RESULTS: GROUND STATE ENERGY

Ground state energy per particle in units of (2 /Mr02)/(nr0

2)3/2 as a function of parameter nr0

2: symbols – DMC results, lines – best fit. Inset: energy per particle in units of 2 /Mr0

0 200 400 600 800 1000

liquid solid

0 200 400 600 800 10000.0

5.0x104

1.0x105

1.5x105 liquid solid

(nr0)3/2

PHASE TRANSITIONAt 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 r0

2) 3/2+a2(n r0

2) 5/4 +a3(n r02)

a1=4.536(8), a2=4.38(4), a3=1.2(3) - liquid

a1=4.43(1), a2=4. 80(3), a3=2.5(2) - solid

Classical crystal limit is recovered at large density: Etriang/N = 4.446… (n r0

2) 3/2.

Transition point is estimated as ncr02 =29030.

PIMC estimation [1] ncr02 = 320140

GFMC estimation [2] ncr02 = 23020

nr02 256

nr02 358

[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

PAIR DISTRIBUTION: LIQUID PHASE

Pair distribution function )0(ˆ)(ˆ)(ˆ)0(ˆ)(2 rrr

MEAN-FIELD THEORYIn 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 [2]:

[1] E. Lieb, R. Seiringer, J. Yngvason Commun. Math. Phys. 224, 17 (2001)[2] M. Schick, Phys. Rev. A 3, 1067 (1971)

)/1ln(

...)/1ln(

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 nas>10-250 (1% error).

[1] S. Pilati, J. Boronat, J. Casulleras, S. Giorgini, PRA 71, 023605 (2005).

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

10-2 10-1 100 101

Mean-field Dipoles Hard-Disks

E / EMF

MF DMC

ONE-BODY DENSITY MATRIX: GAS

One-body density matrix for different densities. Finite asymptotic value is a manifestation of Off-Diagonal Long-Range Order

0 1 2 3 40,0

2-14=6.1 10-5

2-5=3.1 10-2

2-10=10-3

n1/2 r

0 =20=1

0 =2-2=0.25

0 = 22=4

0 = 24=16

0 = 26=64

0 = 28=256

)0(ˆ)(ˆ)(1 rrg

CONDENSATE FRACTION

Condensate fraction n0/n as a function of gas parameter na2.

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 1030,0

DMC weakly interacting

Bose-gas

n0 / n

SUPERSOLID?There are several ways to define a supersolid:

1) Spatial order of a solid + finite superfluid density2) Spatial order of a solid (diagonal order in OBDM)+ off-diagonal long-range order (finite long-range asymptotic of one-body density matrix)

Literature overview:[1] 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.

WINDING NUMBER

Diffusion coefficient of center of masses D as a function of imaginary time τ in a crystal at critical density na2=290 for different system size.

0.000 0.005 0.010 0.0150

2x10-4

4x10-4

6x10-4

8x10-4

1x10-3

Symmetrized w.f. N=12 N=16 N=24

Non-symmetrized w.f. N=12

SUPERFLUID FRACTION:ADDING VACANCIES

Superfluid fraction nS/n as a function of concentration of vacancies for N=30 particles in crystal phase.

0.00 0.03 0.06

nS / n

Superfluid fraction N=30

ONE-BODY DENSITY MATRIX: SOLID

One-body density matrix in a crystal at critical density na2=290 for different system sizes, symmetrized and non-symmetrized w.f.

0.0 0.1 0.210-5

r / r0

Symmetrized w.f. N=30 N=48 N=56 N=80 N=108

Non symmetrized w.f. N=12

CONDENSATE AND SUPERFLUID FRACTIONS

Superfluid fraction (blue) and condensate fraction (red) as a function of vacancy concentration.

0.00 0.03 0.06

Superfluid fraction N=30

Condensate fraction N=30 N=56

CONCLUSIONSDiffusion Monte Carlo method was used to study the properties of a dipolar two-dimensional Bose system at T=0.-) the ground state energy, pair distribution function, one-bode density matrix are calculated in a wide range of densities -) fit to the energy (10-100<nr0

2 <1024.) can be used for LDA-) gas-solid quantum phase transition is found at density nr0

2 = 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.

THANK YOU FOR YOUR ATTENTION!

INTRODUCTIONWhy 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-Thouless * 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

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 = E2α2, where α 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)

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 К) may form a sort of quantum liquid– degenerated bose gas and might experience Bose condensation. Spatial separation betweenelectron and hole increases exciton lifetime.

If the size of an exciton pair is much smaller than the distance between exciton such a pairacts as a dipole.

In this case Cdd = e2 D2 / ε, where e is the charge of an electron, ε is the dielectric constant of the semiconductor, and D is the separation between electron and hole.

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, γ=0.577… Long-range behavior defines the scattering length a:

The scattering length on a 2D dipole-dipole potential is

int constrdrV

)/ln()( arr

00 17.3)2exp( rra

OPTIMIZATION OF PARAMETERS

Variational energy has two minima for densities nr02 8 :

1) α = 0 – no localization, i.e. liquid 2) α > 0 – localized system, i.e. crystal

0 20 40120

0 10 20 30 40 50

0 400 800

0 5000 10000 15000

136000

140000

144000

0 = 16

0 = 128

0 = 1024

FINITE SIZE DEPENDENCE

Energy (solid phase) for nr02 =256 as a function 1/N. Symbols – DMC+tail,

solid line – best fit, dashed line – extrapolation to thermodynamic limit.

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

RESULTS: GROUND STATE ENERGY

Ground state energy per particle in units of 2 /Mr02 as a function of nr0

2: symbols – DMC results, solid lines – best fit, dashed – classical crystal.

LINDEMANN RATIOThe 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 γ = 0.230(6).

Comparison to other two-dimensional systems:γ = 0.279 – hard-disks, L. Xing, Phys. Rev. B 42, 8426 (1990),γ = 0.235(15) – 2D Yukawa bosons, W. R. Magro and D. M. Ceperley, Phys. Rev. B 48, 411 (1993)γ = 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 γ at transition is generally larger, for example γ = 0.28 for 3D Yukawa potential, D. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B 17, 1070 (1978)

ii arr /)( 2

CORRELATION FUNCTIONSThe 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

drdrrr

drdrrrr

NNrrrg

..|),..,(|

..|),..,,0,(|)1()0(ˆ)(ˆ)(ˆ)0(ˆ)(

20 ]1)()[(π1 rdrrgkrJnSk

PAIR DISTRIBUTION FUNCTION: LIQUID PHASE

Pair distribution functions at densities (liquid phase)

0,0 0,5 1,0 1,5 2,0 2,5 3,00,0

0 = 2-30 (na2= 9.4 10-9)

0 = 2-14 (na2= 6.1 10-4)

0 = 2-7 (na2= 7.9 10-2)

0 = 2-4 (na2= 0.63)

0 = 20 (na2=10)

0 = 24 (na2=103)

n1/2 r

PAIR DISTRIBUTION FUNCTION:SOLID PHASE

Pair distribution functions at densities nr02 =384, 512, 768 (solid phase)

0 1 2 3 4 5 6 7 80,0

n1/2 r

0 = 384

0 = 512

0 = 768

STATIC STRUCTURE FACTOR

The static structure factor is a continuous function in the liquid phase. In the solid phase a δ-peak appears at a momentum, corresponding to the inverse lattice spacing.

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

0 3 6 9 120,0

na2= 9.4 10-9

na2= 6.1 10-4

na2= 7.9 10-2

na2= 0.63

na2=10

k / n1/2

mckSk 2/22 +k|na \ln|n/16/ kSk

The static structure factor is a continuous function in the liquid phase. In the solid phase a δ-peak appears at a momentum, corresponding to the inverse lattice spacing.

0 5 10 15 200.0

k / n1/2

liquid:

0 = 16

0 = 32

0 = 64

0 = 128

0 = 256

solid:

0 = 256

EXCITATION SPECTRUM

Upper bound to the excitation spectrum obtained from Sk by Feynman relation: Roton minimum appears for nr0

0 2 4 6 8 100

k / n1/2

0 = 4 nr2

0 = 64

0 = 8 nr2

0 = 128

0 = 16 nr2

0 = 256

0 = 32 free particle

ONE-BODY DENSITY MATRIXImportant 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)const>0 as |r1’-r1| . The off-diagonal long-range order (ODLRO) is present in the system.

In the above expression ψ†(r) and ψ(r) denote the creation (annihilation)operator of spin-up particles.

drdrrr

drdrrrrrrr

Nrrrrg

..|),..,(|

..),..,,'(),..,,()'(ˆ)(ˆ)',(

2202*0

GROUND STATE ENERGY (DILUTE GAS)

Ground state energy per particle in units of 2 /Mr02 as function of nr0

2. Symbols – DMC results, green line – mean field prediction.

10-4 10-3 10-2 10-1 100 101 1021E-5

fit - liquid phase low density mean-field expansion DMC

EOS: MEAN-FIELD GPEIn 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 contribution:

[1] M.Schick, Phys.Rev.A 3, 1067 (1971)

Lieb et al. (2002) rigorously prove that 2D Gross-Pitaevskiiequation has coupling constant

thus recovering MF energy

)/1ln(

EOS: BEYOND MEAN-FIELDA number of beyond mean-field corrections exist in literature. Iterating Schick’s expression for thechemical potential0)

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 ……

n)/1ln(

)/1ln( 2

)/1ln(

)/1ln(ln)/1ln(

ss nana

)4ln()/1ln(ln)/1ln(

ss nana

TESTING ln ln 1/na2 TERM IN EXPANSION

Beyond MF terms: red line ,blue line: fit

3.0 3.5 4.0 4.5 5.0 5.50.0

2 ) DMC fit, coef. fixed to 1 fit, coef. free

ln ln 1/na2

constnas )/1ln(ln 2

32.2)/1ln(ln88.0 2 sna

Analytic expansions:

where a is a “cut-off” length

Numerical fit:

1) Dipoles, na2=10-100-10-10

2) Hard spheres [1], na2=10-8-10-2

[1] S.Pilati et al., PRA 71, 023605 (2005)

53.2)/1ln(ln)4ln()/1ln(ln 22 nana

32.2)/1ln(ln88.0 2 sna

53.3)/1ln(ln1)4ln()/1ln(ln 22 nana

03.3)/1ln(ln2/1)4ln()/1ln(ln 22 nana

26.2)/1ln(ln86.0 2 sna

Analytic expansions vs DMC data. Results for hard-disks, soft-disks, pseudopotential are taken from S.Pilati et al., PRA 71, 023605 (2005)

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

10-7 10-6 10-5 10-4 10-3 10-2 10-10.0

Hard-disks Soft-disks

I Spatially separated indirect excitons. r0 = m e2 D2 / (ε 2),

1) Timofeev et al., n = 2.5 1010 cm-2, m=0.22 me, D = 14 nm r0=6 10-6 cm,n r0

2 = 0.8

2) D.W. Snoke et al, n = 5 109 cm-2, m=0.14 me, D = 10.5 nm

r0=2 10-6 cm,n r0

2 = 0.03

3) Butov et al, n = 1 1010 cm-2 - 3 1010 cm-2, m=0.27 me, D = 12.5 nm me is mass of a free electron

r0 = 6 10-6 cm,n r0

2 = 0.4 - 1.2

SOME TYPICAL NUMBERS (EXCITONS)

II Atoms with permanent moments:

r0=m 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 r0

2 = 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 r0

2 and 2 /m as2 is of the order of one.

Note that for 87Rb the same ratio is <0.001.

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.

SOME TYPICAL NUMBERS (ATOMIC GASES)

*SUPERSOLIDSuperfluid density was calculated using Path Integral Monte Carlomethod in [1]. Superfluid fraction was found to be equal to

1) unity in gas phase2) zero in solid phase

Still, absence of a superfluid fraction in a solid (i.e. supersolid) is notconclusive as a critical temperature of a (possible) supersolid can bemuch 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 of0.0007 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)

CONCLUSIONSDiffusion Monte Carlo method was used to study the properties of a dipolar Bose system at T=0.

-) the ground state energy is calculated in a wide range of densities The constructed fit (10-100<nr0

2 <1024.) can be used for local density approximation. -) quantum phase transition from liquid to crystal is found at density nr0

2 = 290(30). -) Lindemann ratio at transition point is γ = 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

DIFFUSION MONTE CARLO METHODTime-dependent Schrödinger equation in imaginary time

for the function

At large times

Observables extracted from averages over

),(])([)],()([2

),( 22

RRRRFRR

RR fEEfm

/it),()(),( RRR Tf ),...,,( 21 NrrrR

)()()( 1 RRR TTL HE

)()(2)( 1 RRRF R TT

local energy

drift force

),( Rf

0),( Rf

Ground state of bosons:

- probability distribution

Fermions or excited stateif nodes of and T coincide

0),()(),( RRR Tf

)()()()(),( 00)( RRRRR

EEnnT cecf refn

TAIL ENERGYHomogeneous 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-rj|=L/2

Finite size effects can be significantly reduced by adding the tail energy:

where the pair distribution functiong2 (r) can be approximated by its limiting value g2 (r) n, r > L/2. Etail / N ~ 1 / N1/2

0.00 0.01 0.02 0.030

DMC DMC + tail 1 / N

2 π2)()(2

tail rdrrgrVN

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 exp{1.35 ln(nr02)+0.0120 ln(nr0

DMC fit

DILUTE GAS: EXPANSION OF EOSIn 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 contribution:

[1] M.Schick, Phys.Rev.A 3, 1067 (1971)2) The beyond mean-field correction was expressed in

perturbative form:

[2] 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:

)/1ln(

)]/1[ln(

)]/1ln[ln(2

)]/1ln()4/1ln[()/1ln(

ss nanam

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:

Extrpl.

SCATTERING IN A QUASI-2D GEOMETRY

Mean-field energy in 2D system

The mean-field approximation for the coupling constant:

Presence 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)

BEYOND MEAN-FIELD CORRECTION

Using diagramatic approach one can derive following expression:

for the dimensionless energy

Formula (1) should be solved in a self-consistent way.

We do it iteratively, taking as starting approximation the mean-field expression:

The resulting formula leads to series expansion reported in the talk

)0(1 1

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

0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.00500

60000 solid, DMC liquid, DMC solid, fit liquid, fit Maxwell double-tangent

construction

1 / nr2

CONDENSATE FRACTION (HIGH DENSITY)

Condensate fraction n0/n as a function of dimensionless density nr02.

Condensate depletion is large for considered densities.

0 50 100 150 200 250 3000.00

n0 / n

Beyond MF terms: red line ,blue line: fit

3.0 3.5 4.0 4.5 5.0 5.50.0

2 ) DMC fit, coef. fixed to 1 fit, coef. free

ln ln 1/na2

constnas )/1ln(ln 2

32.2)/1ln(ln88.0 2 sna

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

10-7 10-6 10-5 10-4 10-3 10-2 10-10.0

Hard-disks Soft-disks

I Spatially separated indirect excitons. r0 = m e2 D2 / (ε 2),

1) Timofeev et al., n = 2.5 1010 cm-2, m=0.22 me, D = 14 nm r0=6 10-6 cm,n r0

2 = 0.8

2) D.W. Snoke et al, n = 5 109 cm-2, m=0.14 me, D = 10.5 nm

r0=2 10-6 cm,n r0

2 = 0.03

3) Butov et al, n = 1 1010 cm-2 - 3 1010 cm-2, m=0.27 me, D = 12.5 nm me is mass of a free electron

r0 = 6 10-6 cm,n r0

2 = 0.4 - 1.2

SOME TYPICAL NUMBERS (EXCITONS)

r0=m Cdd /2 .

r0 = 2 10-7 cm,n r0

2 = 2 10-3

PHASE TRANSITION: CRITICAL DENSITY