Path-Integral-Monte-Carlo simulation has been used to calculate the
properties of a two-dimensional (2D) interacting Bose system. The bosons
interact with hard-core potentials and are confined to a harmonic trap. Results
for the density profiles, the condensate fraction, and the superfluid density
are presented. By comparing with the ideal gas we easily observe the effects of
finite size and the depletion of the condensate because of interactions. The
system is known to have no phase transition to a Bose-Einstein condensation in
2D, but the finite system shows that a significant fraction of the particles
are in the lowest state at low temperatures.Comment: six pages, two figures; Contribution to QFS98; To be published in
  Journ. Low. Temp. Phy

Heinrichs, Stefan

Mullin, William J.

English

arXiv

Path-Integral-Monte-Carlo simulation has been used to calculate the properties of a two-dimensional (2D) interacting Bose system. The bosons interact with hard-core potentials and are confined to a harmonic trap. Results for the density profiles, the condensate fraction, and the superfluid density are presented. By comparing with the ideal gas we easily observe the effects of finite size and the depletion of the condensate because of interactions. The system is known to have no phase transition to a Bose-Einstein condensation in 2D, but the finite system shows that a significant fraction of the particles are in the lowest state at low temperatures

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University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics1998Quantum-Monte-Carlo Calculations for Bosons ina Two-Dimensional Harmonic TrapStefan HeinrichsUniversity of KonstanzWilliam J. MullinUniversity of Massachusetts - AmherstFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationHeinrichs, Stefan and Mullin, William J., "Quantum-Monte-Carlo Calculations for Bosons in a Two-Dimensional Harmonic Trap"(1998). Physics Department Faculty Publication Series. 10.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/10arXiv:cond-mat/9807331v1  [cond-mat.stat-mech]  24 Jul 1998Quantum-Monte-Carlo Calculations for Bosons ina Two-Dimensional Harmonic TrapStefan Heinrichs† and William J. Mullin∗†Department of Physics, University of Konstanz, Germany∗Department of Physics and Astronomy, Hasbrouck Laboratory, University ofMassachusetts, Amherst MA 01003-3720, USAPath-Integral-Monte-Carlo simulation has been used to calculate the proper-ties of a two-dimensional (2D) interacting Bose system. The bosons interactwith hard-core potentials and are confined to a harmonic trap. Results forthe density profiles, the condensate fraction, and the superfluid density arepresented. By comparing with the ideal gas we easily observe the effects offinite size and the depletion of the condensate because of interactions. Thesystem is known to have no phase transition to a Bose-Einstein condensationin 2D, but the finite system shows that a significant fraction of the particlesare in the lowest state at low temperatures.PACS numbers:03.75.Fi,02.70.Lq,05.30.Jp,05.70.Fh,32.80.Pj,67.40.Db1. INTRODUCTIONRecent experiments on alkali atoms cooled by laser methods and evap-oration in a magnetic trap have allowed the observation of Bose-Einsteincondensation (BEC) in a harmonic potential formed by an external mag-netic field. By varying the trapping field so that it is very narrow in onedimension, it is possible to separate the single-particle states in the oscilla-tor potential into well-defined bands. By occupying only states in the lowestband one has an effective two-dimensional system. In contrast to such aquasi-2D system, the system considered here is genuinely two dimensional.In this paper we will investigate the behaviour of harmonic Bose sys-tems in 2D using the very powerful finite-temperature Path-Integral MonteCarlo (PIMC) simulation technique.1 The PIMC technique is in principlecapable of describing systems of arbitrary interaction strength and densityS. Heinrichs and W. J. Mullinand allows one to study the static properties of the condensed gases. Theonly fundamental uncertainty arises from the choice of the interaction poten-tial. Here a hard-core potential appropriate for the s-wave scattering lengthof 87Rb was chosen. In this case the hard-core parameter is a0 = 0.0043 inthe dimensionless units of Ref. 2. The analogous three dimensional case hasbeen studied previously.3The Hamiltonian for the system under consideration isH =N∑i=1p2i2m+∑i,jV (ri − rj) + m2N∑i=1(ω2xri,x + ω2yri,y). (1)The density matrix for N Bose particles at inverse temperature β =1/kbT can be written as a convolution withM intermediate density matricesor ”time slices” at inverse temperature τ = β/M . Then the probabilitydensity for finding a many-particle configuration R = (r1, · · · , rN ) isρ(R,R;β) =1N !∑P∫· · ·∫ρ(R,R1; τ)×× ρ( R1,R2; τ) · · · ρ(RM−1,RP ; τ)dR1dR2 · · · dRM−1where RP denotes a vector with permuted particle labels. The Metropolisalgorithm is used to sample from this distribution. With larger M or corre-spondingly larger temperatures for each time slice, the intermediate densitymatrices approach the classical limit which leads to the primitive approxi-mation. In order to make the computation for many particles feasible, Mcan be reduced by several orders of magnitude by calculating the densitymatrix ρ2 for the interaction involving just two particles and approximatingeach time slice by1ρ(R,R′; τ) =N∏i=1ρ1(ri, r′i; τ)∏i<jρ2(ri, rj ; r′i, r′j ; τ)ρ1(ri, r′i; τ)ρ1(rj , r′j ; τ). (2)As long as only two particles interact, this is equivalent to using the primitiveapproximation.To calculate ρ2, we note that in the harmonic potential the Hamiltonianfor two particles decouples into a center of mass and a relative motion termwith the latter given byHrel =p22µ+ V (r)︸ ︷︷ ︸HHC+µ2(ω2xr2x + ω2yr2y) (3)PIMC Calculations for Bosons in a 2D Harmonic Trapwhere µ = m/2 is the reduced mass. With the Trotter breakup, the quotientin (2), which plays the role of a correction term, can be transformed toρHC(r, r′)X(r)X(r′)ρ1,µ(r, r′)(4)with X(r) = exp(−τµ(ω2xr2x + ω2yr2y)/4) and the relative coordinates r =ri− rj and r′ = r′i− r′j . ρHC(r, r′) is the density matrix for the HamiltonianHHC (Eq. (3)) describing the interaction between two interacting particleswithout confining potential and ρ1,µ(r, r′) is the density matrix for a singleparticle of reduced mass µ in the harmonic potential.4Calculating ρHC(r, r′) for a hard-core potential using an eigenfunctionexpansion yieldsρHC(r, r′; τ) =∞∑l=−∞12pieilϕ∫ ∞0dkRkl(r)Rkl(r′)e−τh¯2k22µ (5)with the radial wave functionsRkl(r) =√k(cos δkl Jl(kr)− sin δklNl(kr)) and tan δkl = Jl(ka0)Nl(ka0)where Jl and Nl are the Bessel and Neumann functions and ϕ is the anglebetween r and r′.Since the numerical evaluation of this function is quite time consuming,values distributed over a mesh with parameters |r|, |r′| and ϕ are computedfor each value of τ . A simple linear interpolation method proved to besufficient to reach the required degree of accuracy for evaluation in the sim-ulation. The full correction factor (Eq. (4)) with dependence on the confin-ing potential can then be calculated very efficiently during the Monte-Carlosimulation.To further speed up the calculation, a boxing algorithm3 is used thatdivides the space of the system into boxes with a size of at least the “healinglength” of the pair interaction. When we compute the effects of the inter-action of a given particle, it is necessary to consider only a small numberof interactions with particles in boxes neighbouring that of the particle inquestion. Furthermore, attempting only permutations with particles fromthe same box, increases the acceptance probability for these moves and re-sults in a more effective sampling of permutation space.3The value of τ has to be chosen carefully for each particle density toaccommodate both the Trotter breakup and the approximation of pair inter-actions (Eq. (2)). For the density used, tests with values for τ ranging overorders of magnitude have been performed, showing that a further decreasebelow τ = 0.01 yield the same results within statistical errors.S. Heinrichs and W. J. Mullin2. DENSITY PROFILES AND CONDENSATE FRACTIONIn 2D, with interactions, there is no phase transition to a Bose condensedstate5 even in a trap, nevertheless there should be a macroscopic fraction ofparticles in the ground state at finite temperature when the particle numberis finite.We tentatively assume that, in 2D, the condensation fraction can bedetermined, as in 3D,3 by observing a macroscopic number of particles withlong exchange cycles in the lowest state of their subsystem. For each tem-perature a characteristic length of permutation cycles l0 can be chosen sothat the density profiles for particles on cycles longer than l0 are essentiallythe same. Particles on shorter cycles will have a broader density profile.Particles on cycles longer than l0 are identified with the condensate (Fig. 1).What is plotted is the average distribution in one coordinate after integrat-ing over the other coordinate. We have fit the lowest temperature curve inFig. 1 with the infinite-N solution of the zero-temperature GP equation.6The interaction coefficient has been determined by the fit since, in 2D, thereis no straightforward connection between a hard-core interaction and a con-tact pseudopotential. We observe no obvious bimodality, due to separatedistributions of condensed and non-condensed particles in the overall den-sity in Fig. 1; if a kink did exist in the density it would likely be washed outby the integration over one coordinate.In two dimensions interactions seem to lead to a comparatively strongerdepletion of the condensate (Fig. 2) than in three dimensions (Cf., Ref. 3).This is expected, since there is no condensate at finite temperature in thethermodynamic limit for the interacting system and therefore the criticaltemperature has to approach zero (for large numbers of particles) wheninteractions are turned on.53. SUPERFLUID FRACTIONRecently one of the authors7 showed that the Hartree-Fock-Bogoliubovequations indicated that there is a phase transition in 2D, although it cannotbe to the BEC state. Perhaps this transition is of the Kosterlitz-Thouless(KT) type8 and involves superfluidity. The superfluid fraction can be relatedto the mean square surface area enclosed by Feynman paths. Sindzingreet. al. have shown that9ρs/ρ =4m2〈A2〉βh2Ic(6)with A the area swept out by the paths and Ic the classical moment ofinertia. The average is taken over configurations in the simulation.PIMC Calculations for Bosons in a 2D Harmonic TrapThe resulting superfluid densities are very small (Fig. 2). A smaller ρsthan in the translationally invariant system is expected because the forma-tion of the superfluid begins in the middle of the potential well where thecontribution to the moment of inertia is small. Paths with different orien-tations will contribute with different signs to the area making cancellationpossible.The simulation results for the 2D system without confining potential10are in agreement with the KT theory for the superfluid transition. If thisdescription is also appropriate with confining potential, the vortex pictureof the KT-transition suggests an additional mechanism leading to a decreaseof ρs8 : Superfluidity is destroyed by dissipation through vortices which arenot paired. At low temperatures vortices form pairs which unbind at highertemperatures. In the non-uniform system the two vortices forming a pairwill in general experience a slightly different potential leading to imperfectpairing and thereby a lowering of the superfluid fraction.30 20 10 0 10 20 300.000.050.10T/Tc=0.84T/Tc=0.21all particlescondensed particlesρxFig. 1. Density profiles for 1000 particles at temperatures T/Tc =0.84, 0.56, 0.42, 0.28, 0.21 for ω = 0.15. The condensate parts are displayedon the left and the profiles for all particles on the right. All profiles arenormalised to unity. Temperatures are given with reference to the criti-cal temperature Tc of the ideal system in the thermodynamic limit. Thesmooth dotted curve through the 0.21 data is the infinite-N solution of theGP equation with interaction strength determined by the fit.S. Heinrichs and W. J. Mullin0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.60.70.80.91.0N = 1000, interact.               TDL                        N = 1000     N0/NT/Tc0.0 0.2 0.4 0.6 0.8 1.010-510-410-310-2T/Tcρ s/ ρFig. 2. Left figure: Condensate fraction for the interacting system withN=1000 and ω = 0.15 and comparison with theoretical prediction for theideal gas in the thermodynamic limit (TDL) and with finite size correctionsfor 1000 particles. Right figure: Dependence of the superfluid fraction ontemperature for 1000 particles.ACKNOWLEDGEMENTSWe thank Werner Krauth for providing his PIMC program for the threedimensional case.3REFERENCES1. D. Ceperley, Rev. Mod. Phys. 67, 279 (1995);E. L. Pollock, and D. M. Ceperley, Phys. Rev. B 30, 2555 (1984).2. F. Dalfovo, S. Stringari, Phys. Rev. A, 53, 2477 (1996).3. W. Krauth, Phys. Rev. Lett. 77, 3695 (1996).4. R. P. Feynman A. R. and Hibbs, Quantum Mechanics and Path Integrals(McGraw-Hill Inc., New York 1965).R. P. Feynman, Statistical Mechanics, (Benjamin, New York, 1972) .5. W. J. Mullin, J. Low Temp. Phys, 106, 615 (1997).6. G. Baym and C. Pethick, Phys. Rev Lett., 76, 6 (1996).7. W. J. Mullin, J. Low Temp. Phys, 110, 167 (1998).8. S. I. Shevchenko, Sov. Phys. JETP, 73, 1009 (1991).S. I. Shevchenko, Sov. J. Low. Temp. Phys., 18, 223 (1992).9. P. Sindzingre, M. Klein, and D. M. Ceperley, Phys. Rev. Lett. 63, 1601 (1989).10. D. M. Ceperley and E. L. Pollock, Phys. Rev. B 39, 2084 (1989).

Quantum-Monte-Carlo Calculations for Bosons in a Two-Dimensional Harmonic Trap

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Quantum-Monte-Carlo Calculations for Bosons in a Two-Dimensional Harmonic Trap

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