Quantized vortices in 4 He droplets: a quantum Monte Carlo study
aa r X i v : . [ c ond - m a t . o t h e r] J un Quantized vortices in He droplets: a quantum Monte Carlo study
E. Sola, J. Casulleras, and J. Boronat
Departament de F´ısica i Enginyeria Nuclear, Campus Nord B4-B5,Universitat Polit`ecnica de Catalunya, E-08034 Barcelona, Spain (Dated: November 20, 2018)We present a diffusion Monte Carlo study of a vortex line excitation attached to the center of a He droplet at zero temperature. The vortex energy is estimated for droplets of increasing numberof atoms, from N = 70 up to 300 showing a monotonous increase with N . The evolution of the coreradius and its associated energy, the core energy, is also studied as a function of N . The core radiusis ∼ σ − ( σ = 2 .
556 ˚A) for N ≥ PACS numbers: 67.55.-s,36.40.-c,02.70.Ss
Quantized vortices are one of the most outstanding sig-natures of superfluidity. They have been widely observedin liquid He at temperatures below the critical temper-ature T λ = 2 .
17 K [1]. More recently, vortices have alsobeen detected in dilute Bose [2] and Fermi [3] gases whenthey are magnetically trapped, their observation beingconsidered the most clear indication of the achievementof their superfluid phases. He droplets produced in freejet expansion experiments [4] are expected to be also su-perfluid due to their very low temperature ( T = 0 .
38 K).However, in this finite system the search for a direct sig-nature of their superfluid character is more difficult. Onthe one hand, indirect evidence of superfluidity has beendriven from the determination of the rotation spectrumof molecules adsorbed into them [5]. This very interestingphenomenon has been considered the microscopic versionof the famous Andronikashvili experiment. On the otherhand, the detection of vortices in droplets would be aneven more conclusive proof of their superfluidity. Never-theless, no signal of straight or circular vortices has beenyet observed in experiments with He droplets, in spiteof some theoretical arguments in favor of their possiblemetastability.The stability of vortex excitations in He droplets wasfirst studied by Bauer et al. [6]. They concluded thatvortices are not stable in droplets due to the high exci-tation energy required, compared to the usual temper-ature at which they are produced. Later on, Lehmannand Schmied [7], studying cold droplets, smaller thanthe ones analyzed in Ref. [6] were led to a differentconclusion. They stated that in small droplets, whereonly surface excitations (ripplons) are relevant, vorticesshould be stable against decay. Finally, density func-tional (DF) calculations [8, 9] have predicted that below acritical atomic number a linear vortex pinned to a dopantatom or molecule can become stable, with a lifetime longenough to allow for its experimental detection.The excitation energy of a vortex line in a He dropletas a function of its number of atoms N is one of its morefundamental properties. An accurate calculation of itsvalue is crucial to elucidate its possible stability and for- mation probability in jet expansions. To our knowledge,there is only one previous microscopic calculation of theexcitation energy associated to a vortex in a droplet.This calculation, performed with the path integral MonteCarlo (PIMC) method [10], was carried out for a N = 500 He droplet and the energy obtained was more than a fac-tor two smaller than DF predictions [8, 9]. In the presentwork, we present zero-temperature diffusion Monte Carlo(DMC) results of vortex energy and vortex structure in He droplets for different number of atoms.A vortex is an excited state of the N -particle Hamil-tonian which corresponds to an eigenstate of the angularmomentum operator. Actually, it is an eigenstate of the z component of the angular momentum, L z , with eigen-value ¯ hN l where l = 1 , , . . . is the quantum of circula-tion. In a general form, the imaginary-time dependentwave function of a vortex in a quantum liquid can bewritten as Ψ v ( R , t ) = e i Ω( R ,t ) Φ( R , t ) . (1)By imposing that Ψ v is an eigenstate of the angular mo-mentum L z , Feynman [11] obtained his famous proposalΩ( R , t ) = lφ F ( R ), with φ F ( R ) = P j =1 ,N θ j , θ j beingthe j -th polar coordinate (in cylindric coordinates).With the decomposition (1), the Schr¨odinger equationfor Ψ v ( R , t ) splits in two coupled equations, one for themodulus − ∂ Φ ∂t = D ( ∇ Ω) Φ − D ∇ Φ + ( V ( R ) − E ) Φ , (2)and one for the phase ∂ Ω ∂t = D (cid:20) ∇ Ω + 2( ∇ Ω) ∇ ΦΦ (cid:21) , (3)with D = ¯ h / m . If a fixed form for the phase Ω is as-sumed, the equation for the modulus, Eq. (2), becomesthe usual Schr¨odinger equation with one additional term(the first one on the right hand side). In this approxi-mation, known as fixed phase (FP), the vortex acts likea static external potential. Using Feynman’s expressionfor the phase, Ω = l P Ni =1 θ i , this potential results V v ( R ) = N X i =1 ¯ h l mρ i , (4)with ρ i the polar coordinate of particle i in cylindric co-ordinates. Therefore, in the FP approximation the prob-lem of having a vortex inside the droplet is reduced to adifferent Hamiltonian ( ˜ H = H + V v ) in the Schr¨odingerequation to be stochastically solved. This approach wasused in the PIMC calculation of the vortex energy in a He droplet [10] and also recently in a Monte Carlo studyof the excitation energy of vortices in trapped dilutedBose gases [12].In the present work, we will assume that the vortexline is fixed in the z direction of the center-of-mass (CM)reference system of the droplet. The vortex state is thenan eigenstate of the L z CM operator, accounting for thetranslational invariance of the Hamiltonian. In this case,the resulting potential ( ¯ V v ) that must be added to theHamiltonian in the droplet geometry is¯ V v ( R ) = ¯ h l m N X j =1 " ρ j − N N X k =1 cos( θ k − θ j ) ρ k ρ j . (5)Coordinates in Eq. (5) and hereafter are referred to thecenter of mass. The potential ¯ V v is very similar to the onefor the bulk (4), but now written using CM coordinates,and a small correction of order 1 /N is introduced.The FP approach to the excitation energy of a vor-tex line with the Feynman’s phase could be thoughtas a too crude approximation for an accurate micro-scopic treatment of the problem. An a priori bettermethod would be to consider the superposition of clock-wise and anticlockwise vortices, which are degenerate inenergy, and using the resulting wave function (orthog-onal to the ground state) as a guiding wave functionin DMC. This leads to a Fermi-like problem due to thenon-positivity of the excited wave function which can beapproximately solved in the fixed-node (FN) approxima-tion. This method was used in Ref. [13] for studying avortex in a two-dimensional geometry. However, the re-sults there obtained showed that FN and FP predictionsare almost compatible. Moreover, both Ref. [13] and Ref.[14] studied the possible improvements upon Feynman’sphase by introducing backflow correlations on it and con-cluded that their effect on the excitation energy is verysmall ( <
1% in the energy per particle). Therefore, itis sound to consider that the FP method is also a goodenough approximation to describe the vortex attached toa droplet.We have carried out DMC simulations of He dropletswith number of atoms N = 70 , , ( R ) = Y i 0, reflecting the repulsive character of the potential ¯ V v ,and approaches 1 far from the core. We have checkedthat the explicit form of the function f ( ρ ) satisfying bothboundary conditions is not really important and that, asexpected, the energy of the system does not depend onit.The other two variational parameters b and α appear-ing in the trial wave function (6) have been optimizedby means of variational Monte Carlo (VMC) calcula-tions. The optimal values are b = 1 . σ and α = 0 . σ − ( σ = 2 . 556 ˚A) and their dependence with the number ofatoms N is negligible in the range studied.The main purpose of this work, i.e., the determinationof the excitation energy associated to a vortex line at-tached to the CM of the He droplet, has been studiedby other groups using mainly density functional theory[7, 8, 9]. Dalfovo et al. [8] proposed a liquid-drop formulafor the energy dependence on the number of particles, E v ( N ) = λN / + βN / log N + γN − / , (9) λ , β and γ being parameters which were obtained byfitting Eq. (9) to their DF results. The final set of pa-rameters was λ = 2 . 868 K, β = 1 . 445 K and γ = 0 . N 70 128 200 300 E [ K ] − . − . − − E [ K ] − . − . − − E v [ K ] 35 . . He droplets with ( E ) and with-out ( E ) a vortex inside, and vortex excitation energy E v . Allenergies are in units of Kelvin; in parenthesis, the statisticalerrors. The specific dependence on N in Eq. (9) is derivedusing the hollow-core model. In this model, the local vortex energy is integrated over all the volume occupiedby particles E v ( N ) = Z V dV ρ , (10)resulting in E v ( N ) = 2 π ¯ h D m (cid:20) R ln (cid:18) Ra (cid:19) − R + a R (cid:21) . (11)Notice that one can rewrite Eq. (11) in terms of N justusing R = r N / , where r = (3 / (4 πD )) / , arriving insuch a way to the same dependence on N as Eq. (9).However, the hollow-core method is a too simple approx-imation to identify the parameters λ , β and γ using, forexample, a constant density D .Calculations performed using DF [8] and MC evalu-ate the vortex energy as the difference between the totalenergies of the droplet with and without (ground-state)a vortex ( E v ( N ) = E ( N ) − E ( N )). In Table I, wepresent our DMC results of the energies E ( N ), E ( N ),and E v ( N ) for He droplets with N = 70 , , E v comes from the difference oftwo energies which increase with the number of atoms N and therefore its statistical error also increases with N .The results contained in Table I show that the exci-tation energy E v increases monotonously with N . Thiscan be more clearly observed in Fig. 1 where the presentDMC results are compared with the liquid drop formula(9) as reported by Dalfovo et al. [8]. One can observethat the DF estimation reproduces better our results forthe larger droplets than for the smaller ones. This iswhat one a priori expects since DF approximations workbetter for large droplets where application of mean fieldtheory is more justified; in spite of this, the difference isonly ∼ 4% for the smallest droplet studied ( N = 70).The repulsive potential induced by the vortex creates a hole when ρ → D ( ρ ) for different values of z along the vortex axis. The slice at z = 0 corresponds tothe one in the center of the droplet and is the density pro-file with the highest peak. When z increases, the radius 30 40 50 60 70 80 50 100 150 200 250 300 E V ( N )[ K ] N FIG. 1: Vortex energies E v for He droplets with N =70 , , 200 and 300. The energy curve as a function of N corresponds to the DF results from Ref. [8]. D ( r )[ s − ] r [ s ] FIG. 2: Density profiles for a N = 200 He droplet witha vortex inside. They are represented as a function of theradial coordinate ρ and for different slices corresponding todifferent z values on the vortex axis. From top to bottomthe curves correspond to increasing values of z from z = 0 inthe center of the droplet up to z values close to the surface.For comparison, we also show the radial density profile of thedroplet without the vortex (solid line). of the slice is progressively smaller and also the oscilla-tions are depressed as z approaches the droplet radius.In the center of the vortex, the density goes to zero inagreement with the DMC results derived en Ref. [13] forhomogeneous 2D liquid He. A relevant parameter in themicroscopic description of a vortex is the size of its core,what is called the core radius ξ . There is not a single def-inition of ξ but different definitions lead to quite similarresults. We have used the criterion of considering for ξ the position of the maximum in the azimuthal circulatingcurrent J θ ( ρ ). If the vortex is described by means of theFeynman approximation, as in the present work, it fol-lows that J θ ( ρ ) = D ( ρ ) /ρ . Therefore, we can get a direct E v ( r ) [ K ] r [ s ] FIG. 3: Local energy E v ( ρ ) for a N = 70 He droplet with avortex inside. From top to bottom the curves correspond toincreasing values of z from z = 0 in the center of the dropletup to z values close to the surface. estimation of ξ from the density profiles D ( ρ ) shown inFig. 2. In the center of the droplet ξ = 1 . ± . z approaches the value of the dropletradius up to ξ = 1 . ± . E v ( ρ ) = A ln( ρ/ξ ) + E c (12)with A a constant, E c the core energy, and ρ the radialcoordinate in cylindrical coordinates. This model, whichreproduces very well DMC results of the vortex energy ina 2D geometry [13], splits the vortex energy in a constantterm E c associated to the hole around the vortex axis anda logarithmic term containing the hydrodynamic tail. Inthe case of droplets, the behavior of the excitation energyis more complex due to their inhomogeneity along the z vortex axis and their finite size. In the function E v ( ρ )one finds, superimposed to the monotonously increasinglaw (12), a decaying trend to zero when the surface ofthe droplet is reached. This behavior is shown in Fig. 3where we have plotted the function E v ( ρ ) for a N = 70 He droplet. The function E v ( ρ ) shows clearly a peakcorresponding to the core of the vortex, especially for theinner slice, and a decay to zero in the surface. The sizeof the droplets studied is too small to see the signatureof the hydrodynamic tail but we can estimate the energyof the core E c by summing the local energy up to theestimated vortex core radius ξ . Extending this sum alongall the vortex axis we obtain an energy which increaseswith N : E c = 9 . N = 70and 300, respectively. Normalizing E c with respect tothe volume of the core, the core energy per volume unit approaches a constant value of 2.8 K σ − for droplets with N ≥ He droplet at zero temperature. The energies obtainedare in good agreement with DF estimations and thereforegive additional confidence on their predictions. More-over, the magnitude of the core radius and core energyhas also been studied for the first time in these inhomoge-neous system. The core radius is ∼ σ − for N ≥ He droplets.We thank Manuel Barranco and Marti Pi for usefuldiscussions. We acknowledge financial support from DGI(Spain) Grant No. FIS2005-04181 and Generalitat deCatalunya Grant No. 2005SGR-00779. [1] R. J. Donnelly, Quantized Vortices in Helium II (Cam-bridge University Press, Cambridge, 1991).[2] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S.Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. , 2498 (1999).[3] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H.Schunck, and W. Ketterle, Nature , 1047 (2005).[4] J. D. Close, F. Federman, K. Hoffman, and N. Quaas, J.Low Temp. Phys. , 661 (1998).[5] S. Grebenev, J. P. Toennies, and A. F. Vilesov, Science , 2083 (1998).[6] G. H. Bauer, R. 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