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

Ground state properties of a homogeneous 2D system of Bosons with dipolar interactions. Departament de F ì sica i Enginyeria Nuclear Universitat Polit è cnica de Catalunya, Barcelona, Spain. RPMBT14. G. E. Astrakharchik J. Boronat J. Casulleras I. L. Kurbakov

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

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  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 RPMBT14 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 / |r|3. 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 densityn r02, with r0=mCdd /4π2. By expressing distances in units of r0 and energy in units of 2 /m r02 Hamiltonian takes a simple formIn 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

  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 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.)

  4. TRIAL WAVEFUNCTION We construct the trial wave function in the following form: where ri, i=1,…,N are particle coordinates and rllatt , l=1,…,Mare 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 widthα 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 st 1 minimum: LIQUID In the regime of high density we find two minima in the variational energy as a function of the localization widthα. E / N 7000 nd 2 minimum: SOLID 1) α=0 translational invariance is preserved. Density profile is flat. This minimum corresponds to a gas/liquid state. 6900 0 200 400 600 800 a 2) α >0 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)3/2+a2(n r02) 5/4 +a3(n r02)½a1=4.536(8), a2=4.38(4), a3=1.2(3) - liquida1=4.43(1), a2=4. 80(3), a3=2.5(2) - solidClassical 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 [2]:[1] 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 nas>10-250(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 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.

  14. 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.

  15. SUPERFLUID FRACTION:ADDING VACANCIES Superfluid fraction nS/n as a function of concentration of vacancies for N=30 particles in crystal phase.

  16. 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.

  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 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<nr02 <1024.) 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-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

  21. ATOMS z y x 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 Misthe magnetic dipole moment. b)If the dipoles are induced by an electric fieldE then the interaction constant is Cdd=E2α2, whereα is a static polarizability. c)Polar molecules + a static electric fieldE + 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 ~ 1К) 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 /ε, where e is the charge of an electron, εis the dielectric constant of the semiconductor, and Dis 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, γ=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) α = 0 – no localization, i.e. liquid 2) α > 0 – localized system, i.e. crystal

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

  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 FUNCTION:SOLID 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 δ-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 δ-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)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.

  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 contribution:[1] 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 Schick’s 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 expansions:1)2)3)where a is a “cut-off” lengthNumerical fit:1) Dipoles, na2=10-100-10-102) Hard spheres [1], na2=10-8-10-2[1] 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 /(ε2), 1) Timofeev et al., n = 2.5 1010 cm-2, m=0.22 me, D = 14 nm r0=6 10-6 cm, nr02= 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, nr02= 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, nr02= 0.4 - 1.2

  45. SOME TYPICAL NUMBERS (ATOMIC GASES) 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, nr02= 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 r02and 2 /m as2is 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.

  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 T=0.-) the ground state energy is calculated in a wide range of densities The constructed fit (10-100<nr02 <1024.) 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 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

  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 Lx Ly 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-rj|=L/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 > 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 exp{1.35 ln(nr02)+0.0120 ln(nr02) 2}

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